https://calculus.subwiki.org/w/api.php?action=feedcontributions&feedformat=atom&user=SebastianCalculus - User contributions [en]2026-04-30T11:00:10ZUser contributionsMediaWiki 1.41.2https://calculus.subwiki.org/w/index.php?title=Real_number&diff=3387Real number2022-04-29T01:16:41Z<p>Sebastian: </p>
<hr />
<div><br />
Calculus is based in the system of real numbers and their properties.<br />
<br />
== Classification ==<br />
<br />
Real numbers are classified as [[rational number]]s (denoted by <math>Q</math>), [[integer]]s (<math>Z</math>), [[whole number]]s (<math>W</math>), [[natural number]]s, and [[irrational number]]s. In order of inclusion, non-irrational real numbers can be ordered as follows:<br />
<br />
<math>N \subseteq W \subseteq Z \subseteq Q</math><br />
<br />
== Properties ==<br />
<br />
=== Distributive Property ===<br />
<br />
<math>a.(b+c)=a.b+a.c</math><br />
<br />
=== Commutative property of addition ===<br />
<br />
<math>a+b=b+a</math><br />
<br />
=== Commutative property of multiplication ===<br />
<br />
<math>a.(b.c)=(a.b).c</math><br />
<br />
=== Aditive identity property ===<br />
<br />
<math>a+0=a</math><br />
<br />
=== Multiplicative identiy property ===<br />
<br />
<math>a.1=a</math><br />
<br />
=== Multiplicative identity property ===<br />
<br />
<math>a.1=a</math><br />
<br />
=== Additive inverse property ===<br />
<br />
<math>a+(-a)=0</math><br />
<br />
=== Multiplicative inverse property ===<br />
<br />
<math>a.\frac{1}{a}=1</math> where <math>a\neq0</math><br />
<br />
=== Zero property of multiplication ===<br />
<br />
<math>a.0=0</math><br />
<br />
=== Closure property of addition ===<br />
<br />
<math>a+b</math> is a real number<br />
<br />
=== Closure property of multiplication ===<br />
<br />
<math>a.b</math> is a real number<br />
<br />
=== Addition property of equality ===<br />
<br />
If <math>a=b</math>, then <math>a+c=b+c</math><br />
<br />
=== Substitution property ===<br />
<br />
If <math>a=b</math>, then <math>a</math> may be substituted for <math>b</math> or conversely<br />
<br />
=== Reflexive (or identity) property of equality ===<br />
<br />
<math>a=a</math><br />
<br />
=== Symmetric property of equality ===<br />
<br />
If <math>a=b</math>, then <math>b=a</math><br />
<br />
=== Transitive property of equality ===<br />
If <math>a=b</math> and <math>b=c</math>, then <math>a=c</math><br />
<br />
=== Law of trichotomy ===<br />
<br />
Exactly one of the following holds: <math>a<b, a=b, a>b</math></div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=Real_number&diff=3386Real number2022-04-29T01:09:52Z<p>Sebastian: /* Multiplicative inverse property */</p>
<hr />
<div><br />
Calculus is based in the system of real numbers and their properties.<br />
<br />
== Classification ==<br />
<br />
Real numbers are classified as [[rational number]]s (denoted by '''Q'''), [[integer]]s ('''Z'''), [[whole number]]s ('''W'''), [[natural number]]s, and [[irrational number]]s. In order of inclusion, non-irrational real numbers can be ordered as follows:<br />
<br />
<math>N \subseteq W \subseteq Z \subseteq Q</math><br />
<br />
== Properties ==<br />
<br />
=== Distributive Property ===<br />
<br />
<math>a.(b+c)=a.b+a.c</math><br />
<br />
=== Commutative property of addition ===<br />
<br />
<math>a+b=b+a</math><br />
<br />
=== Commutative property of multiplication ===<br />
<br />
<math>a.(b.c)=(a.b).c</math><br />
<br />
=== Aditive identity property ===<br />
<br />
<math>a+0=a</math><br />
<br />
=== Multiplicative identiy property ===<br />
<br />
<math>a.1=a</math><br />
<br />
=== Multiplicative identity property ===<br />
<br />
<math>a.1=a</math><br />
<br />
=== Additive inverse property ===<br />
<br />
<math>a+(-a)=0</math><br />
<br />
=== Multiplicative inverse property ===<br />
<br />
<math>a.\frac{1}{a}=1</math> where <math>a\neq0</math><br />
<br />
=== Zero property of multiplication ===<br />
<br />
<math>a.0=0</math><br />
<br />
=== Closure property of addition ===<br />
<br />
<math>a+b</math> is a real number<br />
<br />
=== Closure property of multiplication ===<br />
<br />
<math>a.b</math> is a real number<br />
<br />
=== Addition property of equality ===<br />
<br />
If <math>a=b</math>, then <math>a+c=b+c</math><br />
<br />
=== Substitution property ===<br />
<br />
If <math>a=b</math>, then <math>a</math> may be substituted for <math>b</math> or conversely<br />
<br />
=== Reflexive (or identity) property of equality ===<br />
<br />
<math>a=a</math><br />
<br />
=== Symmetric property of equality ===<br />
<br />
If <math>a=b</math>, then <math>b=a</math><br />
<br />
=== Transitive property of equality ===<br />
If <math>a=b</math> and <math>b=c</math>, then <math>a=c</math><br />
<br />
=== Law of trichotomy ===<br />
<br />
Exactly one of the following holds: <math>a<b, a=b, a>b</math></div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=Real_number&diff=3385Real number2022-04-29T01:09:40Z<p>Sebastian: /* Multiplicative inverse property */</p>
<hr />
<div><br />
Calculus is based in the system of real numbers and their properties.<br />
<br />
== Classification ==<br />
<br />
Real numbers are classified as [[rational number]]s (denoted by '''Q'''), [[integer]]s ('''Z'''), [[whole number]]s ('''W'''), [[natural number]]s, and [[irrational number]]s. In order of inclusion, non-irrational real numbers can be ordered as follows:<br />
<br />
<math>N \subseteq W \subseteq Z \subseteq Q</math><br />
<br />
== Properties ==<br />
<br />
=== Distributive Property ===<br />
<br />
<math>a.(b+c)=a.b+a.c</math><br />
<br />
=== Commutative property of addition ===<br />
<br />
<math>a+b=b+a</math><br />
<br />
=== Commutative property of multiplication ===<br />
<br />
<math>a.(b.c)=(a.b).c</math><br />
<br />
=== Aditive identity property ===<br />
<br />
<math>a+0=a</math><br />
<br />
=== Multiplicative identiy property ===<br />
<br />
<math>a.1=a</math><br />
<br />
=== Multiplicative identity property ===<br />
<br />
<math>a.1=a</math><br />
<br />
=== Additive inverse property ===<br />
<br />
<math>a+(-a)=0</math><br />
<br />
=== Multiplicative inverse property ===<br />
<br />
<math>a.\frac{1}{a})=1</math> where <math>a\neq0</math><br />
<br />
=== Zero property of multiplication ===<br />
<br />
<math>a.0=0</math><br />
<br />
=== Closure property of addition ===<br />
<br />
<math>a+b</math> is a real number<br />
<br />
=== Closure property of multiplication ===<br />
<br />
<math>a.b</math> is a real number<br />
<br />
=== Addition property of equality ===<br />
<br />
If <math>a=b</math>, then <math>a+c=b+c</math><br />
<br />
=== Substitution property ===<br />
<br />
If <math>a=b</math>, then <math>a</math> may be substituted for <math>b</math> or conversely<br />
<br />
=== Reflexive (or identity) property of equality ===<br />
<br />
<math>a=a</math><br />
<br />
=== Symmetric property of equality ===<br />
<br />
If <math>a=b</math>, then <math>b=a</math><br />
<br />
=== Transitive property of equality ===<br />
If <math>a=b</math> and <math>b=c</math>, then <math>a=c</math><br />
<br />
=== Law of trichotomy ===<br />
<br />
Exactly one of the following holds: <math>a<b, a=b, a>b</math></div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=Real_number&diff=3384Real number2022-04-29T01:06:40Z<p>Sebastian: /* Substitution property */</p>
<hr />
<div><br />
Calculus is based in the system of real numbers and their properties.<br />
<br />
== Classification ==<br />
<br />
Real numbers are classified as [[rational number]]s (denoted by '''Q'''), [[integer]]s ('''Z'''), [[whole number]]s ('''W'''), [[natural number]]s, and [[irrational number]]s. In order of inclusion, non-irrational real numbers can be ordered as follows:<br />
<br />
<math>N \subseteq W \subseteq Z \subseteq Q</math><br />
<br />
== Properties ==<br />
<br />
=== Distributive Property ===<br />
<br />
<math>a.(b+c)=a.b+a.c</math><br />
<br />
=== Commutative property of addition ===<br />
<br />
<math>a+b=b+a</math><br />
<br />
=== Commutative property of multiplication ===<br />
<br />
<math>a.(b.c)=(a.b).c</math><br />
<br />
=== Aditive identity property ===<br />
<br />
<math>a+0=a</math><br />
<br />
=== Multiplicative identiy property ===<br />
<br />
<math>a.1=a</math><br />
<br />
=== Multiplicative identity property ===<br />
<br />
<math>a.1=a</math><br />
<br />
=== Additive inverse property ===<br />
<br />
<math>a+(-a)=0</math><br />
<br />
=== Multiplicative inverse property ===<br />
<br />
<math>a.(/frac{1}{a})=1</math> where <math>a\neq0</math><br />
<br />
=== Zero property of multiplication ===<br />
<br />
<math>a.0=0</math><br />
<br />
=== Closure property of addition ===<br />
<br />
<math>a+b</math> is a real number<br />
<br />
=== Closure property of multiplication ===<br />
<br />
<math>a.b</math> is a real number<br />
<br />
=== Addition property of equality ===<br />
<br />
If <math>a=b</math>, then <math>a+c=b+c</math><br />
<br />
=== Substitution property ===<br />
<br />
If <math>a=b</math>, then <math>a</math> may be substituted for <math>b</math> or conversely<br />
<br />
=== Reflexive (or identity) property of equality ===<br />
<br />
<math>a=a</math><br />
<br />
=== Symmetric property of equality ===<br />
<br />
If <math>a=b</math>, then <math>b=a</math><br />
<br />
=== Transitive property of equality ===<br />
If <math>a=b</math> and <math>b=c</math>, then <math>a=c</math><br />
<br />
=== Law of trichotomy ===<br />
<br />
Exactly one of the following holds: <math>a<b, a=b, a>b</math></div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=Real_number&diff=3383Real number2022-04-29T01:06:11Z<p>Sebastian: /* Closure property of addition */</p>
<hr />
<div><br />
Calculus is based in the system of real numbers and their properties.<br />
<br />
== Classification ==<br />
<br />
Real numbers are classified as [[rational number]]s (denoted by '''Q'''), [[integer]]s ('''Z'''), [[whole number]]s ('''W'''), [[natural number]]s, and [[irrational number]]s. In order of inclusion, non-irrational real numbers can be ordered as follows:<br />
<br />
<math>N \subseteq W \subseteq Z \subseteq Q</math><br />
<br />
== Properties ==<br />
<br />
=== Distributive Property ===<br />
<br />
<math>a.(b+c)=a.b+a.c</math><br />
<br />
=== Commutative property of addition ===<br />
<br />
<math>a+b=b+a</math><br />
<br />
=== Commutative property of multiplication ===<br />
<br />
<math>a.(b.c)=(a.b).c</math><br />
<br />
=== Aditive identity property ===<br />
<br />
<math>a+0=a</math><br />
<br />
=== Multiplicative identiy property ===<br />
<br />
<math>a.1=a</math><br />
<br />
=== Multiplicative identity property ===<br />
<br />
<math>a.1=a</math><br />
<br />
=== Additive inverse property ===<br />
<br />
<math>a+(-a)=0</math><br />
<br />
=== Multiplicative inverse property ===<br />
<br />
<math>a.(/frac{1}{a})=1</math> where <math>a\neq0</math><br />
<br />
=== Zero property of multiplication ===<br />
<br />
<math>a.0=0</math><br />
<br />
=== Closure property of addition ===<br />
<br />
<math>a+b</math> is a real number<br />
<br />
=== Closure property of multiplication ===<br />
<br />
<math>a.b</math> is a real number<br />
<br />
=== Addition property of equality ===<br />
<br />
If <math>a=b</math>, then <math>a+c=b+c</math><br />
<br />
=== Substitution property ===<br />
<br />
If <math>a=b, then a may be substituted for b or conversely<br />
<br />
=== Reflexive (or identity) property of equality ===<br />
<br />
<math>a=a<br />
<br />
=== Symmetric property of equality ===<br />
<br />
If <math>a=b</math>, then <math>b=a</math><br />
<br />
=== Transitive property of equality ===<br />
If <math>a=b</math> and <math>b=c</math>, then <math>a=c</math><br />
<br />
=== Law of trichotomy ===<br />
<br />
Exactly one of the following holds: <math>a<b, a=b, a>b</math></div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=Real_number&diff=3382Real number2022-04-29T01:05:50Z<p>Sebastian: /* Multiplicative inverse property */</p>
<hr />
<div><br />
Calculus is based in the system of real numbers and their properties.<br />
<br />
== Classification ==<br />
<br />
Real numbers are classified as [[rational number]]s (denoted by '''Q'''), [[integer]]s ('''Z'''), [[whole number]]s ('''W'''), [[natural number]]s, and [[irrational number]]s. In order of inclusion, non-irrational real numbers can be ordered as follows:<br />
<br />
<math>N \subseteq W \subseteq Z \subseteq Q</math><br />
<br />
== Properties ==<br />
<br />
=== Distributive Property ===<br />
<br />
<math>a.(b+c)=a.b+a.c</math><br />
<br />
=== Commutative property of addition ===<br />
<br />
<math>a+b=b+a</math><br />
<br />
=== Commutative property of multiplication ===<br />
<br />
<math>a.(b.c)=(a.b).c</math><br />
<br />
=== Aditive identity property ===<br />
<br />
<math>a+0=a</math><br />
<br />
=== Multiplicative identiy property ===<br />
<br />
<math>a.1=a</math><br />
<br />
=== Multiplicative identity property ===<br />
<br />
<math>a.1=a</math><br />
<br />
=== Additive inverse property ===<br />
<br />
<math>a+(-a)=0</math><br />
<br />
=== Multiplicative inverse property ===<br />
<br />
<math>a.(/frac{1}{a})=1</math> where <math>a\neq0</math><br />
<br />
=== Zero property of multiplication ===<br />
<br />
<math>a.0=0</math><br />
<br />
=== Closure property of addition ===<br />
<br />
<math>a+b is a real number<br />
<br />
=== Closure property of multiplication ===<br />
<br />
<math>a.b is a real number<br />
<br />
=== Addition property of equality ===<br />
<br />
If <math>a=b</math>, then <math>a+c=b+c</math><br />
<br />
=== Substitution property ===<br />
<br />
If <math>a=b, then a may be substituted for b or conversely<br />
<br />
=== Reflexive (or identity) property of equality ===<br />
<br />
<math>a=a<br />
<br />
=== Symmetric property of equality ===<br />
<br />
If <math>a=b</math>, then <math>b=a</math><br />
<br />
=== Transitive property of equality ===<br />
If <math>a=b</math> and <math>b=c</math>, then <math>a=c</math><br />
<br />
=== Law of trichotomy ===<br />
<br />
Exactly one of the following holds: <math>a<b, a=b, a>b</math></div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=Real_number&diff=3381Real number2022-04-29T01:05:28Z<p>Sebastian: /* Multiplicative inverse property */</p>
<hr />
<div><br />
Calculus is based in the system of real numbers and their properties.<br />
<br />
== Classification ==<br />
<br />
Real numbers are classified as [[rational number]]s (denoted by '''Q'''), [[integer]]s ('''Z'''), [[whole number]]s ('''W'''), [[natural number]]s, and [[irrational number]]s. In order of inclusion, non-irrational real numbers can be ordered as follows:<br />
<br />
<math>N \subseteq W \subseteq Z \subseteq Q</math><br />
<br />
== Properties ==<br />
<br />
=== Distributive Property ===<br />
<br />
<math>a.(b+c)=a.b+a.c</math><br />
<br />
=== Commutative property of addition ===<br />
<br />
<math>a+b=b+a</math><br />
<br />
=== Commutative property of multiplication ===<br />
<br />
<math>a.(b.c)=(a.b).c</math><br />
<br />
=== Aditive identity property ===<br />
<br />
<math>a+0=a</math><br />
<br />
=== Multiplicative identiy property ===<br />
<br />
<math>a.1=a</math><br />
<br />
=== Multiplicative identity property ===<br />
<br />
<math>a.1=a</math><br />
<br />
=== Additive inverse property ===<br />
<br />
<math>a+(-a)=0</math><br />
<br />
=== Multiplicative inverse property ===<br />
<br />
<math>a.(frac{1}{a})=1</math> where <math>a\neq0</math><br />
<br />
=== Zero property of multiplication ===<br />
<br />
<math>a.0=0</math><br />
<br />
=== Closure property of addition ===<br />
<br />
<math>a+b is a real number<br />
<br />
=== Closure property of multiplication ===<br />
<br />
<math>a.b is a real number<br />
<br />
=== Addition property of equality ===<br />
<br />
If <math>a=b</math>, then <math>a+c=b+c</math><br />
<br />
=== Substitution property ===<br />
<br />
If <math>a=b, then a may be substituted for b or conversely<br />
<br />
=== Reflexive (or identity) property of equality ===<br />
<br />
<math>a=a<br />
<br />
=== Symmetric property of equality ===<br />
<br />
If <math>a=b</math>, then <math>b=a</math><br />
<br />
=== Transitive property of equality ===<br />
If <math>a=b</math> and <math>b=c</math>, then <math>a=c</math><br />
<br />
=== Law of trichotomy ===<br />
<br />
Exactly one of the following holds: <math>a<b, a=b, a>b</math></div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=Real_number&diff=3380Real number2022-04-29T01:04:16Z<p>Sebastian: /* Multiplicative inverse property */</p>
<hr />
<div><br />
Calculus is based in the system of real numbers and their properties.<br />
<br />
== Classification ==<br />
<br />
Real numbers are classified as [[rational number]]s (denoted by '''Q'''), [[integer]]s ('''Z'''), [[whole number]]s ('''W'''), [[natural number]]s, and [[irrational number]]s. In order of inclusion, non-irrational real numbers can be ordered as follows:<br />
<br />
<math>N \subseteq W \subseteq Z \subseteq Q</math><br />
<br />
== Properties ==<br />
<br />
=== Distributive Property ===<br />
<br />
<math>a.(b+c)=a.b+a.c</math><br />
<br />
=== Commutative property of addition ===<br />
<br />
<math>a+b=b+a</math><br />
<br />
=== Commutative property of multiplication ===<br />
<br />
<math>a.(b.c)=(a.b).c</math><br />
<br />
=== Aditive identity property ===<br />
<br />
<math>a+0=a</math><br />
<br />
=== Multiplicative identiy property ===<br />
<br />
<math>a.1=a</math><br />
<br />
=== Multiplicative identity property ===<br />
<br />
<math>a.1=a</math><br />
<br />
=== Additive inverse property ===<br />
<br />
<math>a+(-a)=0</math><br />
<br />
=== Multiplicative inverse property ===<br />
<br />
<math>a.(frac{1}{a})=1</math> where <math>a≠0</math><br />
<br />
=== Zero property of multiplication ===<br />
<br />
<math>a.0=0</math><br />
<br />
=== Closure property of addition ===<br />
<br />
<math>a+b is a real number<br />
<br />
=== Closure property of multiplication ===<br />
<br />
<math>a.b is a real number<br />
<br />
=== Addition property of equality ===<br />
<br />
If <math>a=b</math>, then <math>a+c=b+c</math><br />
<br />
=== Substitution property ===<br />
<br />
If <math>a=b, then a may be substituted for b or conversely<br />
<br />
=== Reflexive (or identity) property of equality ===<br />
<br />
<math>a=a<br />
<br />
=== Symmetric property of equality ===<br />
<br />
If <math>a=b</math>, then <math>b=a</math><br />
<br />
=== Transitive property of equality ===<br />
If <math>a=b</math> and <math>b=c</math>, then <math>a=c</math><br />
<br />
=== Law of trichotomy ===<br />
<br />
Exactly one of the following holds: <math>a<b, a=b, a>b</math></div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=Real_number&diff=3379Real number2022-04-29T01:01:38Z<p>Sebastian: </p>
<hr />
<div><br />
Calculus is based in the system of real numbers and their properties.<br />
<br />
== Classification ==<br />
<br />
Real numbers are classified as [[rational number]]s (denoted by '''Q'''), [[integer]]s ('''Z'''), [[whole number]]s ('''W'''), [[natural number]]s, and [[irrational number]]s. In order of inclusion, non-irrational real numbers can be ordered as follows:<br />
<br />
<math>N \subseteq W \subseteq Z \subseteq Q</math><br />
<br />
== Properties ==<br />
<br />
=== Distributive Property ===<br />
<br />
<math>a.(b+c)=a.b+a.c</math><br />
<br />
=== Commutative property of addition ===<br />
<br />
<math>a+b=b+a</math><br />
<br />
=== Commutative property of multiplication ===<br />
<br />
<math>a.(b.c)=(a.b).c</math><br />
<br />
=== Aditive identity property ===<br />
<br />
<math>a+0=a</math><br />
<br />
=== Multiplicative identiy property ===<br />
<br />
<math>a.1=a</math><br />
<br />
=== Multiplicative identity property ===<br />
<br />
<math>a.1=a</math><br />
<br />
=== Additive inverse property ===<br />
<br />
<math>a+(-a)=0</math><br />
<br />
=== Multiplicative inverse property ===<br />
<br />
<math>a.(1/a)=1</math> where <math>a≠0</math><br />
<br />
=== Zero property of multiplication ===<br />
<br />
<math>a.0=0</math><br />
<br />
=== Closure property of addition ===<br />
<br />
<math>a+b is a real number<br />
<br />
=== Closure property of multiplication ===<br />
<br />
<math>a.b is a real number<br />
<br />
=== Addition property of equality ===<br />
<br />
If <math>a=b</math>, then <math>a+c=b+c</math><br />
<br />
=== Substitution property ===<br />
<br />
If <math>a=b, then a may be substituted for b or conversely<br />
<br />
=== Reflexive (or identity) property of equality ===<br />
<br />
<math>a=a<br />
<br />
=== Symmetric property of equality ===<br />
<br />
If <math>a=b</math>, then <math>b=a</math><br />
<br />
=== Transitive property of equality ===<br />
If <math>a=b</math> and <math>b=c</math>, then <math>a=c</math><br />
<br />
=== Law of trichotomy ===<br />
<br />
Exactly one of the following holds: <math>a<b, a=b, a>b</math></div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=Real_number&diff=3378Real number2022-04-28T23:38:47Z<p>Sebastian: </p>
<hr />
<div><br />
Calculus is based in the system of real numbers and their properties.<br />
<br />
== Classification ==<br />
<br />
Real numbers are classified as [[rational number]]s (denoted by '''Q'''), [[integer]]s ('''Z'''), [[whole number]]s ('''W'''), [[natural number]]s, and [[irrational number]]s. In order of inclusion, non-irrational real numbers can be ordered as follows:<br />
<br />
<math>N \subseteq W \subseteq Z \subseteq Q</math><br />
<br />
== Properties ==<br />
<br />
=== Distributive Property ===<br />
<br />
a.(b+c)=a.b+a.c<br />
<br />
=== Commutative property of addition ===<br />
<br />
a+b=b+a<br />
<br />
=== Commutative property of multiplication ===<br />
<br />
a.(b.c)=(a.b).c<br />
<br />
=== Aditive identity property ===<br />
<br />
a+0=a<br />
<br />
=== Multiplicative identiy property ===<br />
<br />
a.1=a<br />
<br />
=== Multiplicative identity property ===<br />
<br />
a.1 =a<br />
<br />
=== Additive inverse property ===<br />
<br />
a+(-a)=0<br />
<br />
=== Multiplicative inverse property ===<br />
<br />
a.(1/a)=1 where a is not 0<br />
<br />
=== Zero property of multiplication ===<br />
<br />
a.0=0<br />
<br />
=== Closure property of addition ===<br />
<br />
a+b is a real number<br />
<br />
=== Closure property of multiplication ===<br />
<br />
a.b is a real number<br />
<br />
=== Addition property of equality ===<br />
<br />
If a=b, then a+c=b+c<br />
<br />
=== Substitution property ===<br />
<br />
If a=b, then a may be substituted for b or conversely<br />
<br />
=== Reflexive (or identity) property of equality ===<br />
<br />
a=a<br />
<br />
=== Symmetric property of equality ===<br />
<br />
If a=b, then b=a<br />
<br />
=== Transitive property of equality ===<br />
If a=b and b=c, then a=c.<br />
<br />
=== Law of trichotomy ===<br />
<br />
Exactly one of the following holds: a<b, a=b, a>b</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=Real_number&diff=3377Real number2022-04-28T23:20:13Z<p>Sebastian: </p>
<hr />
<div><br />
Calculus is based in the system of real numbers and their properties.<br />
<br />
== Classification ==<br />
<br />
Real numbers are classified as [[rational number]]s (denoted by '''Q'''), [[integer]]s ('''Z'''), [[whole number]]s ('''W'''), [[natural number]]s, and [[irrational number]]s. In order of inclusion, non-irrational real numbers can be ordered as follows:<br />
<br />
<math>N \subseteq W \subseteq Z \subseteq Q</math><br />
<br />
== Properties ==<br />
<br />
=== Distributive Property ===<br />
<br />
a.(b+c)=a.b+a.c<br />
<br />
=== Commutative property of addition ===<br />
<br />
a+b=b+a<br />
<br />
=== Commutative property of multiplication ===<br />
<br />
a.(b.c)=(a.b).c<br />
<br />
=== Aditive identity peoperty ===<br />
<br />
a+0=a<br />
<br />
=== Multiplicative identiy property ===<br />
<br />
a.1=a<br />
<br />
=== Multiplicative identity property ===<br />
<br />
a.1 =a<br />
<br />
=== Additive inverse property ===<br />
<br />
a+(-a)=0<br />
<br />
=== Multiplicative inverse property ===<br />
<br />
a.(1/a)=1 where a is not 0</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=Real_number&diff=3376Real number2022-04-28T23:13:44Z<p>Sebastian: /* Properties */</p>
<hr />
<div><br />
Calculus is based in the system of real numbers and their properties.<br />
<br />
== Classification ==<br />
<br />
Real numbers are classified as [[rational number]]s (denoted by '''Q'''), [[integer]]s ('''Z'''), [[whole number]]s ('''W'''), [[natural number]]s, and [[irrational number]]s. In order of inclusion, non-irrational real numbers can be ordered as follows:<br />
<br />
<math>N \subseteq W \subseteq Z \subseteq Q</math><br />
<br />
== Properties ==<br />
<br />
=== Distributive Property ===<br />
<br />
a.(b+c)=a.b+a.c<br />
<br />
=== Commutative property of addition ===<br />
<br />
a+b=b+a<br />
<br />
=== Commutative property of multiplication ===<br />
<br />
a.(b.c)=(a.b).c</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=Real_number&diff=3375Real number2022-04-28T23:11:45Z<p>Sebastian: /* Properties */</p>
<hr />
<div><br />
Calculus is based in the system of real numbers and their properties.<br />
<br />
== Classification ==<br />
<br />
Real numbers are classified as [[rational number]]s (denoted by '''Q'''), [[integer]]s ('''Z'''), [[whole number]]s ('''W'''), [[natural number]]s, and [[irrational number]]s. In order of inclusion, non-irrational real numbers can be ordered as follows:<br />
<br />
<math>N \subseteq W \subseteq Z \subseteq Q</math><br />
<br />
== Properties ==<br />
<br />
=== Distributive Property ===<br />
<br />
a.(b+c)=a.b+a.c<br />
<br />
=== Commutative property of addition ===<br />
<br />
a+b=b+a</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=User:Sebastian&diff=3374User:Sebastian2022-04-28T04:31:37Z<p>Sebastian: </p>
<hr />
<div>Pageview ranking: [https://analytics.vipulnaik.com/top-pages.php?project_title=Calculus%20subwiki&fbclid=IwAR26FVz9b-WAf2FpqyFxO_tItTH2tdmutjFeOLLtH2Rjz-3GUTDkd2VY_E0]<br />
<br />
Vipul's thoughts: [https://github.com/vipulnaik/working-drafts/blob/master/blog/subwiki-selection-and-value-comparison.md?fbclid=IwAR2p2frBrA7z2e1askDlxjP4qANldoFi3Iw8vxkDAiypUe_ZLttNnk2SmIk]<br />
<br />
LaTeX/Mathematics: [https://en.wikibooks.org/wiki/LaTeX/Mathematics?fbclid=IwAR2xZSg9Ib17g5ko49EuJC16fA_vdUD50QHThwInnShQlehx_6s8u5CBAiQ]<br />
<br />
<br />
<br />
Red articles:<br />
<br />
*[[Absolute maximum value]]<br />
*[[Absolute minimum value]]<br />
*[[Antiderivative]]<br />
*[[Closed interval]]<br />
*[[Constant function]]<br />
*[[Continuously differentiable functions]]<br />
*[[Decreasing function]]<br />
*[[Exponential function]]<br />
*[[Fundamental theorem of calculus]]<br />
*[[Identity function]]<br />
*[[Imaginary number]]<br />
*[[Inequality]]<br />
*[[Interval]]<br />
*[[Inverse function]]<br />
*[[Linear function]]<br />
*[[Logarithmic differentiation]] <br />
*[[Natural logarithm]] <br />
*[[Number]]<br />
*[[One-one function]]<br />
*[[Open interval]]<br />
*[[Piecewise linear function]]<br />
*[[Point of inflection]]<br />
*[[Pointwise maximum]]<br />
*[[Product theorem for continuity]]<br />
*[[Range]]<br />
*[[Real number]]<br />
*[[Vertical tangent]]</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=User:Sebastian&diff=3373User:Sebastian2022-04-28T04:31:18Z<p>Sebastian: </p>
<hr />
<div>Pageview ranking: [https://analytics.vipulnaik.com/top-pages.php?project_title=Calculus%20subwiki&fbclid=IwAR26FVz9b-WAf2FpqyFxO_tItTH2tdmutjFeOLLtH2Rjz-3GUTDkd2VY_E0]<br />
Vipul's thoughts: [https://github.com/vipulnaik/working-drafts/blob/master/blog/subwiki-selection-and-value-comparison.md?fbclid=IwAR2p2frBrA7z2e1askDlxjP4qANldoFi3Iw8vxkDAiypUe_ZLttNnk2SmIk]<br />
LaTeX/Mathematics: [https://en.wikibooks.org/wiki/LaTeX/Mathematics?fbclid=IwAR2xZSg9Ib17g5ko49EuJC16fA_vdUD50QHThwInnShQlehx_6s8u5CBAiQ]<br />
<br />
<br />
<br />
Red articles:<br />
<br />
*[[Absolute maximum value]]<br />
*[[Absolute minimum value]]<br />
*[[Antiderivative]]<br />
*[[Closed interval]]<br />
*[[Constant function]]<br />
*[[Continuously differentiable functions]]<br />
*[[Decreasing function]]<br />
*[[Exponential function]]<br />
*[[Fundamental theorem of calculus]]<br />
*[[Identity function]]<br />
*[[Imaginary number]]<br />
*[[Inequality]]<br />
*[[Interval]]<br />
*[[Inverse function]]<br />
*[[Linear function]]<br />
*[[Logarithmic differentiation]] <br />
*[[Natural logarithm]] <br />
*[[Number]]<br />
*[[One-one function]]<br />
*[[Open interval]]<br />
*[[Piecewise linear function]]<br />
*[[Point of inflection]]<br />
*[[Pointwise maximum]]<br />
*[[Product theorem for continuity]]<br />
*[[Range]]<br />
*[[Real number]]<br />
*[[Vertical tangent]]</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=User:Sebastian&diff=3372User:Sebastian2022-04-28T04:23:15Z<p>Sebastian: </p>
<hr />
<div>Pageview ranking: [https://analytics.vipulnaik.com/top-pages.php?project_title=Calculus%20subwiki&fbclid=IwAR26FVz9b-WAf2FpqyFxO_tItTH2tdmutjFeOLLtH2Rjz-3GUTDkd2VY_E0]<br />
<br />
LaTeX/Mathematics: [https://en.wikibooks.org/wiki/LaTeX/Mathematics?fbclid=IwAR2xZSg9Ib17g5ko49EuJC16fA_vdUD50QHThwInnShQlehx_6s8u5CBAiQ]<br />
<br />
<br />
<br />
Red articles:<br />
<br />
*[[Absolute maximum value]]<br />
*[[Absolute minimum value]]<br />
*[[Antiderivative]]<br />
*[[Closed interval]]<br />
*[[Constant function]]<br />
*[[Continuously differentiable functions]]<br />
*[[Decreasing function]]<br />
*[[Exponential function]]<br />
*[[Fundamental theorem of calculus]]<br />
*[[Identity function]]<br />
*[[Imaginary number]]<br />
*[[Inequality]]<br />
*[[Interval]]<br />
*[[Inverse function]]<br />
*[[Linear function]]<br />
*[[Logarithmic differentiation]] <br />
*[[Natural logarithm]] <br />
*[[Number]]<br />
*[[One-one function]]<br />
*[[Open interval]]<br />
*[[Piecewise linear function]]<br />
*[[Point of inflection]]<br />
*[[Pointwise maximum]]<br />
*[[Product theorem for continuity]]<br />
*[[Range]]<br />
*[[Real number]]<br />
*[[Vertical tangent]]</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=One-one_function&diff=3371One-one function2022-04-28T04:12:07Z<p>Sebastian: /* Properties */</p>
<hr />
<div>A [[function]] <math>f</math> is called one-one function if it never adopts the same value twice. This means, one-one functions return a unique [[range]] for each element of their [[domain]].<br />
<br />
Other names given to the one-one function are one to one, or injective function.<br />
<br />
One-one functions are a set of functions which denote the relation between sets, elements or identities. The other two sets are [[surjective function]]s and [[bijective function]]s.<br />
<br />
== Definition ==<br />
The function <math>f(x)</math> is called one-one function when for every value of <math>x</math> in the domain of the function, there will be a unique value of <math>f(x)</math>.<br />
<br />
== Geometric proof ==<br />
Similar to the vertical line test (VLT) for functions, there is a horizontal line test (HLT) to prove the one-one property. A function is one-one if and only if no horizontal line intersects its graph more than once.<br />
<br />
<br />
In the graph below, the function <math>y=x^3</math> is intersected once by the horizontal line. Therefore the function is geometrically proven to be one-one.<br />
<br />
[[File:One-one function geometric proof true.PNG|350px]]<br />
<br />
<br />
In the graph below, the function <math>y=x^2</math> is intersected twice by the horizontal line. Therefore the function is geometrically proven no to be one-one.<br />
<br />
[[File:One-one function geometric proof.PNG|350px]]<br />
<br />
== Properties ==<br />
* The domain of f equals the range of f<sup>-1</sup>.<br />
* <math>f^{-1}(f_(x))=x</math> for every x in the domain of f and f<br />
* The graph of a function and the graph of its inverse are symmetric with respect to the line y=x.<br />
* If f and g are both one-one, then f°g follows injectivity.<br />
* If g°f is one-one, then function f is one-one, but function g may not be.<br />
* A one-one function is either strictly decreasing or strictly increasing.<br />
* A function that is not a one-one is considered as many-to-one.<br />
* [[Parabolic function]]s are not one-one functions.<br />
<br />
== Examples ==<br />
<br />
Examples of one-one functions include:<br />
<br />
* Identity function: f(x) is always injective.</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=One-one_function&diff=3370One-one function2022-04-28T04:11:56Z<p>Sebastian: /* Properties */</p>
<hr />
<div>A [[function]] <math>f</math> is called one-one function if it never adopts the same value twice. This means, one-one functions return a unique [[range]] for each element of their [[domain]].<br />
<br />
Other names given to the one-one function are one to one, or injective function.<br />
<br />
One-one functions are a set of functions which denote the relation between sets, elements or identities. The other two sets are [[surjective function]]s and [[bijective function]]s.<br />
<br />
== Definition ==<br />
The function <math>f(x)</math> is called one-one function when for every value of <math>x</math> in the domain of the function, there will be a unique value of <math>f(x)</math>.<br />
<br />
== Geometric proof ==<br />
Similar to the vertical line test (VLT) for functions, there is a horizontal line test (HLT) to prove the one-one property. A function is one-one if and only if no horizontal line intersects its graph more than once.<br />
<br />
<br />
In the graph below, the function <math>y=x^3</math> is intersected once by the horizontal line. Therefore the function is geometrically proven to be one-one.<br />
<br />
[[File:One-one function geometric proof true.PNG|350px]]<br />
<br />
<br />
In the graph below, the function <math>y=x^2</math> is intersected twice by the horizontal line. Therefore the function is geometrically proven no to be one-one.<br />
<br />
[[File:One-one function geometric proof.PNG|350px]]<br />
<br />
== Properties ==<br />
* The domain of f equals the range of f<sup>-1</sup>.<br />
* <math>f^{-1}(f<_(x))=x</math> for every x in the domain of f and f<br />
* The graph of a function and the graph of its inverse are symmetric with respect to the line y=x.<br />
* If f and g are both one-one, then f°g follows injectivity.<br />
* If g°f is one-one, then function f is one-one, but function g may not be.<br />
* A one-one function is either strictly decreasing or strictly increasing.<br />
* A function that is not a one-one is considered as many-to-one.<br />
* [[Parabolic function]]s are not one-one functions.<br />
<br />
== Examples ==<br />
<br />
Examples of one-one functions include:<br />
<br />
* Identity function: f(x) is always injective.</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=One-one_function&diff=3369One-one function2022-04-28T04:11:34Z<p>Sebastian: /* Properties */</p>
<hr />
<div>A [[function]] <math>f</math> is called one-one function if it never adopts the same value twice. This means, one-one functions return a unique [[range]] for each element of their [[domain]].<br />
<br />
Other names given to the one-one function are one to one, or injective function.<br />
<br />
One-one functions are a set of functions which denote the relation between sets, elements or identities. The other two sets are [[surjective function]]s and [[bijective function]]s.<br />
<br />
== Definition ==<br />
The function <math>f(x)</math> is called one-one function when for every value of <math>x</math> in the domain of the function, there will be a unique value of <math>f(x)</math>.<br />
<br />
== Geometric proof ==<br />
Similar to the vertical line test (VLT) for functions, there is a horizontal line test (HLT) to prove the one-one property. A function is one-one if and only if no horizontal line intersects its graph more than once.<br />
<br />
<br />
In the graph below, the function <math>y=x^3</math> is intersected once by the horizontal line. Therefore the function is geometrically proven to be one-one.<br />
<br />
[[File:One-one function geometric proof true.PNG|350px]]<br />
<br />
<br />
In the graph below, the function <math>y=x^2</math> is intersected twice by the horizontal line. Therefore the function is geometrically proven no to be one-one.<br />
<br />
[[File:One-one function geometric proof.PNG|350px]]<br />
<br />
== Properties ==<br />
* The domain of f equals the range of f<sup>-1</sup>.<br />
* <math>f^{-1}(f<sub>(x)</sub>)=x</math> for every x in the domain of f and f<br />
* The graph of a function and the graph of its inverse are symmetric with respect to the line y=x.<br />
* If f and g are both one-one, then f°g follows injectivity.<br />
* If g°f is one-one, then function f is one-one, but function g may not be.<br />
* A one-one function is either strictly decreasing or strictly increasing.<br />
* A function that is not a one-one is considered as many-to-one.<br />
* [[Parabolic function]]s are not one-one functions.<br />
<br />
== Examples ==<br />
<br />
Examples of one-one functions include:<br />
<br />
* Identity function: f(x) is always injective.</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=Inverse_function&diff=3368Inverse function2022-04-28T03:59:22Z<p>Sebastian: /* Relevant observations */</p>
<hr />
<div>An inverse function is a function that serves to "undo" another function. That is, if '''f(x)''' produces '''y''', then putting '''y''' into the inverse of '''f''' produces the output '''x'''. Not every function has an inverse. <br />
<br />
== Definition == <br />
A function '''g''' is the inverse function of f if '''f(g(x))'''='''x''' for each value of '''x''' in the domain of '''g''', and '''g(f(x))=x''' for each value of '''x''' in the domain of '''f'''. The function '''g''' is denoted as '''f<sup>-1</sup>''' ("inverse of f").<br />
<br />
=== Notation ===<br />
Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, '''f<sup>-1</sup>(x)'''≠'''1/f(x)'''.<br />
<br />
== Relevant observations ==<br />
* If '''g''' is the inverse function of '''f''', then '''f''' is the inverse function of '''g'''.<br />
* The domain of '''f<sup>-1</sup>''' is the range of '''f''' and the range of '''f<sup>-1</sup>''' is the domain of '''f'''.<br />
* A function may not have an inverse function, but if it has, the inverse function is unique.</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=Inverse_function&diff=3367Inverse function2022-04-28T03:58:25Z<p>Sebastian: /* Definition */</p>
<hr />
<div>An inverse function is a function that serves to "undo" another function. That is, if '''f(x)''' produces '''y''', then putting '''y''' into the inverse of '''f''' produces the output '''x'''. Not every function has an inverse. <br />
<br />
== Definition == <br />
A function '''g''' is the inverse function of f if '''f(g(x))'''='''x''' for each value of '''x''' in the domain of '''g''', and '''g(f(x))=x''' for each value of '''x''' in the domain of '''f'''. The function '''g''' is denoted as '''f<sup>-1</sup>''' ("inverse of f").<br />
<br />
=== Notation ===<br />
Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, '''f<sup>-1</sup>(x)'''≠'''1/f(x)'''.<br />
<br />
== Relevant observations ==<br />
* If g is the inverse function of f, then f is the inverse function of g.<br />
* The domain of f<sup>-1</sup> is the range of f and the range of f<sup>-1</sup> is the domain of f.<br />
* A function may not have an inverse function, but if it has, the inverse function is unique.</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=Inverse_function&diff=3366Inverse function2022-04-28T03:58:00Z<p>Sebastian: </p>
<hr />
<div>An inverse function is a function that serves to "undo" another function. That is, if '''f(x)''' produces '''y''', then putting '''y''' into the inverse of '''f''' produces the output '''x'''. Not every function has an inverse. <br />
<br />
== Definition == <br />
A function '''g''' is the inverse function of f if '''f(g(x))=x''' for each value of '''x''' in the domain of '''g''', and '''g(f(x))=x''' for each value of '''x''' in the domain of '''f'''. The function '''g''' is denoted as '''f<sup>-1</sup>''' ("inverse of f").<br />
<br />
=== Notation ===<br />
Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, '''f<sup>-1</sup>(x)'''≠'''1/f(x)'''.<br />
<br />
== Relevant observations ==<br />
* If g is the inverse function of f, then f is the inverse function of g.<br />
* The domain of f<sup>-1</sup> is the range of f and the range of f<sup>-1</sup> is the domain of f.<br />
* A function may not have an inverse function, but if it has, the inverse function is unique.</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=Inverse_function&diff=3365Inverse function2022-04-28T03:57:18Z<p>Sebastian: /* Definition */</p>
<hr />
<div>An inverse function is a function that serves to "undo" another function. That is, if f(x) produces y, then putting y into the inverse of f produces the output x. Not every function has an inverse. <br />
<br />
== Definition == <br />
A function '''g''' is the inverse function of f if '''f(g(x))=x''' for each value of '''x''' in the domain of '''g''', and '''g(f(x))=x''' for each value of '''x''' in the domain of '''f'''. The function '''g''' is denoted as '''f<sup>-1</sup>''' ("inverse of f").<br />
<br />
=== Notation ===<br />
Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, '''f<sup>-1</sup>(x)'''≠'''1/f(x)'''.<br />
<br />
== Relevant observations ==<br />
* If g is the inverse function of f, then f is the inverse function of g.<br />
* The domain of f<sup>-1</sup> is the range of f and the range of f<sup>-1</sup> is the domain of f.<br />
* A function may not have an inverse function, but if it has, the inverse function is unique.</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=Inverse_function&diff=3364Inverse function2022-04-28T03:55:44Z<p>Sebastian: </p>
<hr />
<div>An inverse function is a function that serves to "undo" another function. That is, if f(x) produces y, then putting y into the inverse of f produces the output x. Not every function has an inverse. <br />
<br />
== Definition == <br />
A function g is the inverse function of f if <math>f(g(x))=x</math> for each value of x in the domain of g, and g(f(x))=x for each value of x in the domain of f. The function g is denoted as f<sup>-1</sup> ("inverse of f").<br />
<br />
=== Notation ===<br />
Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, f<sup>-1</sup>(x) is not equal to 1/f(x).<br />
<br />
== Relevant observations ==<br />
* If g is the inverse function of f, then f is the inverse function of g.<br />
* The domain of f<sup>-1</sup> is the range of f and the range of f<sup>-1</sup> is the domain of f.<br />
* A function may not have an inverse function, but if it has, the inverse function is unique.</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=Inverse_function&diff=3363Inverse function2022-04-28T03:54:23Z<p>Sebastian: /* Definition */</p>
<hr />
<div>An inverse function is a function that serves to "undo" another function. That is, if f(x) produces y, then putting y into the inverse of f produces the output x. Not every function has an inverse. <br />
<br />
== Definition == <br />
A function <math>g</math> is the inverse function of <math>f</math> if <math>f(g(x))=x</math> for each value of <math>x</math> in the domain of <math>g</math>, and <math>g(f(x))=x</math> for each value of <math>x</math> in the domain of <math>f</math>. The function <math>g</math> is denoted as <math>f</math><sup>-1</sup> ("inverse of f").<br />
<br />
=== Notation ===<br />
Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, <math>f<sup>-1</sup>(x)</math> is not equal to 1/f(x).<br />
<br />
== Relevant observations ==<br />
* If g is the inverse function of f, then f is the inverse function of g.<br />
* The domain of f<sup>-1</sup> is the range of f and the range of f<sup>-1</sup> is the domain of f.<br />
* A function may not have an inverse function, but if it has, the inverse function is unique.</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=Inverse_function&diff=3362Inverse function2022-04-28T03:53:40Z<p>Sebastian: /* Definition */</p>
<hr />
<div>An inverse function is a function that serves to "undo" another function. That is, if f(x) produces y, then putting y into the inverse of f produces the output x. Not every function has an inverse. <br />
<br />
== Definition == <br />
A function <math>g</math> is the inverse function of <math>f</math> if <math>f(g(x))=x</math> for each value of <math>x</math> in the domain of <math>g</math>, and <math>g(f(x))=x</math> for each value of <math>x</math> in the domain of <math>f</math>. The function <math>g</math> is denoted as f<sup>-1</sup> ("inverse of f").<br />
<br />
=== Notation ===<br />
Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, <math>f</math><sup>-1</sup>(x) is not equal to 1/f(x).<br />
<br />
== Relevant observations ==<br />
* If g is the inverse function of f, then f is the inverse function of g.<br />
* The domain of f<sup>-1</sup> is the range of f and the range of f<sup>-1</sup> is the domain of f.<br />
* A function may not have an inverse function, but if it has, the inverse function is unique.</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=Inverse_function&diff=3361Inverse function2022-04-28T03:52:02Z<p>Sebastian: /* Definition */</p>
<hr />
<div>An inverse function is a function that serves to "undo" another function. That is, if f(x) produces y, then putting y into the inverse of f produces the output x. Not every function has an inverse. <br />
<br />
== Definition == <br />
A function g is the inverse function of f if <math>f(g(x))=x</math> for each value of x in the domain of g, and g(f(x))=x for each value of x in the domain of f. The function g is denoted as f<sup>-1</sup> ("inverse of f").<br />
<br />
=== Notation ===<br />
Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, f<sup>-1</sup>(x) is not equal to 1/f(x).<br />
<br />
== Relevant observations ==<br />
* If g is the inverse function of f, then f is the inverse function of g.<br />
* The domain of f<sup>-1</sup> is the range of f and the range of f<sup>-1</sup> is the domain of f.<br />
* A function may not have an inverse function, but if it has, the inverse function is unique.</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=Inverse_function&diff=3360Inverse function2022-04-27T23:40:18Z<p>Sebastian: /* Definition */</p>
<hr />
<div>An inverse function is a function that serves to "undo" another function. That is, if f(x) produces y, then putting y into the inverse of f produces the output x. Not every function has an inverse. <br />
<br />
== Definition == <br />
A function g is the inverse function of f if f(g(x))=x for each value of x in the domain of g, and g(f(x))=x for each value of x in the domain of f. The function g is denoted as f<sup>-1</sup> ("inverse of f").<br />
<br />
=== Notation ===<br />
Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, f<sup>-1</sup>(x) is not equal to 1/f(x).<br />
<br />
== Relevant observations ==<br />
* If g is the inverse function of f, then f is the inverse function of g.<br />
* The domain of f<sup>-1</sup> is the range of f and the range of f<sup>-1</sup> is the domain of f.<br />
* A function may not have an inverse function, but if it has, the inverse function is unique.</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=Inverse_function&diff=3359Inverse function2022-04-27T23:31:08Z<p>Sebastian: /* Definition */</p>
<hr />
<div>An inverse function is a function that serves to "undo" another function. That is, if f(x) produces y, then putting y into the inverse of f produces the output x. Not every function has an inverse. <br />
<br />
== Definition == <br />
A function g is the inverse function of f if f(g(x))=x for each value of x in the domain of g, and g(f(x))=x for each value of x in the domain of f. The function g is denoted as f<sup>-1</sup> ("inverse of f").<br />
<br />
=== Notation ===<br />
Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, f<sup>-1</sup>(x) is not equal to 1/f(x).</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=Inverse_function&diff=3358Inverse function2022-04-27T23:26:25Z<p>Sebastian: Created page with "An inverse function is a function that serves to "undo" another function. That is, if f(x) produces y, then putting y into the inverse of f produces the output x. Not every fu..."</p>
<hr />
<div>An inverse function is a function that serves to "undo" another function. That is, if f(x) produces y, then putting y into the inverse of f produces the output x. Not every function has an inverse. <br />
<br />
== Definition == <br />
A function g is the inverse function of f if f(g(x))=x for each value of x in the domain of g, and g(f(x))=x for each value of x in the domain of f. The function g is denoted as f<sup>-1</sup> ("inverse of f").</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=One-one_function&diff=3357One-one function2022-04-27T22:58:21Z<p>Sebastian: /* Geometric proof */</p>
<hr />
<div>A [[function]] <math>f</math> is called one-one function if it never adopts the same value twice. This means, one-one functions return a unique [[range]] for each element of their [[domain]].<br />
<br />
Other names given to the one-one function are one to one, or injective function.<br />
<br />
One-one functions are a set of functions which denote the relation between sets, elements or identities. The other two sets are [[surjective function]]s and [[bijective function]]s.<br />
<br />
== Definition ==<br />
The function <math>f(x)</math> is called one-one function when for every value of <math>x</math> in the domain of the function, there will be a unique value of <math>f(x)</math>.<br />
<br />
== Geometric proof ==<br />
Similar to the vertical line test (VLT) for functions, there is a horizontal line test (HLT) to prove the one-one property. A function is one-one if and only if no horizontal line intersects its graph more than once.<br />
<br />
<br />
In the graph below, the function <math>y=x^3</math> is intersected once by the horizontal line. Therefore the function is geometrically proven to be one-one.<br />
<br />
[[File:One-one function geometric proof true.PNG|350px]]<br />
<br />
<br />
In the graph below, the function <math>y=x^2</math> is intersected twice by the horizontal line. Therefore the function is geometrically proven no to be one-one.<br />
<br />
[[File:One-one function geometric proof.PNG|350px]]<br />
<br />
== Properties ==<br />
* The domain of f equals the range of f<sup>-1</sup>.<br />
* f<sup>-1</sup>(f<sub>(x)</sub>)=x for every x in the domain of f and f<br />
* The graph of a function and the graph of its inverse are symmetric with respect to the line y=x.<br />
* If f and g are both one-one, then f°g follows injectivity.<br />
* If g°f is one-one, then function f is one-one, but function g may not be.<br />
* A one-one function is either strictly decreasing or strictly increasing.<br />
* A function that is not a one-one is considered as many-to-one.<br />
* [[Parabolic function]]s are not one-one functions.<br />
<br />
== Examples ==<br />
<br />
Examples of one-one functions include:<br />
<br />
* Identity function: f(x) is always injective.</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=User:Sebastian&diff=3356User:Sebastian2022-04-27T05:35:22Z<p>Sebastian: </p>
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<br />
Red articles:<br />
<br />
*[[Absolute maximum value]]<br />
*[[Absolute minimum value]]<br />
*[[Antiderivative]]<br />
*[[Closed interval]]<br />
*[[Constant function]]<br />
*[[Continuously differentiable functions]]<br />
*[[Decreasing function]]<br />
*[[Exponential function]]<br />
*[[Fundamental theorem of calculus]]<br />
*[[Identity function]]<br />
*[[Imaginary number]]<br />
*[[Inequality]]<br />
*[[Interval]]<br />
*[[Inverse function]]<br />
*[[Linear function]]<br />
*[[Logarithmic differentiation]] <br />
*[[Natural logarithm]] <br />
*[[Number]]<br />
*[[One-one function]]<br />
*[[Open interval]]<br />
*[[Piecewise linear function]]<br />
*[[Point of inflection]]<br />
*[[Pointwise maximum]]<br />
*[[Product theorem for continuity]]<br />
*[[Range]]<br />
*[[Real number]]<br />
*[[Vertical tangent]]</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=User:Sebastian&diff=3355User:Sebastian2022-04-27T05:34:09Z<p>Sebastian: </p>
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<div>Pageview ranking: [https://analytics.vipulnaik.com/top-pages.php?project_title=Calculus%20subwiki&fbclid=IwAR26FVz9b-WAf2FpqyFxO_tItTH2tdmutjFeOLLtH2Rjz-3GUTDkd2VY_E0]<br />
<br />
Red articles:<br />
<br />
*[[Absolute maximum value]]<br />
*[[Absolute minimum value]]<br />
*[[Antiderivative]]<br />
*[[Closed interval]]<br />
*[[Constant function]]<br />
*[[Continuously differentiable functions]]<br />
*[[Decreasing function]]<br />
*[[Exponential function]]<br />
*[[Fundamental theorem of calculus]]<br />
*[[Identity function]]<br />
*[[Imaginary number]]<br />
*[[Inequality]]<br />
*[[Inverse function]]<br />
*[[Linear function]]<br />
*[[Logarithmic differentiation]] <br />
*[[Natural logarithm]] <br />
*[[Number]]<br />
*[[One-one function]]<br />
*[[Open interval]]<br />
*[[Piecewise linear function]]<br />
*[[Point of inflection]]<br />
*[[Pointwise maximum]]<br />
*[[Product theorem for continuity]]<br />
*[[Range]]<br />
*[[Real number]]<br />
*[[Vertical tangent]]</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=User:Sebastian&diff=3354User:Sebastian2022-04-27T05:27:31Z<p>Sebastian: </p>
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<div>Pageview ranking: [https://analytics.vipulnaik.com/top-pages.php?project_title=Calculus%20subwiki&fbclid=IwAR26FVz9b-WAf2FpqyFxO_tItTH2tdmutjFeOLLtH2Rjz-3GUTDkd2VY_E0]<br />
<br />
Red articles:<br />
<br />
*[[Absolute maximum value]]<br />
*[[Absolute minimum value]]<br />
*[[Antiderivative]]<br />
*[[Closed interval]]<br />
*[[Constant function]]<br />
*[[Continuously differentiable functions]]<br />
*[[Decreasing function]]<br />
*[[Exponential function]]<br />
*[[Fundamental theorem of calculus]]<br />
*[[Identity function]]<br />
*[[Imaginary number]]<br />
*[[Inverse function]]<br />
*[[Linear function]]<br />
*[[Logarithmic differentiation]] <br />
*[[Natural logarithm]] <br />
*[[One-one function]]<br />
*[[Open interval]]<br />
*[[Piecewise linear function]]<br />
*[[Point of inflection]]<br />
*[[Pointwise maximum]]<br />
*[[Product theorem for continuity]]<br />
*[[Range]]<br />
*[[Real number]]<br />
*[[Vertical tangent]]</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=User:Sebastian&diff=3353User:Sebastian2022-04-27T05:24:20Z<p>Sebastian: </p>
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<div>Pageview ranking: [https://analytics.vipulnaik.com/top-pages.php?project_title=Calculus%20subwiki&fbclid=IwAR26FVz9b-WAf2FpqyFxO_tItTH2tdmutjFeOLLtH2Rjz-3GUTDkd2VY_E0]<br />
<br />
Red articles:<br />
<br />
*[[Absolute maximum value]]<br />
*[[Absolute minimum value]]<br />
*[[Closed interval]]<br />
*[[Constant function]]<br />
*[[Continuously differentiable functions]]<br />
*[[Decreasing function]]<br />
*[[Exponential function]]<br />
*[[Fundamental theorem of calculus]]<br />
*[[Identity function]]<br />
*[[Imaginary number]]<br />
*[[Inverse function]]<br />
*[[Logarithmic differentiation]] <br />
*[[Natural logarithm]] <br />
*[[One-one function]]<br />
*[[Piecewise linear function]]<br />
*[[Point of inflection]]<br />
*[[Pointwise maximum]]<br />
*[[Product theorem for continuity]]<br />
*[[Range]]<br />
*[[Real number]]<br />
*[[Vertical tangent]]</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=One-one_function&diff=3352One-one function2022-04-27T04:27:27Z<p>Sebastian: /* Definition */</p>
<hr />
<div>A [[function]] <math>f</math> is called one-one function if it never adopts the same value twice. This means, one-one functions return a unique [[range]] for each element of their [[domain]].<br />
<br />
Other names given to the one-one function are one to one, or injective function.<br />
<br />
One-one functions are a set of functions which denote the relation between sets, elements or identities. The other two sets are [[surjective function]]s and [[bijective function]]s.<br />
<br />
== Definition ==<br />
The function <math>f(x)</math> is called one-one function when for every value of <math>x</math> in the domain of the function, there will be a unique value of <math>f(x)</math>.<br />
<br />
== Geometric proof ==<br />
A function is one-one if and only if no horizontal line intersects its graph more than once.<br />
<br />
<br />
In the graph below, the function <math>y=x^3</math> is intersected once by the horizontal line. Therefore the function is geometrically proven to be one-one.<br />
<br />
[[File:One-one function geometric proof true.PNG|350px]]<br />
<br />
<br />
In the graph below, the function <math>y=x^2</math> is intersected twice by the horizontal line. Therefore the function is geometrically proven no to be one-one.<br />
<br />
[[File:One-one function geometric proof.PNG|350px]]<br />
<br />
== Properties ==<br />
* The domain of f equals the range of f<sup>-1</sup>.<br />
* f<sup>-1</sup>(f<sub>(x)</sub>)=x for every x in the domain of f and f<br />
* The graph of a function and the graph of its inverse are symmetric with respect to the line y=x.<br />
* If f and g are both one-one, then f°g follows injectivity.<br />
* If g°f is one-one, then function f is one-one, but function g may not be.<br />
* A one-one function is either strictly decreasing or strictly increasing.<br />
* A function that is not a one-one is considered as many-to-one.<br />
* [[Parabolic function]]s are not one-one functions.<br />
<br />
== Examples ==<br />
<br />
Examples of one-one functions include:<br />
<br />
* Identity function: f(x) is always injective.</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=One-one_function&diff=3351One-one function2022-04-27T04:27:17Z<p>Sebastian: /* Definition */</p>
<hr />
<div>A [[function]] <math>f</math> is called one-one function if it never adopts the same value twice. This means, one-one functions return a unique [[range]] for each element of their [[domain]].<br />
<br />
Other names given to the one-one function are one to one, or injective function.<br />
<br />
One-one functions are a set of functions which denote the relation between sets, elements or identities. The other two sets are [[surjective function]]s and [[bijective function]]s.<br />
<br />
== Definition ==<br />
The function <math>f(x)</math> is called one-one function when for every value of <math>x</math> in the domain of the function, there will be a unique value of <math>f(x)</math><br />
<br />
== Geometric proof ==<br />
A function is one-one if and only if no horizontal line intersects its graph more than once.<br />
<br />
<br />
In the graph below, the function <math>y=x^3</math> is intersected once by the horizontal line. Therefore the function is geometrically proven to be one-one.<br />
<br />
[[File:One-one function geometric proof true.PNG|350px]]<br />
<br />
<br />
In the graph below, the function <math>y=x^2</math> is intersected twice by the horizontal line. Therefore the function is geometrically proven no to be one-one.<br />
<br />
[[File:One-one function geometric proof.PNG|350px]]<br />
<br />
== Properties ==<br />
* The domain of f equals the range of f<sup>-1</sup>.<br />
* f<sup>-1</sup>(f<sub>(x)</sub>)=x for every x in the domain of f and f<br />
* The graph of a function and the graph of its inverse are symmetric with respect to the line y=x.<br />
* If f and g are both one-one, then f°g follows injectivity.<br />
* If g°f is one-one, then function f is one-one, but function g may not be.<br />
* A one-one function is either strictly decreasing or strictly increasing.<br />
* A function that is not a one-one is considered as many-to-one.<br />
* [[Parabolic function]]s are not one-one functions.<br />
<br />
== Examples ==<br />
<br />
Examples of one-one functions include:<br />
<br />
* Identity function: f(x) is always injective.</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=One-one_function&diff=3350One-one function2022-04-27T02:36:12Z<p>Sebastian: </p>
<hr />
<div>A [[function]] <math>f</math> is called one-one function if it never adopts the same value twice. This means, one-one functions return a unique [[range]] for each element of their [[domain]].<br />
<br />
Other names given to the one-one function are one to one, or injective function.<br />
<br />
One-one functions are a set of functions which denote the relation between sets, elements or identities. The other two sets are [[surjective function]]s and [[bijective function]]s.<br />
<br />
== Definition ==<br />
The function <math>f(x)</math><br />
<br />
== Geometric proof ==<br />
A function is one-one if and only if no horizontal line intersects its graph more than once.<br />
<br />
<br />
In the graph below, the function <math>y=x^3</math> is intersected once by the horizontal line. Therefore the function is geometrically proven to be one-one.<br />
<br />
[[File:One-one function geometric proof true.PNG|350px]]<br />
<br />
<br />
In the graph below, the function <math>y=x^2</math> is intersected twice by the horizontal line. Therefore the function is geometrically proven no to be one-one.<br />
<br />
[[File:One-one function geometric proof.PNG|350px]]<br />
<br />
== Properties ==<br />
* The domain of f equals the range of f<sup>-1</sup>.<br />
* f<sup>-1</sup>(f<sub>(x)</sub>)=x for every x in the domain of f and f<br />
* The graph of a function and the graph of its inverse are symmetric with respect to the line y=x.<br />
* If f and g are both one-one, then f°g follows injectivity.<br />
* If g°f is one-one, then function f is one-one, but function g may not be.<br />
* A one-one function is either strictly decreasing or strictly increasing.<br />
* A function that is not a one-one is considered as many-to-one.<br />
* [[Parabolic function]]s are not one-one functions.<br />
<br />
== Examples ==<br />
<br />
Examples of one-one functions include:<br />
<br />
* Identity function: f(x) is always injective.</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=One-one_function&diff=3349One-one function2022-04-27T01:52:46Z<p>Sebastian: /* Properties */</p>
<hr />
<div>A [[function]] <math>f</math> is called one-one function if it never adopts the same value twice. This means, one-one functions return a unique [[range]] for each element of their [[domain]].<br />
<br />
Other names given to the one-one function are one to one, or injective function.<br />
<br />
One-one functions are a set of functions which denote the relation between sets, elements or identities. The other two sets are [[surjective function]]s and [[bijective function]]s.<br />
<br />
== Definition ==<br />
The function <math>f<sub>(x)</sub> </math><br />
<br />
== Geometric proof ==<br />
A function is one-one if and only if no horizontal line intersects its graph more than once.<br />
<br />
<br />
In the graph below, the function <math>y=x^3</math> is intersected once by the horizontal line. Therefore the function is geometrically proven to be one-one.<br />
<br />
[[File:One-one function geometric proof true.PNG|350px]]<br />
<br />
<br />
In the graph below, the function <math>y=x^2</math> is intersected twice by the horizontal line. Therefore the function is geometrically proven no to be one-one.<br />
<br />
[[File:One-one function geometric proof.PNG|350px]]<br />
<br />
== Properties ==<br />
* The domain of f equals the range of f<sup>-1</sup>.<br />
* f<sup>-1</sup>(f<sub>(x)</sub>)=x for every x in the domain of f and f<br />
* The graph of a function and the graph of its inverse are symmetric with respect to the line y=x.<br />
* If f and g are both one-one, then f°g follows injectivity.<br />
* If g°f is one-one, then function f is one-one, but function g may not be.<br />
* A one-one function is either strictly decreasing or strictly increasing.<br />
* A function that is not a one-one is considered as many-to-one.<br />
* [[Parabolic function]]s are not one-one functions.<br />
<br />
== Examples ==<br />
<br />
Examples of one-one functions include:<br />
<br />
* Identity function: f(x) is always injective.</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=User:Sebastian&diff=3348User:Sebastian2022-04-27T01:50:04Z<p>Sebastian: </p>
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Red articles:<br />
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*[[Real number]]<br />
*[[Imaginary number]]<br />
*[[one-one function]]<br />
*[[inverse function]]<br />
*[[range]]<br />
*[[decreasing function]]<br />
*[[point of inflection]]<br />
*[[vertical tangent]]<br />
*[[natural logarithm]] <br />
*[[exponential function]]<br />
*[[continuously differentiable functions]]<br />
*[[Logarithmic differentiation]] <br />
*[[product theorem for continuity]]<br />
*[[piecewise linear function]]<br />
*[[pointwise maximum]]<br />
*[[closed interval]]<br />
*[[absolute maximum value]]<br />
*[[absolute minimum value]]<br />
*[[identity function]]<br />
*[[Constant function]]</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=One-one_function&diff=3347One-one function2022-04-27T01:45:02Z<p>Sebastian: /* Properties */</p>
<hr />
<div>A [[function]] <math>f</math> is called one-one function if it never adopts the same value twice. This means, one-one functions return a unique [[range]] for each element of their [[domain]].<br />
<br />
Other names given to the one-one function are one to one, or injective function.<br />
<br />
One-one functions are a set of functions which denote the relation between sets, elements or identities. The other two sets are [[surjective function]]s and [[bijective function]]s.<br />
<br />
== Definition ==<br />
The function <math>f<sub>(x)</sub> </math><br />
<br />
== Geometric proof ==<br />
A function is one-one if and only if no horizontal line intersects its graph more than once.<br />
<br />
<br />
In the graph below, the function <math>y=x^3</math> is intersected once by the horizontal line. Therefore the function is geometrically proven to be one-one.<br />
<br />
[[File:One-one function geometric proof true.PNG|350px]]<br />
<br />
<br />
In the graph below, the function <math>y=x^2</math> is intersected twice by the horizontal line. Therefore the function is geometrically proven no to be one-one.<br />
<br />
[[File:One-one function geometric proof.PNG|350px]]<br />
<br />
== Properties ==<br />
* The domain of f equals the range of f<sup>-1</sup>.<br />
* f<sup>-1</sup>(f<sub>(x)</sub>)=x for every x in the domain of f and f<br />
* The graph of a function and the graph of its inverse are symmetric with respect to the line y=x.<br />
* If f and g are both one-one, then f°g follows injectivity.<br />
* If g°f is one-one, then function f is one-one, but function g may not be.<br />
<br />
== Examples ==<br />
<br />
Examples of one-one functions include:<br />
<br />
* Identity function: f(x) is always injective.</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=One-one_function&diff=3346One-one function2022-04-27T01:43:31Z<p>Sebastian: </p>
<hr />
<div>A [[function]] <math>f</math> is called one-one function if it never adopts the same value twice. This means, one-one functions return a unique [[range]] for each element of their [[domain]].<br />
<br />
Other names given to the one-one function are one to one, or injective function.<br />
<br />
One-one functions are a set of functions which denote the relation between sets, elements or identities. The other two sets are [[surjective function]]s and [[bijective function]]s.<br />
<br />
== Definition ==<br />
The function <math>f<sub>(x)</sub> </math><br />
<br />
== Geometric proof ==<br />
A function is one-one if and only if no horizontal line intersects its graph more than once.<br />
<br />
<br />
In the graph below, the function <math>y=x^3</math> is intersected once by the horizontal line. Therefore the function is geometrically proven to be one-one.<br />
<br />
[[File:One-one function geometric proof true.PNG|350px]]<br />
<br />
<br />
In the graph below, the function <math>y=x^2</math> is intersected twice by the horizontal line. Therefore the function is geometrically proven no to be one-one.<br />
<br />
[[File:One-one function geometric proof.PNG|350px]]<br />
<br />
== Properties ==<br />
* The domain of f equals the range of f<sup>-1</sup>.<br />
* f<sup>-1</sup>(f<sub>(x)</sub>)=x for every x in the domain of f and f<br />
* The graph of a function and the graph of its inverse are symmetric with respect to the line y=x.<br />
* If f and g are both one-one, then f°g follows injectivity.<br />
* If g°f is one-one, then function f is one-one, but function g may not be.</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=One-one_function&diff=3345One-one function2022-04-27T01:40:32Z<p>Sebastian: </p>
<hr />
<div>A [[function]] <math>f</math> is called one-one function if it never adopts the same value twice. This means, one-one functions return a unique [[range]] for each element of their [[domain]].<br />
<br />
Other names given to the one-one function are one to one, or injective function.<br />
<br />
== Definition ==<br />
The function <math>f<sub>(x)</sub> </math><br />
<br />
== Geometric proof ==<br />
A function is one-one if and only if no horizontal line intersects its graph more than once.<br />
<br />
<br />
In the graph below, the function <math>y=x^3</math> is intersected once by the horizontal line. Therefore the function is geometrically proven to be one-one.<br />
<br />
[[File:One-one function geometric proof true.PNG|350px]]<br />
<br />
<br />
In the graph below, the function <math>y=x^2</math> is intersected twice by the horizontal line. Therefore the function is geometrically proven no to be one-one.<br />
<br />
[[File:One-one function geometric proof.PNG|350px]]<br />
<br />
== Properties ==<br />
* The domain of f equals the range of f<sup>-1</sup>.<br />
* f<sup>-1</sup>(f<sub>(x)</sub>)=x for every x in the domain of f and f<br />
* The graph of a function and the graph of its inverse are symmetric with respect to the line y=x.<br />
* If f and g are both one-one, then f°g follows injectivity.<br />
* If g°f is one-one, then function f is one-one, but function g may not be.</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=One-one_function&diff=3344One-one function2022-04-27T01:38:05Z<p>Sebastian: /* Properties */</p>
<hr />
<div>A [[function]] <math>f</math> is called one-one function if it never adopts the same value twice. This means, one-one functions return a unique [[range]] for each element of their [[domain]].<br />
<br />
== Definition ==<br />
The function <math>f<sub>(x)</sub> </math><br />
<br />
== Geometric proof ==<br />
A function is one-one if and only if no horizontal line intersects its graph more than once.<br />
<br />
<br />
In the graph below, the function <math>y=x^3</math> is intersected once by the horizontal line. Therefore the function is geometrically proven to be one-one.<br />
<br />
[[File:One-one function geometric proof true.PNG|350px]]<br />
<br />
<br />
In the graph below, the function <math>y=x^2</math> is intersected twice by the horizontal line. Therefore the function is geometrically proven no to be one-one.<br />
<br />
[[File:One-one function geometric proof.PNG|350px]]<br />
<br />
== Properties ==<br />
* The domain of f equals the range of f<sup>-1</sup>.<br />
* f<sup>-1</sup>(f<sub>(x)</sub>)=x for every x in the domain of f and f<br />
* The graph of a function and the graph of its inverse are symmetric with respect to the line y=x.<br />
* If f and g are both one-one, then f°g follows injectivity.<br />
* If g°f is one-one, then function f is one-one, but function g may not be.</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=One-one_function&diff=3343One-one function2022-04-27T00:30:53Z<p>Sebastian: </p>
<hr />
<div>A [[function]] <math>f</math> is called one-one function if it never adopts the same value twice. This means, one-one functions return a unique [[range]] for each element of their [[domain]].<br />
<br />
== Definition ==<br />
The function <math>f<sub>(x)</sub> </math><br />
<br />
== Geometric proof ==<br />
A function is one-one if and only if no horizontal line intersects its graph more than once.<br />
<br />
<br />
In the graph below, the function <math>y=x^3</math> is intersected once by the horizontal line. Therefore the function is geometrically proven to be one-one.<br />
<br />
[[File:One-one function geometric proof true.PNG|350px]]<br />
<br />
<br />
In the graph below, the function <math>y=x^2</math> is intersected twice by the horizontal line. Therefore the function is geometrically proven no to be one-one.<br />
<br />
[[File:One-one function geometric proof.PNG|350px]]<br />
<br />
== Properties ==<br />
* The domain of f equals the range of f<sup>-1</sup>.<br />
* f<sup>-1</sup>(f<sub>(x)</sub>)=x for every x in the domain of f and f<br />
* The graph of a function and the graph of its inverse are symmetric with respect to the line y=x.<br />
* If f and g are both one-one, then f°g follows injectivity.</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=One-one_function&diff=3342One-one function2022-04-27T00:15:34Z<p>Sebastian: </p>
<hr />
<div>A [[function]] <math>f</math> is called one-one function if it never adopts the same value twice.<br />
<br />
== Geometric proof ==<br />
A function is one-one if and only if no horizontal line intersects its graph more than once.<br />
<br />
<br />
In the graph below, the function <math>y=x^3</math> is intersected once by the horizontal line. Therefore the function is geometrically proven to be one-one.<br />
<br />
[[File:One-one function geometric proof true.PNG|350px]]<br />
<br />
<br />
In the graph below, the function <math>y=x^2</math> is intersected twice by the horizontal line. Therefore the function is geometrically proven no to be one-one.<br />
<br />
[[File:One-one function geometric proof.PNG|350px]]<br />
<br />
== Properties ==<br />
* The domain of f equals the range of f<sup>-1</sup>.<br />
* f<sup>-1</sup>(f<sub>(x)</sub>)=x for every x in the domain of f and f<br />
* The graph of a function and the graph of its inverse are symmetric with respect to the line y=x.<br />
* If f and g are both one-one, then f°g follows injectivity.</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=One-one_function&diff=3341One-one function2022-04-27T00:11:12Z<p>Sebastian: /* Properties */</p>
<hr />
<div>A [[function]] <math>f</math> is called one-one function if it never adopts the same value twice.<br />
<br />
== Geometric proof ==<br />
A function is one-one if and only if no horizontal line intersects its graph more than once.<br />
<br />
<br />
In the graph below, the function <math>y=x^3</math> is intersected once by the horizontal line. Therefore the function is geometrically proven to be one-one.<br />
<br />
[[File:One-one function geometric proof true.PNG|350px]]<br />
<br />
<br />
In the graph below, the function <math>y=x^2</math> is intersected twice by the horizontal line. Therefore the function is geometrically proven no to be one-one.<br />
<br />
[[File:One-one function geometric proof.PNG|350px]]<br />
<br />
== Properties ==<br />
* The domain of f equals the range of f<sup>-1</sup>.<br />
* f<sup>-1</sup>(f<sub>(x)</sub>)=x</div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=One-one_function&diff=3340One-one function2022-04-27T00:08:12Z<p>Sebastian: /* Geometric proof */</p>
<hr />
<div>A [[function]] <math>f</math> is called one-one function if it never adopts the same value twice.<br />
<br />
== Geometric proof ==<br />
A function is one-one if and only if no horizontal line intersects its graph more than once.<br />
<br />
<br />
In the graph below, the function <math>y=x^3</math> is intersected once by the horizontal line. Therefore the function is geometrically proven to be one-one.<br />
<br />
[[File:One-one function geometric proof true.PNG|350px]]<br />
<br />
<br />
In the graph below, the function <math>y=x^2</math> is intersected twice by the horizontal line. Therefore the function is geometrically proven no to be one-one.<br />
<br />
[[File:One-one function geometric proof.PNG|350px]]<br />
<br />
== Properties ==<br />
* The domain of f equals the range of f<sup>-1</sup></div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=One-one_function&diff=3339One-one function2022-04-27T00:07:35Z<p>Sebastian: /* Geometric proof */</p>
<hr />
<div>A [[function]] <math>f</math> is called one-one function if it never adopts the same value twice.<br />
<br />
== Geometric proof ==<br />
A function is one-one if and only if no horizontal line intersects its graph more than once.<br />
<br />
<br />
In the graph below, the function <math>y=x^3</math> is intersected once by the horizontal line. Therefore the function is geometrically proven to be one-one.<br />
<br />
[[File:One-one function geometric proof true.PNG|350px]]<br />
<br />
<br />
In the graph below, the function <math>y=x^2</math> is intersected twice by the horizontal line. Therefore the function is geometrically proven no to be one-one.<br />
<br />
[[File:One-one function geometric proof.PNG|350px]]<br />
<br />
=== Properties ===<br />
* The domain of f equals the range of f<sup>-1</sup></div>Sebastianhttps://calculus.subwiki.org/w/index.php?title=One-one_function&diff=3337One-one function2022-04-22T15:39:11Z<p>Sebastian: /* Geometric proof */</p>
<hr />
<div>A [[function]] <math>f</math> is called one-one function if it never adopts the same value twice.<br />
<br />
== Geometric proof ==<br />
A function is one-one if and only if no horizontal line intersects its graph more than once.<br />
<br />
<br />
In the graph below, the function <math>y=x^3</math> is intersected once by the horizontal line. Therefore the function is geometrically proven to be one-one.<br />
<br />
[[File:One-one function geometric proof true.PNG|350px]]<br />
<br />
<br />
In the graph below, the function <math>y=x^2</math> is intersected twice by the horizontal line. Therefore the function is geometrically proven no to be one-one.<br />
<br />
[[File:One-one function geometric proof.PNG|350px]]</div>Sebastian