Domain - Revision history

Vipul at 00:43, 29 April 2014

2014-04-29T00:43:12Z

← Older revision		Revision as of 00:43, 29 April 2014
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			{{core term}}
	==Definition==		==Definition==

Vipul: /* Computation of maximum possible domain from algebraic expression */

2011-09-19T22:04:36Z

Computation of maximum possible domain from algebraic expression

← Older revision		Revision as of 22:04, 19 September 2011
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	\| [[pointwise product of functions\|pointwise product]] \|\| <math>f \cdot g</math> (sometimes denoted <math>fg</math>) is the function <math>x \mapsto f(x)g(x)</math><br><math>f_1 \cdot f_2 \cdot \dots f_n</math> (sometimes denoted <math>f_1f_2\dots f_n</math> is the function <math>x \mapsto f_1(x)f_2(x) \dots f_n(x)</math> \|\| Intersection of the domains of all the functions being multiplied \|\| Similar to the note for sums.<br> <toggledisplay>Note that this definition is quasi-paradoxical, in that the function defined as <math>f \cdot g</math> where <math>f(x) = x - 2</math> and <math>g(x) = 1/(x - 2)</math> is 1 everywhere ''except'' at 2. This is because even though the expressions for <math>f</math> and <math>g</math> cancel each other, evaluating at <math>x = 2</math> does not make sense for the expression ''prior'' to algebraic simplication by cancellation.</toggledisplay>		\| [[pointwise product of functions\|pointwise product]] \|\| <math>f \cdot g</math> (sometimes denoted <math>fg</math>) is the function <math>x \mapsto f(x)g(x)</math><br><math>f_1 \cdot f_2 \cdot \dots f_n</math> (sometimes denoted <math>f_1f_2\dots f_n</math> is the function <math>x \mapsto f_1(x)f_2(x) \dots f_n(x)</math> \|\| Intersection of the domains of all the functions being multiplied \|\| Similar to the note for sums.<br> <toggledisplay>Note that this definition is quasi-paradoxical, in that the function defined as <math>f \cdot g</math> where <math>f(x) = x - 2</math> and <math>g(x) = 1/(x - 2)</math> is 1 everywhere ''except'' at 2. This is because even though the expressions for <math>f</math> and <math>g</math> cancel each other, evaluating at <math>x = 2</math> does not make sense for the expression ''prior'' to algebraic simplication by cancellation.</toggledisplay>
	\|-		\|-
	\| [[pointwise quotient of functions\|pointwise quotient]] \|\| <math>f/g</math> is the function <math>x \mapsto f(x)/g(x)</math> \|\| Intersection of the domain of <math>f</math> with the ''subset'' of the domain of <math>g</math> comprising those points <math>x</math> where <math>g(x) \ne 0</math>.		\| [[pointwise quotient of functions\|pointwise quotient]] \|\| <math>f/g</math> is the function <math>x \mapsto f(x)/g(x)</math> \|\| Intersection of the domain of <math>f</math> with the ''subset'' of the domain of <math>g</math> comprising those points <math>x</math> where <math>g(x) \ne 0</math>. \|\|
	\|-		\|-
	\| [[composite of two functions]] \|\| <math>f \circ g</math> is the function <math>x \mapsto f(g(x))</math> \|\| Set of those values <math>x</math> for which <math>g(x)</math> lies inside the domain of <math>f</math>. \|\|		\| [[composite of two functions]] \|\| <math>f \circ g</math> is the function <math>x \mapsto f(g(x))</math> \|\| Set of those values <math>x</math> for which <math>g(x)</math> lies inside the domain of <math>f</math>. \|\|

Vipul: /* Computation of maximum possible domain from algebraic expression */

2011-09-19T22:03:42Z

Computation of maximum possible domain from algebraic expression

← Older revision		Revision as of 22:03, 19 September 2011
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	\| [[scalar multiple of function\|scalar multiple]] by a constant \|\| <math>af</math> is the function <math>x \mapsto af(x)</math> where <math>a</math> is a real number \|\| Same as the domain of the original function \|\|		\| [[scalar multiple of function\|scalar multiple]] by a constant \|\| <math>af</math> is the function <math>x \mapsto af(x)</math> where <math>a</math> is a real number \|\| Same as the domain of the original function \|\|
	\|-		\|-
	\| [[pointwise product of functions\|pointwise product]] \|\| <math>f \cdot g</math> (sometimes denoted <math>fg</math>) is the function <math>x \mapsto f(x)g(x)</math><br>f_1 \cdot f_2 \cdot \dots f_n</math> (sometimes denoted <math>f_1f_2\dots f_n</math> is the function <math>x \mapsto f_1(x)f_2(x) \dots f_n(x)</math> \|\| Intersection of the domains of all the functions being multiplied \|\| Similar to the note for sums.<br> <toggledisplay>Note that this definition is quasi-paradoxical, in that the function defined as <math>f \cdot g</math> where <math>f(x) = x - 2</math> and <math>g(x) = 1/(x - 2)</math> is 1 everywhere ''except'' at 2. This is because even though the expressions for <math>f</math> and <math>g</math> cancel each other, evaluating at <math>x = 2</math> does not make sense for the expression ''prior'' to algebraic simplication by cancellation.</toggledisplay>		\| [[pointwise product of functions\|pointwise product]] \|\| <math>f \cdot g</math> (sometimes denoted <math>fg</math>) is the function <math>x \mapsto f(x)g(x)</math><br><math>f_1 \cdot f_2 \cdot \dots f_n</math> (sometimes denoted <math>f_1f_2\dots f_n</math> is the function <math>x \mapsto f_1(x)f_2(x) \dots f_n(x)</math> \|\| Intersection of the domains of all the functions being multiplied \|\| Similar to the note for sums.<br> <toggledisplay>Note that this definition is quasi-paradoxical, in that the function defined as <math>f \cdot g</math> where <math>f(x) = x - 2</math> and <math>g(x) = 1/(x - 2)</math> is 1 everywhere ''except'' at 2. This is because even though the expressions for <math>f</math> and <math>g</math> cancel each other, evaluating at <math>x = 2</math> does not make sense for the expression ''prior'' to algebraic simplication by cancellation.</toggledisplay>
	\|-		\|-
	\| [[pointwise quotient of functions\|pointwise quotient]] \|\| <math>f/g</math> is the function <math>x \mapsto f(x)/g(x)</math> \|\| Intersection of the domain of <math>f</math> with the ''subset'' of the domain of <math>g</math> comprising those points <math>x</math> where <math>g(x) \ne 0</math>.		\| [[pointwise quotient of functions\|pointwise quotient]] \|\| <math>f/g</math> is the function <math>x \mapsto f(x)/g(x)</math> \|\| Intersection of the domain of <math>f</math> with the ''subset'' of the domain of <math>g</math> comprising those points <math>x</math> where <math>g(x) \ne 0</math>.

Vipul: /* Computation of maximum possible domain from algebraic expression */

2011-09-19T22:03:20Z

Computation of maximum possible domain from algebraic expression

← Older revision		Revision as of 22:03, 19 September 2011
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	\| [[scalar multiple of function\|scalar multiple]] by a constant \|\| <math>af</math> is the function <math>x \mapsto af(x)</math> where <math>a</math> is a real number \|\| Same as the domain of the original function \|\|		\| [[scalar multiple of function\|scalar multiple]] by a constant \|\| <math>af</math> is the function <math>x \mapsto af(x)</math> where <math>a</math> is a real number \|\| Same as the domain of the original function \|\|
	\|-		\|-
	\| [[pointwise product of functions\|pointwise product]] \|\| <math>f \cdot g</math> (sometimes denoted <math>fg</math>) is the function <math>x \mapsto f(x)g(x)</math><br>f_1 \cdot f_2 \cdot \dots f_n</math> (sometimes denoted <math>f_1f_2\dots f_n</math> is the function <math>x \mapsto \|\| Intersection of the domains of all the functions being multiplied \|\| Similar to the note for sums.<br> <toggledisplay>Note that this definition is quasi-paradoxical, in that the function defined as <math>f \cdot g</math> where <math>f(x) = x - 2</math> and <math>g(x) = 1/(x - 2)</math> is 1 everywhere ''except'' at 2. This is because even though the expressions for <math>f</math> and <math>g</math> cancel each other, evaluating at <math>x = 2</math> does not make sense for the expression ''prior'' to algebraic simplication by cancellation.</toggledisplay>		\| [[pointwise product of functions\|pointwise product]] \|\| <math>f \cdot g</math> (sometimes denoted <math>fg</math>) is the function <math>x \mapsto f(x)g(x)</math><br>f_1 \cdot f_2 \cdot \dots f_n</math> (sometimes denoted <math>f_1f_2\dots f_n</math> is the function <math>x \mapsto f_1(x)f_2(x) \dots f_n(x)</math> \|\| Intersection of the domains of all the functions being multiplied \|\| Similar to the note for sums.<br> <toggledisplay>Note that this definition is quasi-paradoxical, in that the function defined as <math>f \cdot g</math> where <math>f(x) = x - 2</math> and <math>g(x) = 1/(x - 2)</math> is 1 everywhere ''except'' at 2. This is because even though the expressions for <math>f</math> and <math>g</math> cancel each other, evaluating at <math>x = 2</math> does not make sense for the expression ''prior'' to algebraic simplication by cancellation.</toggledisplay>
			\|-
			\| [[pointwise quotient of functions\|pointwise quotient]] \|\| <math>f/g</math> is the function <math>x \mapsto f(x)/g(x)</math> \|\| Intersection of the domain of <math>f</math> with the ''subset'' of the domain of <math>g</math> comprising those points <math>x</math> where <math>g(x) \ne 0</math>.
			\|-
			\| [[composite of two functions]] \|\| <math>f \circ g</math> is the function <math>x \mapsto f(g(x))</math> \|\| Set of those values <math>x</math> for which <math>g(x)</math> lies inside the domain of <math>f</math>. \|\|
			\|-
			\| [[inverse function]] of a [[one-one function]] \|\| <math>f^{-1}</math> sends <matH>x</math> to the unique <math>y</math> such that <math>f(y) = x</math> \|\| Same as the [[range]] of the original one-one function <math>f</math> \|\|
			\|-
			\| [[piecewise definition of functions\|piecewise definition]] \|\| {{fillin}} \|\| Union of the domain of definition for each piece. This domain is usually given explicitly or requires finding the subset of an explicitly specified set where an explicit expression makes sense. \|\|
	\|}		\|}

Vipul at 21:55, 19 September 2011

2011-09-19T21:55:21Z

← Older revision		Revision as of 21:55, 19 September 2011
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	** <math>g</math> is not a [[one-one function]], but the restriction <math>f</math> is a one-one function.		** <math>g</math> is not a [[one-one function]], but the restriction <math>f</math> is a one-one function.
	** <math>g</math> attains its absolute minimum value, but the restriction <math>f</math> does not.		** <math>g</math> attains its absolute minimum value, but the restriction <math>f</math> does not.

			==Computation of domain==

			===Computation of maximum possible domain from algebraic expression===

			We discuss below how we can compute the domain of a function obtained by using a [[pointwise combination of functions\|pointwise combination]], [[composite of two functions\|composite]], [[inverse function]], and [[piecewise definition of functions\|piecewise definition]]:

			{\| class="sortable" border="1"
			! Method for constructing new functions from old !! In symbols !! Domain of the new function in terms of domains of the old functions !! Additional notes
			\|-
			\| [[pointwise sum of functions\|pointwise sum]] \|\| <math>f + g</math> is the function <math>x \mapsto f(x) + g(x)</math><br><math>f_1 + f_2 + \dots + f_n</math> is the function <math>x \mapsto f_1(x) + f_2(x) + \dots + f_n(x)</math> \|\| Intersection of the domains of all the functions being added \|\| <toggledisplay>Note that this definition is quasi-paradoxical, in that the function defined as <math>f + g</math> where <math>f(x) = 1/(x-1)</math> and <math>g(x) = -1/(x-1)</math> is the function <math>0</math> everywhere except at <math>x = 1</math>, where it is undefined. This is because even though the expressions for <math>f</math> and <math>g</math> cancel each other, evaluating at <math>x = 1</math> does not make sense for the expression ''prior'' to algebraic simplication by cancellation.</toggledisplay>
			\|-
			\| [[pointwise difference of functions\|pointwise difference]] \|\| <math>f - g</math> is the function <math>x \mapsto f(x) - g(x)</math>\|\| Intersection of the domains of the functions being subtracted \|\| Similar to the note for sums
			\|-
			\| [[scalar multiple of function\|scalar multiple]] by a constant \|\| <math>af</math> is the function <math>x \mapsto af(x)</math> where <math>a</math> is a real number \|\| Same as the domain of the original function \|\|
			\|-
			\| [[pointwise product of functions\|pointwise product]] \|\| <math>f \cdot g</math> (sometimes denoted <math>fg</math>) is the function <math>x \mapsto f(x)g(x)</math><br>f_1 \cdot f_2 \cdot \dots f_n</math> (sometimes denoted <math>f_1f_2\dots f_n</math> is the function <math>x \mapsto \|\| Intersection of the domains of all the functions being multiplied \|\| Similar to the note for sums.<br> <toggledisplay>Note that this definition is quasi-paradoxical, in that the function defined as <math>f \cdot g</math> where <math>f(x) = x - 2</math> and <math>g(x) = 1/(x - 2)</math> is 1 everywhere ''except'' at 2. This is because even though the expressions for <math>f</math> and <math>g</math> cancel each other, evaluating at <math>x = 2</math> does not make sense for the expression ''prior'' to algebraic simplication by cancellation.</toggledisplay>
			\|}

Vipul: /* Definition */

2011-09-19T21:41:34Z

Definition

← Older revision		Revision as of 21:41, 19 September 2011
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	===General definition===		===General definition===

	The '''domain''' of a function is the set of inputs allowed for the function, i.e., the set of values that can be fed into the function to give a valid output.		The '''domain''' of a [[defining ingredient::function]] is the set of inputs allowed for the function, i.e., the set of values that can be fed into the function to give a valid output.

	If <math>f:A \to B</math> is a function, the domain of <math>f</math> is the set <math>A</math>.		If <math>f:A \to B</math> is a function, the domain of <math>f</math> is the set <math>A</math>.

Vipul: Created page with "==Definition== ===General definition=== The '''domain''' of a function is the set of inputs allowed for the function, i.e., the set of values that can be fed into the function ..."

2011-09-19T21:41:12Z

Created page with "==Definition== ===General definition=== The '''domain''' of a function is the set of inputs allowed for the function, i.e., the set of values that can be fed into the function ..."

New page

==Definition==

===General definition===

The '''domain''' of a function is the set of inputs allowed for the function, i.e., the set of values that can be fed into the function to give a valid output.

If <math>f:A \to B</math> is a function, the domain of <math>f</math> is the set <math>A</math>.

===For a function described by an expression or procedure without explicit domain specification===

If a function of one variable (i.e., a function whose domain is a subset of the reals) is defined by means of an expression or procedure, but the domain is not ''explicitly specified'', the domain is taken to be the ''maximum possible'' domain, i.e., the largest subset of the reals on which that expression or procedure makes sense and yields a meaningful answer.

On the other hand, the domain of the function may be specified ''explicitly'' to be a proper subset of the maximum possible domain that we could infer from the expression alone.

==Caveats==

===Domain and restriction of domain===

The study of a function depends crucially on the [[domain]] on which the function is being studied. If a function of one variable is defined solely by means of an expression or procedure, the domain of the function is taken to be the largest possible subset of the reals on which that expression or procedure makes sense and gives a valid answer. However, we can also consider functions restricted to domains that are strictly smaller than the maximum possible domain on which the expression being used for the function makes sense. The behavior of the function, as well as answers to questions like whether it is increasing or decreasing and what its extreme values are, depends on what domain we are considering the function on.

Here are some examples:

* Consider the function <math>f</math> defined as follows: ''<math>f(d)</math> is defined as the area of a circle with diameter <math>d</math>''. Ignoring the boundary case of point circles and line circles, the only possible inputs for this function are positive reals, so <math>f</math> is a function from the positive reals to the positive reals given by the expression <math>f(d) := \pi d^2/4</math>. However, if we look ''only'' at the expression for <math>f</math>, then that expression makes sense for ''all'' real numbers, including zero and negative real numbers as well as positive real numbers. Call the latter function <math>g</math>, i.e., <math>g(d) := \pi d^2/4</math> for all <math>d \in \R</math>. Then, <math>f</math> is the restriction of <math>g</math> to the subdomain <math>(0,\infty)</math>. Note that:
** <math>g</math> is not an [[increasing function]], but the restriction <math>f</math> is an increasing function.
** <math>g</math> is not a [[one-one function]], but the restriction <math>f</math> is a one-one function.
** <math>g</math> attains its absolute minimum value, but the restriction <math>f</math> does not.