Vipul: /* Key data */

2011-08-29T12:37:31Z

Key data

← Older revision		Revision as of 12:37, 29 August 2011
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	\| [[second derivative]] \|\| <math>x \mapsto 2/x^3</math>		\| [[second derivative]] \|\| <math>x \mapsto 2/x^3</math>
	\|-		\|-
	\| higher derivatives \|\| The <math>n^{th}</math> derivative is <math>x \mapsto \frac{(-1)^nn!}x^{n+1}}</math>		\| higher derivatives \|\| The <math>n^{th}</math> derivative is <math>x \mapsto \frac{(-1)^nn!}{x^{n+1}}</math>
	\|-		\|-
	\| [[first antiderivative]] \|\| <math>x \mapsto \ln\|x\| + C</math> where <math>\ln</math> is the [[natural logarithm]]. Note that the value <math>C</math> is constant in <math>(0,\infty)</math> and also constant in <math>(-\infty,0)</math>, but the constant values could differ ''between'' these two intervals.		\| [[first antiderivative]] \|\| <math>x \mapsto \ln\|x\| + C</math> where <math>\ln</math> is the [[natural logarithm]]. Note that the value <math>C</math> is constant in <math>(0,\infty)</math> and also constant in <math>(-\infty,0)</math>, but the constant values could differ ''between'' these two intervals.
	\|}		\|}

Vipul: Created page with "{{particular function}} ==Definition== The '''reciprocal function''' is a function defined on the set of nonzero reals, that sends every real number to its reciprocal, i.e...."

2011-08-29T12:30:30Z

Created page with "{{particular function}} ==Definition== The '''reciprocal function''' is a function defined on the set of nonzero reals, that sends every real number to its reciprocal, i.e...."

New page

{{particular function}}

==Definition==

The '''reciprocal function''' is a [[function]] defined on the set of nonzero reals, that sends every real number to its reciprocal, i.e., its multiplicative inverse. Explicitly, it is the function:

<math>x \mapsto \frac{1}{x}</math>

==Key data==

{| class="sortable" border="1"
! Item !! Value
|-
| default [[domain]] || all nonzero real numbers, i.e., <math>\R \setminus \{ 0 \}</math>, which can also be written as <math>(-\infty,0) \cup (0,\infty)</math>.
|-
| [[range]] || all nonzero real numbers, i.e., <math>\R \setminus \{ 0 \}</math>, which can also be written as <math>(-\infty,0) \cup (0,\infty)</math>.
|-
| [[inverse function]] || the reciprocal function itself. In other words, this function ''equals its own inverse''. Another way of putting this is that the reciprocal of the reciprocal of a number is the original number.
|-
| [[local minimum value]]s and points of attainment || no local minimum values
|-
| [[local maximum value]]s and points of attainment || no local maximum values
|-
| [[horizontal asymptote]]s || The <math>x</math>-axis (<math>y = 0</math>) is a horizontal asymptote both as <math>x \to -\infty</math> (with approach from below) and as <math>x \to \infty</math> (with approach from above).
|-
| [[vertical asymptote]]s || The <math>y</math>-axis (<math>x = 0</math>) is a vertical asymptote from both left (i.e., as <math>x \to 0^-</math>, <math>y \to -\infty</math>) and right (i.e., as <math>x \to 0^+</math>, <math>y \to \infty</math>).
|-
| [[first derivative]] || <math>x \mapsto -1/x^2</math>
|-
| [[second derivative]] || <math>x \mapsto 2/x^3</math>
|-
| higher derivatives || The <math>n^{th}</math> derivative is <math>x \mapsto \frac{(-1)^nn!}x^{n+1}}</math>
|-
| [[first antiderivative]] || <math>x \mapsto \ln|x| + C</math> where <math>\ln</math> is the [[natural logarithm]]. Note that the value <math>C</math> is constant in <math>(0,\infty)</math> and also constant in <math>(-\infty,0)</math>, but the constant values could differ ''between'' these two intervals.
|}

Reciprocal function - Revision history

Vipul: /* Key data */

Vipul: Created page with "{{particular function}} ==Definition== The '''reciprocal function''' is a function defined on the set of nonzero reals, that sends every real number to its reciprocal, i.e...."