Difference between revisions of "Absolute value function"

From Calculus
Jump to: navigation, search
(Created page with "{{particular function}} ==Definition== The '''absolute value function''', denoted as <math>\operatorname{abs}</math> or <math>| \cdot |</math>, is defined as the function that ...")
 
(Definition)
 
Line 5: Line 5:
 
The '''absolute value function''', denoted as <math>\operatorname{abs}</math> or <math>| \cdot |</math>, is defined as the function that sends a real number to itself if it is nonnegative and to its negative if it is negative.
 
The '''absolute value function''', denoted as <math>\operatorname{abs}</math> or <math>| \cdot |</math>, is defined as the function that sends a real number to itself if it is nonnegative and to its negative if it is negative.
  
The absolute value of <math>x</math> is denoted <math>\operatorname{abs}(x)</math> or <math>|x|</math>. It is defined as:
+
The absolute value of <math>x</math> is denoted <math>\operatorname{abs}(x)</math> or <math>|x|</math>. It can be defined in many equivalent ways.
 +
 
 +
===As a piecewise linear function===
 +
 
 +
The function is a [[piecewise linear function]] with the following [[piecewise definition of function|piecewise definition]]:
  
 
<math>|x| := \left\lbrace \begin{array}{rl}x, & x \ge 0 \\ -x, & x < 0 \\\end{array} \right.</math>
 
<math>|x| := \left\lbrace \begin{array}{rl}x, & x \ge 0 \\ -x, & x < 0 \\\end{array} \right.</math>
 +
 +
===As a max of two functions===
 +
 +
The absolute value function can be defined as:
 +
 +
<math>|x| := \max \{ x, -x \}</math>
 +
 +
In other words, it is obtained by applying the [[pointwise maximum of functions|pointwise maximum]] to the following two functions: the identity function and the negative of the identity function.
  
 
==Graph==
 
==Graph==
  
 
[[File:Absolutevalue.png|500px]]
 
[[File:Absolutevalue.png|500px]]

Latest revision as of 22:53, 21 September 2011

This article is about a particular function from a subset of the real numbers to the real numbers. Information about the function, including its domain, range, and key data relating to graphing, differentiation, and integration, is presented in the article.
View a complete list of particular functions on this wiki

Definition

The absolute value function, denoted as \operatorname{abs} or | \cdot |, is defined as the function that sends a real number to itself if it is nonnegative and to its negative if it is negative.

The absolute value of x is denoted \operatorname{abs}(x) or |x|. It can be defined in many equivalent ways.

As a piecewise linear function

The function is a piecewise linear function with the following piecewise definition:

|x| := \left\lbrace \begin{array}{rl}x, & x \ge 0 \\ -x, & x < 0 \\\end{array} \right.

As a max of two functions

The absolute value function can be defined as:

|x| := \max \{ x, -x \}

In other words, it is obtained by applying the pointwise maximum to the following two functions: the identity function and the negative of the identity function.

Graph

Absolutevalue.png