Math Guide

Absolute value functions


Absolutevalue function definition:

An absolute value  function is a function that is positive for all values of x. It is  another way of representing a piecewise function. The parent absolute  value function would look like this:

Y = f(x), such that f(x)= |x|. The two parallel bars represent absolute  value. This functionis sometimes also written as f(x) = abs(x), which means absolutevalue of the terms inside the bracket ( ).

In piecewise form (also  called the split function form), the absolute value function can be  written like this:

f(x) = x, if x ≥0

= -x, if x <0

A table of values for some values of x for theabove piece wise function would be like this:

X

-4

-3

-2

-1

0

1

2

3

4

y

4

3

2

1

0

1

2

3

4


Graphing absolute value functions for the above table we get a graph that looks like this:

  


Note that the graph makes a right angle at the  origin (0,0). This absolute value parent function is always an even  function. The domain of this function is all real numbers denoted byR or in interval notation as (-∞, ∞+). The range of this functionis all non negative real numbers, denoted by U {0} or in interval         notation as [0, ∞+).


Absolute value functionsand graphs of absolute value functions are frequently used in calculus and algebra. Usually for logarithmic equations, we use  absolute values as logarithm of negative numbers do not exist.


Derivative of absolutevalue function:


The absolute value function defined over theset of all real numbers, (that means domain = all real numbers) is  continuous for all values of x. It is also differentiable over the  said domain (that is for all values of x) except at x = 0. The  absolute value function decreases monotonically for x values from -∞to 0, that is over the interval (-∞, 0]. The absolute value  function increases monotonically for x values in the domain portion  from x = 0 to x = ∞+, that is over the interval [0, ∞+). Everyreal number and its negative, would always have the same absolute  value, we can say that the absolute value function is an even  function. For obvious reasons, the absolute value function does nothave an inverse. The absolute value function for real numbers and forcomplex numbers is idempotent and convex.


Derivative is as follows:


|x| = = -1, for x <0

= undefined , for x = 0

= 1, for x > 0