**Absolute value functions**

**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