Chapter 6 Elementary Functions

Section 6.2 Linear Functions and Polynomials

6.2.5 Absolute Value Functions

In Module 2 the absolute value of a real number $x$ was introduced in the following way:

${\displaystyle |x|=\{\begin{array}{cc}\hfill x& \text{for}\mathrm{\hspace{0.5em}\hspace{0.5em}}x\ge 0\hfill \\ \hfill -x& \text{for}\mathrm{\hspace{0.5em}\hspace{0.5em}}x<0\hspace{0.5em}.\hfill \end{array}}$
In the context of this module, the absolute value can be regarded as a function. This results in the absolute value function:

${\displaystyle b:\mathrm{\hspace{0.5em}\hspace{0.5em}}\{\begin{array}{ccc}\hfill ℝ& \hfill \to \hfill & ℝ\hfill \\ \hfill x& \hfill ⟼\hfill & |x|\hspace{0.5em}.\hfill \end{array}}$


Due to the definition by cases

${\displaystyle b(x)=|x|=\{\begin{array}{cc}\hfill x& \text{for}\mathrm{\hspace{0.5em}\hspace{0.5em}}x\ge 0\hfill \\ \hfill -x& \text{for}\mathrm{\hspace{0.5em}\hspace{0.5em}}x<0\hspace{0.5em},\hfill \end{array}}$
the absolute value function is an example of a piecewise defined function. If absolute values are defined according to different cases, then it is also said that the absolute value is resolved. Then, the graph of the absolute value function $b$ looks as follows:

One property of the graph of the absolute value function, which most of the more general functions involving absolute values have in common, is the kink at $x=0$. The absolute value function $b$ defined above is only the simplest case of a function involving an absolute value. More complicated examples of functions can be constructed, involving one or several absolute values, e.g.

${\displaystyle f:\mathrm{\hspace{0.5em}\hspace{0.5em}}\{\begin{array}{ccc}\hfill ℝ& \hfill \to \hfill & ℝ\hfill \\ \hfill x& \hfill ⟼\hfill & |2x-1|\hspace{0.5em}.\hfill \end{array}}$

For such functions, it is a relevant task to get an idea of how the graph of the function looks. To do this, we use the piecewise definition of the absolute value, and the approach is similar to the one for the solution of absolute value equations and inequalities. Here, we demonstrate this approach for the example of the function $f$ defined above: