Express the absolute value function [tex]$f(x) = 22|x|$[/tex] as a piecewise-defined function.

[tex]
f(x) = \left\{
\begin{array}{ll}
22x, & x \geq 0 \\
-22x, & x \ \textless \ 0
\end{array}
\right.
[/tex]



Answer :

To express the absolute value function [tex]\( f(x) = 22|x| \)[/tex] as a piecewise-defined function, let's analyze the scenarios based on the value of [tex]\( x \)[/tex].

The absolute value function, denoted as [tex]\( |x| \)[/tex], is defined piecewise as follows:
- [tex]\( |x| = x \)[/tex] if [tex]\( x \geq 0 \)[/tex]
- [tex]\( |x| = -x \)[/tex] if [tex]\( x < 0 \)[/tex]

Given [tex]\( f(x) = 22|x| \)[/tex], we can substitute the definitions of [tex]\( |x| \)[/tex] into the function:

1. For [tex]\( x \geq 0 \)[/tex]:
[tex]\[ f(x) = 22|x| = 22x \][/tex]

2. For [tex]\( x < 0 \)[/tex]:
[tex]\[ f(x) = 22|x| = 22(-x) \][/tex]

Hence, the piecewise-defined function for [tex]\( f(x) = 22|x| \)[/tex] can be expressed as:
[tex]\[ f(x) = \left\{ \begin{array}{ll} 22x, & x \geq 0 \\ 22(-x), & x < 0 \end{array} \right. \][/tex]

Let's summarize the piecewise function:

[tex]\[ f(x) = \left\{ \begin{array}{ll} 22x, & x \geq 0 \\ 22(-x), & x < 0 \end{array} \right. \][/tex]

Therefore, the piecewise-defined function for [tex]\( f(x) = 22|x| \)[/tex] is:

[tex]\[ f(x) = \left\{ \begin{array}{ll} 22x, & x \geq 0 \\ 22(-x), & x < 0 \end{array} \right. \][/tex]