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]
