To express the absolute value function [tex]\(f(x) = 20|x|\)[/tex] as a piecewise-defined function, we need to carefully consider the definition of the absolute value function. The absolute value [tex]\( |x| \)[/tex] can be expressed piecewise as follows:
[tex]\[
|x| = \begin{cases}
x, & \text{if } x \geq 0 \\
-x, & \text{if } x < 0
\end{cases}
\][/tex]
Given this, we can rewrite [tex]\( f(x) = 20|x| \)[/tex] by substituting the piecewise definition of [tex]\( |x| \)[/tex].
When [tex]\( x \geq 0 \)[/tex]:
[tex]\[
|x| = x
\][/tex]
Thus,
[tex]\[
f(x) = 20|x| = 20x
\][/tex]
When [tex]\( x < 0 \)[/tex]:
[tex]\[
|x| = -x
\][/tex]
Thus,
[tex]\[
f(x) = 20|x| = 20(-x) = -20x
\][/tex]
Combining these two pieces, we can express [tex]\( f(x) = 20|x| \)[/tex] as follows:
[tex]\[
f(x) = \begin{cases}
20x, & \text{if } x \geq 0 \\
-20x, & \text{if } x < 0
\end{cases}
\][/tex]
So, the piecewise-defined function is:
[tex]\[
f(x) = \left\{
\begin{array}{ll}
20x, & x \geq 0 \\
-20x, & x<0
\end{array}
\right.
\][/tex]
