Express the absolute value function [tex]$f(x)=20|x|$[/tex] as a piecewise-defined function.

[tex]
f(x)=\left\{
\begin{array}{ll}
20x, & x \geq 0 \\
-20x, & x \ \textless \ 0
\end{array}
\right.
[/tex]



Answer :

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]