To express the absolute value function [tex]\( f(x) = 19|x| \)[/tex] as a piecewise-defined function, we need to consider the definition of the absolute value function [tex]\( |x| \)[/tex]. The absolute value function can be defined piecewise as follows:
- When [tex]\( x \)[/tex] is greater than or equal to 0, [tex]\( |x| = x \)[/tex].
- When [tex]\( x \)[/tex] is less than 0, [tex]\( |x| = -x \)[/tex].
Using this piecewise definition of [tex]\( |x| \)[/tex], we can rewrite the function [tex]\( f(x) = 19|x| \)[/tex] accordingly.
So:
- For [tex]\( x \geq 0 \)[/tex], [tex]\( f(x) = 19|x| = 19x \)[/tex].
- For [tex]\( x < 0 \)[/tex], [tex]\( f(x) = 19|x| = 19(-x) = -19x \)[/tex].
Therefore, the piecewise-defined function for [tex]\( f(x) \)[/tex] is:
[tex]\[
f(x) = \left\{\begin{array}{ll}
19x, & x \geq 0 \\
19(-x), & x < 0
\end{array}\right.
\][/tex]
