Question

Express the absolute value function f(x)=9|x| as a​ piecewise-defined function.

Answer 1

`f\left(x\right)=\begin{cases}-9x&x\: < \:0\\ \:\:\:9x&x > \:0\end{cases}`

Explanation:

The first thing to note is that absolute value functions are piecewise functions.

A piecewise function is a function in which more than one formula is used to define the output over different pieces of the domain or inputs(in this case the x-values).

In this case, the absolute value function is

`f(x) = 9|x|`

To write this function as a piecewise function you have to use the definition of
`f(x) = |x|` :

`f\left(x\right)=\begin{cases}-x&x\: < \:0\\ \:\:\:x&x > \:0\end{cases}`

Applying this to
`9|x|` we multiply each of the parts by 9 to get

`f\left(x\right)=\begin{cases}-9x&x\: < \:0\\ \:\:\:9x&x > \:0\end{cases}`

At
`|x| = 0, f(x) = 9|x| = 0\\`

The graph of the function is attached for easier understanding