# How do you graph the function f(x)=abs(2x) and its inverse?

Sep 24, 2017

See below.

#### Explanation:

Graph $f \left(x\right) = | 2 x |$ piecewise:

Notice that $f \left(x\right) = | 2 x |$ is equivalent to $f \left(x\right) = 2 x$ for $x \ge 0$
and $f \left(x\right) = - 2 x$ for $x \le 0$ ,because the absolute value is always positive.

The graph will look like this:

graph{|2x| [-22.8, 22.8, -11.4, 11.4]}

The function $f \left(x\right) = | 2 x |$ maps two elements in the domain to one element in the range:

example: $- 2$ and $2$ both map to $4$.

The inverse would then need to map $4$ onto both $- 2$ and $2$ . This is a one many relationship, and is therefore NOT A FUNCTION. However inverse functions can be found by restricting the domain of the function:

Example: $f \left(x\right) = | 2 x |$ for domain $\left\{x \in {\mathbb{R}}^{+}\right\}$

would have ${f}^{-} 1 \left(x\right) = \left(\frac{x}{2}\right)$

And: $f \left(x\right) = | 2 x |$ for domain $\left\{x \in {\mathbb{R}}^{-}\right\}$

would have ${f}^{-} 1 \left(x\right) = - \left(\frac{x}{2}\right)$

graph:

Inverse for domain ${\mathbb{R}}^{+}$

Inverse for domain ${\mathbb{R}}^{-}$