Chapter 11.6, Problem 22E

To determine

To explain:

The inverse of the following function is function or not.

  $f\left(x\right)=\frac{2}{x}-4$

Answer to Problem 22E

This graph passes the horizontal line test, thus the function has inverse that is the function.

Explanation of Solution

Given information:

  $f\left(x\right)=\frac{2}{x}-4$

Replace the $f\left(x\right)$ by $y$ :

  $y=\frac{2}{x}-4$

Switch $x$ and $y$ :

  $x=\frac{2}{y}-4$

Determine $y$ as a function of $x$ :

  $\begin{array}{l}x+4=\frac{2}{y}\\ \left(x+4\right)y=2\\ y=\frac{2}{x+4}\end{array}$

Replace $y$ by $f^{-1}\left(x\right)$ :

  $f^{-1}\left(x\right)=\frac{2}{x+4}$

Graph the function and then perform the horizontal line test:

  

Graph: $f\left(x\right)=\frac{2}{x}-4$

Therefore, this graph passes the horizontal line test, thus the function has inverse that is the function.