Chapter 7.5, Problem 38E

To determine

To find: Whether the inverse of $f$ is a function and then find the inverse.

Answer to Problem 38E

The inverse of $f$ is a function and inverse is $g\left(x\right)=\frac{7}{x}-6$ .

Explanation of Solution

Given:

The function is $f\left(x\right)=\frac{7}{x+6}$ .

Calculation:

Plot the graph of the function $f\left(x\right)=\frac{7}{x+6}$ as follows.

  

It is noticed that no horizontal line intersects the graph more than once. So, the inverse of $f$ is a function.

Find the inverse of the function.

Set $y$ equal to $f\left(x\right)$ in the equation $f\left(x\right)=\frac{7}{x+6}$ .

  $y=\frac{7}{x+6}$

Interchange $x$ and $y$ in the equation $y=\frac{7}{x+6}$ and simplify for $y$ .

  $\begin{array}{c}x=\frac{7}{y+6}\\ y+6=\frac{7}{x}\\ y=\frac{7}{x}-6\end{array}$

Therefore, the inverse of $f$ is $g\left(x\right)=\frac{7}{x}-6$ .