Chapter 7.2, Problem 19PPS

To determine

The inverse of the function $y=-\frac{5}{3}x-8$ and graph the function and its inverse.

Answer to Problem 19PPS

The inverse of function $y=-\frac{5}{3}x-8$ where $y=f\left(x\right)$ is $f^{-1}\left(x\right)=-\frac{3}{5}\left(x+8\right)$ .

Explanation of Solution

Given information: The given function is,

  $y=-\frac{5}{3}x-\mathrm{8......}\left(\text{where }y=f\left(x\right)\right)$

Concept used: In order to find the inverse of a given function $f\left(x\right)$ write $f\left(x\right)=y$ and exchange the variables $\left(x\right)\text{ and }\left(y\right)$ and solve for the variable $\left(y\right)$ . At last write $y=f^{-1}\left(x\right)$ which is the required inverse of the function $f\left(x\right)$ .

Calculation: For the given function $y=-\frac{5}{3}x-8$ , the inverse is evaluated as,

  $\begin{array}{c}y=-\frac{5}{3}x-8\\ \text{or, }\left(x\right)=-\frac{5}{3}\left(y\right)-\mathrm{8......}\left(\text{Exchanging variables}\right)\\ \text{or, }y=-\frac{3}{5}\left(x+8\right)\mathrm{......}\left(\text{Solving for variable }\left(y\right)\right)\\ f^{-1}\left(x\right)=-\frac{3}{5}\left(x+8\right)\mathrm{......}\left(\text{Plugging }y=f^{-1}\left(x\right)\right)\end{array}$

Hence, the inverse of function $y=-\frac{5}{3}x-8$ is $f^{-1}\left(x\right)=-\frac{3}{5}\left(x+8\right)$ .

Graph: The graph for the function $y=-\frac{5}{3}x-8$ and its inverse function $f^{-1}\left(x\right)=-\frac{3}{5}\left(x+8\right)$ is shown as,