Chapter 4.7, Problem 48PFA

To determine

To find the value of $f^{-1}\left(-15\right)$ .

Answer to Problem 48PFA

  $f^{-1}\left(-15\right)=-18$

Explanation of Solution

Given:

  $f\left(x\right)=\frac{2}{3}x-3$

Calculation:

Let us consider,

  $y=\frac{2}{3}x-3$

To find inverse function, replace $y$ by $x$ and then solve for $y$ in terms of $x$ , that is

  $\begin{array}{l}x=\frac{2}{3}y-3\\ \Rightarrow \frac{2}{3}y=x+3\\ \Rightarrow y=\frac{3}{2}\left(x+3\right)\\ \therefore f^{-1}\left(x\right)=\frac{3}{2}\left(x+3\right)\end{array}$

Now put $x=-15$ in $f^{-1}\left(x\right)=\frac{3}{2}\left(x+3\right)$ , we get

  $\begin{array}{l}{f}^{-1}\left(-15\right)=\frac{3}{2}\left(-15+3\right)\\ =\frac{3}{2}\left(-12\right)\\ =-18\end{array}$