Chapter 4.7, Problem 38HP

To determine

To calculate: the value of a.

Answer to Problem 38HP

The value of a is 15.

Explanation of Solution

Given:

  $\begin{array}{l}f\left(x\right)=5x+a\\ f^{-1}\left(10\right)=-1\end{array}$

Calculation:

To find value of a, find inverse by replace f(x) by y and replace every x with y and vice versa.

  $\begin{array}{l}y=5x+a\\ x=5y+a\end{array}$

Solve for y.

  $\begin{array}{l}5y=x-a\\ y=\frac{x-a}{5}\end{array}$

Replace y with $f^{-1}\left(x\right)$ .

  $f^{-1}\left(x\right)=\frac{x-a}{5}$

Now given that $f^{-1}\left(10\right)=-1$ therefore substitute 10 for x in last step equation.

  $\begin{array}{l}{f}^{-1}\left(10\right)=\frac{10-a}{5}\\ -1=\frac{10-a}{5}\\ -5=10-a\\ a=15\end{array}$

Therefore the value of a is 15.