Chapter 4.7, Problem 39HP

To determine

To calculate: the value of a and b.

Answer to Problem 39HP

The value of a is 2 and b is 14.

Explanation of Solution

Given: $f\left(x\right)=\frac{1}{a}x+7$ , $f^{-1}\left(x\right)=2x-b$

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

  $\begin{array}{l}y=\frac{1}{a}x+7\\ x=\frac{1}{a}y+7\end{array}$

Solve for y.

  $\begin{array}{l}x-7=\frac{1}{a}y\\ y=a\left(x-7\right)\end{array}$

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

  $f^{-1}\left(x\right)=a\left(x-7\right)$

In the question given that $f^{-1}\left(x\right)=2x-b$

Equate both values.

  $\begin{array}{l}a\left(x-7\right)=2x-b\\ ax-7a=2x-b\end{array}$

Compare both sides.

  $a=2,$

And,

  $\begin{array}{l}b=7a\\ =7\cdot 2\\ =14\end{array}$

Therefore a = 2 and b = 14.