Chapter 11.5, Problem 29E

To determine

To calculate:

Inverse of the given function.

Answer to Problem 29E

Inverse of the function $y=2^{x-5}$ is $f^{-1}\left(x\right)={\log }_2\left(x+5\right)$ .

Explanation of Solution

Given information:

  $y=2^{x-5}$

Calculation:

It is given that,

  $y=2^{x-5}$

Now interchange $x$ and $y$ ,

  $x=2^{y-5}$

Add $5$ to both sides,

  $x+5=2^y$

Take the log to the both sides,

  ${\log }_2\left(x+5\right)={\log }_2{2}^y$

Now simplify,

  $\begin{array}{l}{\log }_2\left(x+5\right)={\log }_2{2}^y\\ {\log }_2\left(x+5\right)=y\\ y={\log }_2\left(x+5\right)\end{array}$

  $f^{-1}\left(x\right)={\log }_2\left(x+5\right)$

Hence, Inverse of the given function is $f^{-1}\left(x\right)={\log }_2\left(x+5\right)$ .