Chapter 3, Problem 34SGA

To determine

To find: $f^{-1}\left(x\right)$ of $f\left(x\right)=3{\left(x+7\right)}^4$ . Then state whether $f^{-1}\left(x\right)$ is a function.

Answer to Problem 34SGA

  $\text{It is NOT a function}$

Explanation of Solution

Given information: $f\left(x\right)=3{\left(x+7\right)}^4$

Calculation:

  $y=3{\left(x+7\right)}^4$

To find the inverse, switch x with y and solve for y.

  $\begin{array}{l}x=3{\left(y\text{'}+7\right)}^4\\ {\left(y\text{'}+7\right)}^4=\frac{x}{3}\\ y\text{'}+7=\pm \sqrt[4]{\frac{x}{3}}\\ y\text{'}=-7\pm \sqrt[4]{\frac{x}{3}}\end{array}$

  

When you take the “even root”, you must remember it’s $\pm ,$ just like when you take the square root. You can see from the inverse that it is NOT a function.