Chapter 1.9, Problem 73E

To determine

To find: inverse of the function f and then to state the domains of the function f and its

inverse.

Answer to Problem 73E

The inverse of the given function is $f^{-1}(x)=\sqrt{x}-6$ on domain $x\ge -6$ .

Explanation of Solution

Given information:

The function $f(x)={(x+6)}^2$

Concept used:

The function has inverse function if and only if the function is one to one.

Calculation:

The function $f(x)={(x+6)}^2$ is not one to one function.

To make one to one the restrict the domain to $x\ge -6$

  $f(x)={(x+6)}^2$

Let $y={(x+6)}^2$

  $\begin{array}{l}\sqrt{y}=x+6\\ x=\sqrt{y}-6\\ f^{-1}(y)=\sqrt{y}-6\\ f^{-1}(x)=\sqrt{x}-6\end{array}$

Hence, the inverse of the given function is $f^{-1}(x)=\sqrt{x}-6$ on domain $x\ge -6$ .