Chapter 1.9, Problem 61E

To determine

Whether the given function $f(x)={\left(x+3\right)}^2,x\ge -3$ has an inverse function and if it does, find the inverse function.

Answer to Problem 61E

The inverse function of given function is $f^{-1}(x)=\sqrt{x}-3$

Explanation of Solution

Given information :

Given function $f(x)={\left(x+3\right)}^2,x\ge -3$

A function $f(x)$ has an inverse function, if it is both one-one and onto function.

Since $f(x)={\left(x+3\right)}^2,x\ge -3$ is one-one as each value of dependent variable corresponds to exactly one value of the independent variable.

Thus, the function $f(x)={\left(x+3\right)}^2,x\ge -3$ has an inverse function.

To find inverse function:

Step 1: Replace $y=f(x)$ in $f(x)={\left(x+3\right)}^2$ , we get

  $y={\left(x+3\right)}^2$

Step 2: interchange role of $x\text{and}y$ . And solve for $y$ .

  $\begin{array}{l}x={\left(y+3\right)}^2\\ \Rightarrow \sqrt{x}=y+3\\ \Rightarrow y=\sqrt{x}-3\end{array}$

Step 3: replace $y$ with $f^{-1}(x)$ in new equation, we get

  $f^{-1}(x)=\sqrt{x}-3$

Therefore, the inverse function of given function is $f^{-1}(x)=\sqrt{x}-3$