Chapter 1, Problem 54RE

(a)

To determine

To find:The inverse of the given function and also verify that $\left(f\circ f^{-1}\right)\left(x\right)=\left(f^{-1}\circ f\right)\left(x\right)=x$ .

Answer to Problem 54RE

Theinverse of the given function is $f^{-1}\left(x\right)=\sqrt{x}-2$ and it is verified that $\left(f\circ f^{-1}\right)\left(x\right)=\left(f^{-1}\circ f\right)\left(x\right)=x$ .

Explanation of Solution

Given information:The function is $f\left(x\right)={\left(x+2\right)}^2$ for $r\ge -2$ .

Calculation:

Substitute $y$ for $f\left(x\right)$ in the given function and solve the equation for $x$ in terms of $y$ .

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

Interchange $x$ and $y$ , and substitute $f^{-1}\left(x\right)$ for $y$ .

  $\begin{array}{c}y=\sqrt{x}-2\\ f^{-1}\left(x\right)=\sqrt{x}-2\end{array}$

Verify:

The composite function $\left(f\circ f^{-1}\right)\left(x\right)$ can be written as:

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

Substitute $\sqrt{x}-2$ for $f^{-1}\left(x\right)$ to find $\left(f\circ f^{-1}\right)\left(x\right)$ .

  $\begin{array}{c}\left(f\circ f^{-1}\right)\left(x\right)=f\left(\sqrt{x}-2\right)\\ ={\left[\left(\sqrt{x}-2\right)+2\right]}^2\\ ={\left(\sqrt{x}\right)}^2\\ =x\end{array}$

The composite function $\left(f^{-1}\circ f\right)\left(x\right)$ can be written as:

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

Substitute ${\left(x+2\right)}^2$ for $f\left(x\right)$ to find $\left(f^{-1}\circ f\right)\left(x\right)$ .

  $\begin{array}{c}\left(f^{-1}\circ f\right)\left(x\right)=f^{-1}\left({\left(x+2\right)}^2\right)\\ =\sqrt{{\left(x+2\right)}^2}-2\\ =x+2-2\\ =x\end{array}$

It is verified that $\left(f\circ f^{-1}\right)\left(x\right)=\left(f^{-1}\circ f\right)\left(x\right)=x$ .

Therefore, the inverse of the given function is $f^{-1}\left(x\right)=\sqrt{x}-2$ and it is verified that $\left(f\circ f^{-1}\right)\left(x\right)=\left(f^{-1}\circ f\right)\left(x\right)=x$ .

(b)

To determine

To graph:The function $f\left(x\right)={\left(x+2\right)}^2$ and its inverse in the same viewing window.

Explanation of Solution

Given information:The function is $f\left(x\right)={\left(x+2\right)}^2$ for $r\ge -2$ .

Graph:

From part (a), the inverse of the given function is $f^{-1}\left(x\right)=\sqrt{x}-2$ .

To plot the graph of $f$ and $f^{-1}$ in the same viewing window, use the following steps of graphing calculator.

First press “ON” button on graphical calculator and press $\boxed{\text{Y}=}$ key. Write the right hand side of the function $f\left(x\right)={\left(x+2\right)}^2$ after ${\text{Y}}_{\text{1}}$ . Enter the keystrokes $\boxed{\text{(}}\boxed{\text{X}}\boxed{+}\boxed{\text{2}}\boxed{)}\boxed{\wedge }\boxed{\text{2}}$ .

Press $\boxed{\text{ENTER}}$ and write the right hand side of the function $f^{-1}\left(x\right)=\sqrt{x}-2$ after ${\text{Y}}_{\text{2}}$ . Enter the keystrokes $\boxed{2\text{nd}}\boxed{x^2}\boxed{\text{X}}\boxed{)}\boxed{-}\boxed{2}$ . Adjust the window at $\left[-2,6\right]$ by $\left[-4,4\right]$ .

Now, press the $\boxed{\text{GRAPH}}$ key to produce the graph of given function in the same viewing window as shown below.

  

Figure (1)

Interpretation: From the above graph it can be observed that the graph of functions $f$ and $f^{-1}$ is symmetric about the line $y=x$ .