Chapter 2.2, Problem 1QR

To determine

To find: The function $f^{-1}$ for $f\left(x\right)=2x-3$ and also graph the function $f$ , $f^{-1}$ and $y=x$ in the same square viewing window.

Answer to Problem 1QR

The function $f^{-1}$ for $f\left(x\right)=2x-3$ is given as $f^{-1}\left(x\right)=\frac{x+3}{2}$ and the graph of function $f$ , $f^{-1}$ and $y=x$ are shown in Figure (1).

Explanation of Solution

Given information:

The function is $f\left(x\right)=2x-3$.

Calculation:

Equate the given function to $y$.

  $\begin{array}{c}y=2x-3\\ y+3=2x-3+3\\ y+3=2x\\ x=\frac{y+3}{2}\end{array}$

Interchange $x$ and $y$ in above equation.

  $\begin{array}{c}y=\frac{x+3}{2}\\ f^{-1}\left(x\right)=\frac{x+3}{2}\end{array}$

So, the inverse of the function $f\left(x\right)=2x-3$ is $f^{-1}\left(x\right)=\frac{x+3}{2}$.

To graph the functions $y=2x-3$ , $y=\frac{x+3}{2}$ and $y=x$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=2x-3$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{2}\boxed{\text{X}}\boxed{-}\boxed{3}$ . Now, enter right hand side of the equation $y=\frac{x+3}{2}$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{(}\boxed{\text{X}}\boxed{+}\boxed{3}\boxed{)}\boxed{÷}\boxed{2}$ . Enter right hand side of the equation $y=x$ after the symbol ${\text{Y}}_3$ . Enter the keystrokes $\boxed{\text{X}}$.

The display will show the equations,

  $\begin{array}{l}{\text{Y}}_{\text{1}}=2\text{X}-3\\ {\text{Y}}_2=\left(\text{X}+3\right)/2\\ {\text{Y}}_3=\text{X}\end{array}$

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

  

Figure (1)

Therefore, the function $f^{-1}$ for $f\left(x\right)=2x-3$ is given as $f^{-1}\left(x\right)=\frac{x+3}{2}$ and the graph of function $f$ , $f^{-1}$ and $y=x$ are shown in Figure (1).