Chapter 1.5, Problem 13E

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 13E

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

Calculation:

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

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

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

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

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 $\frac{x-3}{2}$ for $f^{-1}\left(x\right)$ to find $\left(f\circ f^{-1}\right)\left(x\right)$ .

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

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 $2x+3$ for $f\left(x\right)$ to find $\left(f^{-1}\circ f\right)\left(x\right)$ .

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

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 $f^{-1}\left(x\right)=\frac{x-3}{2}$ and it is verified that $\left(f\circ f^{-1}\right)\left(x\right)=\left(f^{-1}\circ f\right)\left(x\right)=x$ .