Chapter 4.7, Problem 29PPS

To determine

To find: The inverse of the function $f\left(x\right)=4x+4$ and match the function with its inverse graph.

Answer to Problem 29PPS

The inverse of the function $f\left(x\right)=4x+4$ is $f^{-1}\left(x\right)=\frac{1}{4}x-1$ .and it matches with graph B.

Explanation of Solution

Given:

The function is $f\left(x\right)=4x+4$ .

Concept used:

To obtain the inverse of any function $f\left(x\right)=ax+b$ , following steps are used:

• Replace $f\left(x\right)$ with $y$ .
• Interchange $x$ and $y$ .
• Solve for $y$ .
• Again, replace $y$ with $f\left(x\right)$ .

Calculation:

Replace $f\left(x\right)$ by $y$ in the given equation.

$y=4x+4$

Interchange $x$ and $y$ in the given expression $y=4x+4$ .

$x=4y+4$

Solve for $y$ in expression $x=4y+4$ as follows:

$\begin{array}{c}x=4y+4\\ x-4=4y\\ y=\frac{1}{4}x-1\end{array}$

Replace $y$ with $f^{-1}\left(x\right)$ in above equation to obtain the inverse of the function.

$f^{-1}\left(x\right)=\frac{1}{4}x-1$

Consider x = 0,

$\begin{array}{l}{f}^{-1}\left(x\right)=\frac{1}{4}\left(0\right)-1\\ f^{-1}\left(x\right)=-1\end{array}$

Thus, the coordinate is (0, -1).

Consider x = 4,

$\begin{array}{l}{f}^{-1}\left(x\right)=\frac{1}{4}\left(4\right)-1\\ f^{-1}\left(x\right)=0\end{array}$

Thus, the coordinate is (4, 0).

The graph can be drawn as follows:

Thus, the inverse of the function $f\left(x\right)=4x+4$ is $f^{-1}\left(x\right)=\frac{1}{4}x-1$ .and it matches with graph B.