Chapter 4.7, Problem 30PPS

To determine

To find: The inverse of the function $f\left(x\right)=\frac{1}{4}x+1$ and match the function with its inverse graph.

Answer to Problem 30PPS

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

Explanation of Solution

Given:

The function is $f\left(x\right)=\frac{1}{4}x+1$ .

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.

  $y=\frac{1}{4}x+1$

Interchange $x$ and $y$ in the given expression $y=\frac{1}{4}x+1$ .

  $x=\frac{1}{4}y+1$

Solve for $y$ in expression $x=\frac{1}{4}y+1$ as follows:

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

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

  $f^{-1}\left(x\right)=4x-4$

Consider x = 0,

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

Thus, the coordinate is (0, -4).

Consider x = 1,

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

Thus, the coordinate is (1, 0).

The graph can be drawn as follows:

  

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