Chapter 3.4, Problem 21E

Solution

The inverse of the function whose graph is given.

  

Given information:

The graph of a function,

  

Concept used:

• The graph of the function $y=f\left(x\right)$ and its inverse $f^{-1}\left(x\right)$ are symmetric to the line $y=x$ .
• For any point $\left(a,b\right)$ on the graph of $f\left(x\right)$ , the corresponding point on the graph of inverse function $f^{-1}\left(x\right)$ is $\left(b,a\right)$ .
• If a function is one-to-one, then it has an inverse.

Calculation:

Note that the graph of the given function passes through the points $\left(0,-4\right)\text{ and }\left(6,0\right)$ . So, the equation of the line represented by the given graph is

  $\begin{array}{c}y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)\\ y-\left(-4\right)=\frac{0-\left(-4\right)}{6-0}\left(x-0\right)\\ y+4=\frac{4}{6}\left(x\right)\\ y=\frac{2}{3}x-4\end{array}$

Now, to find the inverse of this function, first interchange the x and y and then solve for y as shown below

  $\begin{array}{c}y=\frac{2}{3}x-4\\ x=\frac{2}{3}y-4\left(\text{interchang }x\text{ and }y\right)\\ x+4=\frac{2}{3}y\left(\text{Solve for }y\right)\\ \frac{3}{2}\cdot x+\frac{3}{2}\cdot 4=y\\ \frac{3}{2}x+6=y\\ \Rightarrow f^{-1}\left(x\right)=\frac{3}{2}x+6\end{array}$

Thus, inverse of the given function is $g\left(x\right)=\frac{3}{2}x+6$ .