Chapter 3.3, Problem 3DC

Solution

How to test to see if the function obtained in exercise 1 is indeed the inverse of the original function.

The points $\left(a,b\right)$ on the graph of the function has corresponding points of the form $\left(b,a\right)$ on the graph of the inverse function.

Given information:

The functions $f\left(x\right)=3x+2,f\left(x\right)=\frac{x-1}{6},f\left(x\right)=4-\frac{3}{2}x$ .

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:

Recall that a function f and its inverse $f^{-1}$ are symmetric about the line $y=x$ . So, for any point $\left(a,b\right)$ on the graph of $f\left(x\right)$ , the corresponding point on the graph of inverse function $f^{-1}\left(x\right)$ must be $\left(b,a\right)$ .

Note that all inverse function obtained in exercise 1 has a corresponding point of the form $\left(b,a\right)$ for every point $\left(a,b\right)$ on the graph of the function.