Chapter 3, Problem 5VE

Solution

To explain how to use the horizontal line test to determine whether the inverse of a function f is also a function.

Given information:

A function and its inverse.

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 the graph of a function $y=f\left(x\right)$ and its inverse $f^{-1}\left(x\right)$ are symmetric to the line $y=x$ . If there is a point $\left(a,b\right)$ on the graph of $f\left(x\right)$ , then the corresponding point on the graph of inverse function $f^{-1}\left(x\right)$ will be $\left(b,a\right)$ .

A relation is a function when it satisfies the vertical line test and since the inverse function is the image of function about the line $y=x$ , to check whether the inverse is a function as well, use the horizontal line test.

If a horizontal line at any point of the graph of the inverse of a function does not intersect the graph at more that one point, the inverse of the function is also a function.