Chapter 7, Problem 1GYR

To determine

To explain: The type of functions that have inverses.

Explanation of Solution

If the graph of a function $y=f\left(x\right)$ is intersects each horizontal line at most one. Then the function has an inverse.

The function is one-to-one on domain. Then the function has an inverse.

To determine

To describe: The method to find if two function having inverse of one another and provide examples of function that are (are not) inverses of one another.

Explanation of Solution

In algebraically:

Two functions f and g are inverses of one another if $f\left(g\left(x\right)\right)=g\left(f\left(x\right)\right)=x$.

In graphically:

If the functions f and g must be symmetric with about the line y = x, then the two functions f and g are inverses of one another.

Example:

Consider the functions $f\left(x\right)=3x-2\text{ and g}\left(x\right)=\frac{x+2}{3}$.

Now check $f\left(g\left(x\right)\right)=g\left(f\left(x\right)\right)$:

$\begin{array}{l}f\left(g\left(x\right)\right)=g\left(f\left(x\right)\right)\\ f\left(\frac{x+2}{3}\right)=g\left(3x-2\right)\\ 3\left(\frac{x+2}{3}\right)-2=\frac{3x-2+2}{3}\\ x=x\end{array}$

This two functions are inverses of one another.

Consider the functions $f\left(x\right)=3x-2\text{ and g}\left(x\right)=\frac{1}{3}x+2$.

Now check $f\left(g\left(x\right)\right)=g\left(f\left(x\right)\right)$:

$\begin{array}{l}f\left(g\left(x\right)\right)=g\left(f\left(x\right)\right)\\ f\left(\frac{1}{3}x+2\right)=g\left(3x-2\right)\\ 3\left(\frac{1}{3}x+2\right)-2=\frac{1}{3}\left(3x-2\right)+2\\ x+4\ne x+\frac{4}{3}\end{array}$

Thus, the two functions are not inverses of one another.