Q. 98 - Precalculus Enhanced with Graphing Utilities

Question

Can a one-to-one function and its inverse be equal? What must be true about the graph of f for this to happen? Give some examples to support your conclusion.

Step-by-Step Solution

Answer

Yes, a one-to-one function and its inverse are equal to each other. For this, the given function and its inverse must be symmetric about the line $y=x$

Example:

$$\begin{gathered} f=\left\{\left(4,4\right),\left(6,6\right),\left(8,8\right),\left(12,12\right)\right\} \\ {f}^{-1}=\left\{\left(4,4\right),\left(6,6\right),\left(8,8\right),\left(12,12\right)\right\} \end{gathered}$$

Step 1. Explanation

Yes, a one-to-one function and its inverse are equal to each other. For this, the given function and its inverse must be symmetric about the line $y=x$

Step 2. Examples

Let's take an example:

$f=\left\{\left(4,4\right),\left(6,6\right),\left(8,8\right),\left(12,12\right)\right\}$

The inverse of the function is :

$f^{-1}=\left\{\left(4,4\right),\left(6,6\right),\left(8,8\right),\left(12,12\right)\right\}$

As we can see that both are same.