Chapter 5.2, Problem 98AYU

To determine

To find: Whether it is possible that the one to one function and it’s inverse are equal and determine the conditionfor the graph $f$ for this to be true, determine with example for the conclusion.

Answer to Problem 98AYU

Thegraph of the function and its inverse can be equal and in that case the graph of the function coincide. The example for this is that the graph of the function $f$ has the points $\left\{\left(1,1\right)\left(2,2\right)\left(3,3\right)\left(4,4\right)\right\}$ and its inverse has the points $\left\{\left(1,1\right)\left(2,2\right)\left(3,3\right)\left(4,4\right)\right\}$ .

Explanation of Solution

Consider the function $f=\left\{\left(1,1\right)\left(2,2\right)\left(3,3\right)\left(4,4\right)\right\}$ .

The graph of the function $f\left(x\right)$ for the point $\left(x,y\right)$ , there is a symmetric graph for the points $\left(y,x\right)$ which will be on the graph of the function $f^{-1}\left(x\right)$ .

For the function $f=\left\{\left(1,1\right)\left(2,2\right)\left(3,3\right)\left(4,4\right)\right\}$ the inverse is,

  $f^{-1}=\left\{\left(1,1\right)\left(2,2\right)\left(3,3\right)\left(4,4\right)\right\}$

As the inverse and the original function are same, the graph and the function will coincide.