Chapter 1, Problem 109CR

To determine

ToFind: whether the statement that there is no function of $f$ such that $f=f^{-1}$ is true or false.

Answer to Problem 109CR

false.

Explanation of Solution

Given:

Given statement is $f=f^{-1}$ .

Calculation:

Any function has inverse function if it is one-to -one.

One-to-one means no two elements in the domain of function corresponds to the same element in the range of the function.

For example,

Let a function $f\left(x\right)=\sqrt{x}+1$ .

Let a and b be non-negative real numbers such that $f\left(a\right)=f\left(b\right)$ .

  $\begin{array}{l}\sqrt{a}+1=\sqrt{b}+1\\ \sqrt{a}=\sqrt{b}\\ a=b\end{array}$

  $f\left(a\right)=f\left(b\right)$ implies that $a=b$ therefore it is a one-to-one function and it has inverse.

Therefore, given statement is false, it can exist if $f$ is one-to-one function.