Chapter 1.9, Problem 76E

To determine

To find: inverse of the function f and then to state the domains of the function f and its

inverse.

Answer to Problem 76E

The inverse of the given function is $f^{-1}(x)=\sqrt{2x+2}$ on domain $x\ge 0.$ .

Explanation of Solution

Given information:

The function $f(x)=\frac{1}{2}{x}^2-1$

Concept used:

The function has inverse function if and only if the function is one to one.

Calculation:

The function $f(x)=\frac{1}{2}{x}^2-1$ is not one to one function.

To make one to one the restrict the domain to $x\ge 0$

Find the inverse function.

  $f(x)=\frac{1}{2}{x}^2-1$

Let $y=\frac{1}{2}{x}^2-1$

  $\begin{array}{l}y+1=\frac{1}{2}{x}^2\\ 2y+2=x^2\\ x=\sqrt{2y+2}\\ f^{-1}(y)=\sqrt{2y+2}\\ f^{-1}(x)=\sqrt{2x+2}\end{array}$

Hence, the inverse of the given function is $f^{-1}(x)=\sqrt{2x+2}$ on domain $x\ge 0$ .