Chapter 6, Problem 21SGR

To determine

To Find and Graph :

Find the inverse of the function and graph the function and its inverse.

Answer to Problem 21SGR

  $f^{-1}(x)=\pm \sqrt{x}$

Explanation of Solution

Given Information:

Given function:

  $f(x)=x^2$

Concept Used:

Inverse Function :

Two functions f and g are inverse functions if and only if both of their compositions are the identity function.

That is ,

f(x) and g(x) are inverses if and only if $[f\circ g](x)=x\text{ and }[g\circ f](x)=x$ .

Calculation:

  $\begin{array}{l}Given\\ f(x)=x^2\\ \text{Replace f(x) by y:}\\ y=x^2\\ \text{Interchange x and y , then solve for y to get }\\ \text{the inverse function:}\\ x=y^2\\ \text{⇒}y=\pm \sqrt{x}\mathrm{............}[\text{taking square root each side]}\\ \text{⇒}g(x)=f^{-1}(x)=\pm \sqrt{x}\\ Hence,\text{ f(x) and g(x) are inverse functions}\text{.}\end{array}$

Graph: