Chapter 7, Problem 19SGR

To determine

To Find and Graph :

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

Answer to Problem 19SGR

  $f^{-1}(x)=2x-6$

Explanation of Solution

Given Information:

Given function:

  $f(x)=\frac{1}{2}x+3$

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:

  $\begin{array}{l}Given\\ f(x)=\frac{1}{2}x+3\\ \text{Replace f(x) by y:}\\ y=\frac{1}{2}x+3\\ \text{Interchange x and y , then solve for y to get }\\ \text{the inverse function:}\\ x=\frac{1}{2}y+3\\ \Rightarrow x-3=\frac{1}{2}y\mathrm{............}[subtract\text{ 3 from each side]}\\ \Rightarrow 2(x-3)=y\mathrm{............}[\text{multiply each side by 2]}\\ \Rightarrow y=2x-6\\ \Rightarrow g(x)=f^{-1}(x)=2x-6\\ Hence,\text{ f(x) and g(x) are inverse functions}\text{.}\end{array}$

Graph: