Chapter 7, Problem 15STP

(a)

To determine

To Find :

How the graphs of f(x) and g(x) would appear on a coordinate grid.

Answer to Problem 15STP

If f(x) and g(x) are inverse functions , the points (x,y) and (y,x) will be symmetric along the line y = x .

Explanation of Solution

Given :

f(x) and g(x) are inverse functions.

Calculation:

  $\begin{array}{l}Let\\ \text{f(x) = }x+1\\ \text{Find its inverse, replace f(x) by 'y':}\\ \text{y=x+1}\\ Interchange\text{ x and y:}\\ x=y+1\\ \text{Solve for y:}\\ \text{y+1-1=x-1}\mathrm{...............}\text{[subtract 1 from each side]}\\ y=x-1\\ So,\text{ the inverse of the function f(x) = }x+1\text{ is }\\ g(x)=x-1\\ \text{Represent both the functions on the graph and also graph the equation }\\ y=x\end{array}$

Graph :

  

So, we observe that if f(x) and g(x) are inverse functions , the points (x,y) and (y,x) will be symmetric along the line y = x .

(b)

To determine

To Find :

The value of the composition $f\left[g(2)\right]$

Answer to Problem 15STP

  $f\left[g(2)\right]=2$

Explanation of Solution

Given :

f(x) and g(x) are inverse functions.

Calculation:

  $\begin{array}{l}\text{We know that if f and g are inverse functions , then}:\\ \left[f\circ g\right](x)=x\\ \left[g\circ f\right](x)=x\\ So,\\ f\left[g(2)\right]=2\\ \text{Since the input is 2 , apply g and then the inverse f reverses the }\\ \text{changes applied by g on 2, hence the left output is 2}\text{.}\\ \text{For example, from step (a)}\\ \text{f(x)=x+1}\\ \text{g(x)=x-1}\\ f\left[g(2)\right]=f\left(\text{2-1}\right)=f\left(1\right)=1+1=2\end{array}$