Chapter 1.9, Problem 22E

a .

To determine

To verify: algebraically that the functions $f\left(x\right)=2x$ and $g\left(x\right)=\frac{x}{2}$ are the inverse functions.

Answer to Problem 22E

f And g are inverse functions.

Explanation of Solution

Concept Used:

Inverse functions: two functions f and g are said to be the inverse functions if:

  $f\left(g\left(x\right)\right)=x$ for every x in the domain of g

And $g\left(f\left(x\right)\right)=x$ for every x in the domain of f

Calculation:

Consider,

  $\begin{array}{c}f\left(g\left(x\right)\right)=f\left(\frac{x}{2}\right)\\ =2\left(\frac{x}{2}\right)\\ =x\end{array}$

And

  $\begin{array}{c}g\left(f\left(x\right)\right)=g\left(2x\right)\\ =\frac{2x}{2}\\ =x\end{array}$

Clearly $f\left(g\left(x\right)\right)=g\left(f\left(x\right)\right)=x$ , so f and g are the inverse functions.

b.

To determine

To verify: graphically that the functions $f\left(x\right)=2x$ and $g\left(x\right)=\frac{x}{2}$ are the inverse functions.

Answer to Problem 22E

f And g are inverse functions.

Explanation of Solution

Concept Used:

The graph of a function and its inverse is symmetric about the line $y=x$ .

First draw the graphs of $f\left(x\right)=2x$ , $g\left(x\right)=\frac{x}{2}$ and $y=x$ on the same graph.

The above graph shows that graphs of f and g are reflections of each other about the line $y=x$ .

Now to further verify this reflective property, test a few points on each graph.

(For each point $\left(a,b\right)$ on the graph of f , the point $\left(b,a\right)$ should be there on the graph of g.)

Graph of $f\left(x\right)=2x$	Graph of $g\left(x\right)=\frac{x}{2}$
$\left(-4,-8\right)$	$\left(-8,-4\right)$
$\left(-2,-4\right)$	$\left(-4,-2\right)$
$\left(0,0\right)$	$\left(0,0\right)$
$\left(2,4\right)$	$\left(4,2\right)$
$\left(4,8\right)$	$\left(8,4\right)$

Hence proved that graphs of f and g are reflections of each other about the line $y=x$ , so they are inverse functions of each other.