Chapter 1.9, Problem 24E

a .

To determine

To verify: algebraically that the functions $f\left(x\right)=3-4x$ and $g\left(x\right)=\frac{3-x}{4}$ are the inverse functions.

Answer to Problem 24E

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,

  $\begin{array}{c}f\left(g\left(x\right)\right)=f\left(\frac{3-x}{4}\right)\\ =3-4\left(\frac{3-x}{4}\right)\\ =3-\left(3-x\right)\end{array}$

  $\begin{array}{c}=3-3+x\\ =x\end{array}$

And

  $\begin{array}{c}g\left(f\left(x\right)\right)=g\left(3-4x\right)\\ =\frac{3-\left(3-4x\right)}{4}\\ =\frac{3-3+4x}{4}\end{array}$

  $\begin{array}{c}=\frac{4x}{4}\\ =x\end{array}$

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 $f\left(x\right)=3-4x$ and $g\left(x\right)=\frac{3-x}{4}$ are the inverse functions.

Answer to Problem 24E

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 $f\left(x\right)=3-4x$ , $g\left(x\right)=\frac{3-x}{4}$ 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 $f\left(x\right)=3-4x$	Graph of $g\left(x\right)=\frac{3-x}{4}$
$\left(-1,7\right)$	$\left(7,-1\right)$
$\left(0,3\right)$	$\left(3,0\right)$
$\left(1,-1\right)$	$\left(-1,1\right)$
$\left(2,-5\right)$	$\left(-5,2\right)$
$\left(3,-9\right)$	$\left(-9,3\right)$

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.