Q 41. - Precalculus Enhanced with Graphing Utilities

Question

Verify that the functions $f\left(x\right)=\frac{2x+3}{x+4}$ and $g\left(x\right)=\frac{4x-3}{2-x}$ are inverses of each other by showing that $f\left(g\left(x\right)\right)=x$ and $g\left(f\left(x\right)\right)=x$. Give any values of x that need to be excluded from the domain of f and the domain of g.

Step-by-Step Solution

Answer

Functions f and g are inverses of one another .Exclude $x=-4$ from domain of f and $x=2$ from domain of g.

Step 1. Given information.

Given functions $f\left(x\right)=\frac{2x+3}{x+4}$ and $g\left(x\right)=\frac{4x-3}{2-x}$.

Step 2. Verify that the functions f and g and are inverses of each other.

Note that functions f and g are inverses of each other if $f\left(g\left(x\right)\right)=x$ and $g\left(f\left(x\right)\right)=x$.

Prove $f\left(g\left(x\right)\right)=x$ as follows.

$\begin{array}{rcl}f\left(g\left(x\right)\right)& =& f\left(\frac{4x-3}{2-x}\right)\\ & =& \frac{2\left(\frac{4x-3}{2-x}\right)+3}{\left(\frac{4x-3}{2-x}\right)+4}\\ & =& \frac{8x-6+6-3x}{4x-3+8-4x}\\ & =& \frac{5x}{5}\\ & =& x\end{array}$

Prove $g\left(f\left(x\right)\right)=x$ as follows.

$\begin{array}{rcl}g\left(f\left(x\right)\right)& =& g\left(\frac{2x+3}{x+4}\right)\\ & =& \frac{4\left(\frac{2x+3}{x+4}\right)-3}{2-\left(\frac{2x+3}{x+4}\right)}\\ & =& \frac{8x+12-3x-12}{2x+8-2x-3}\\ & =& \frac{5x}{5}\\ & =& x\end{array}$

Therefore, f and g are inverses of each other.

Step 3. Find values of x that need to be excluded from domain of f and g .

Note that domain of  f  is $x\ne -4$and that of g is $x\ne 2$.

It follows that f exists for all real numbers except for $-4$ and g exists for all real numbers except for 2.

Therefore,  $x=-4$ and $x=2$ need to be excluded from domain of f and g respectively.