Chapter 5.2, Problem 41AYU

In Problems 35-44, verify that the functions $f$ and $g$ are inverses of each other by showing that $f(g(x))=x$ and $g(f\left(x\right))=x$. Give any values of $x$ that need to be excluded from the domain of $f$ and the domain of $g$.

$f(x)=\frac{2x+3}{x+4}$; $g(x)=\frac{4x-3}{2-x}$

To determine

To find: Verify that the functions $f$ and $g$ are inverses of each other by showing that $f(g(x))=x$ and $g(f\left(x\right))=x$. Give any values of $x$ that need to be excluded from the domain of $f$ and the domain of $g$.

Answer to Problem 41AYU

Solution:

The given functions $f$ and $g$ are verified.

The values of $x$ to be excluded from the domain of $f$ and the domain of $g$ are $-4\&2$.

Explanation of Solution

Given:

By domain of a Composite Function,

The domain of a composite function $f\left(g\left(x\right)\right)$ is the set of those inputs $x$ in the domain of $g$ for which $g\left(x\right)$ is in the domain of $f$.

$g\left(x\right)=\frac{4x-3}{2-x}$

Calculation:

The domain of $g$ is $\left[- \infty ,-4\right)$.

$f\left(x\right)=\frac{2x+3}{x+4}$

The domain of $\left[2,\infty \right)$.

From those inputs, $x$, in the domain of $g$ for which $g\left(x\right)$ is in the domain of $f$. That is, exclude those inputs, $x$, from the domain of $g$ for which $g\left(x\right)$ is not in the domain of $f$. The resulting set is the domain of $f\circ g$. i.e., $-4\&2$.

We know the composition of $f$ and $g$ is defined as.

$\left(f\cdot g\right)\left(x\right)=f\left(g\left(x\right)\right)$

$\left(f\cdot g\right)\left(x\right)=f\left(\frac{4x-3}{2-x}\right)$

$\left(f\cdot g\right)\left(x\right)=\frac{2\left(\frac{4x-3}{2-x}\right)-3}{\left(\frac{4x-3}{2-x}\right)+4}$

$\left(f\cdot g\right)\left(x\right)=\frac{\frac{8x-6+3\left(2-x\right)}{2-x}}{\frac{4x-3+4\left(2-x\right)}{2-x}}$

$\left(f\cdot g\right)\left(x\right)=\frac{\frac{8x-6+6\mp 3x}{2-x}}{\frac{4x-3+8-4x}{2-x}}$

$\left(f\cdot g\right)\left(x\right)=\frac{\frac{5x}{2-x}}{\frac{5}{2-x}}$

$\left(f\cdot g\right)\left(x\right)=\frac{5x}{5}$

$\left(f\cdot g\right)\left(x\right)=x$

Also $\left(g\cdot f\right)\left(x\right)=g\left(f\left(x\right)\right)$

$\left(g\cdot f\right)\left(x\right)=g\left(\frac{2x+3}{x+4}\right)$

$\left(g\cdot f\right)\left(x\right)=\frac{4\left(\frac{2x+3}{x+4}\right)-3}{2-\left(\frac{2x+3}{x+4}\right)}$

$\left(g\cdot f\right)\left(x\right)=\frac{\frac{8x+12-3x-12}{x+4}}{\frac{2x+8-2x-3}{x+4}}$

$\left(g\cdot f\right)\left(x\right)=\frac{\frac{5x}{x+4}}{\frac{5}{x+4}}$

$\left(g\cdot f\right)\left(x\right)=\frac{5x}{5}$

$\left(g\cdot f\right)\left(x\right)=x$