Question

let (f(x)=\frac{x}{x - 4}) and (g(x)=\frac{6}{x}). find the following functions. simplify your answers. (f(g(x))=)(g(f(x))=)

Explanation:

Step1: Find $f(g(x))$

Substitute $g(x)=\frac{6}{x}$ into $f(x)$. So $f(g(x))=\frac{\frac{6}{x}}{\frac{6}{x}-4}$.

Step2: Simplify $f(g(x))$

Multiply numerator and denominator by $x$: $\frac{6}{6 - 4x}$.

Step3: Find $g(f(x))$

Substitute $f(x)=\frac{x}{x - 4}$ into $g(x)$. So $g(f(x))=\frac{6}{\frac{x}{x - 4}}$.

Step4: Simplify $g(f(x))$

By the rule of dividing by a fraction, $g(f(x))=\frac{6(x - 4)}{x}=\frac{6x-24}{x}$.

Answer:

$f(g(x))=\frac{6}{6 - 4x}$
$g(f(x))=\frac{6x - 24}{x}$