Question

let $f(x) = 5 - x$ and $g(x) = \frac{1}{x}$. perform the function operation and then find the domain of the result.
$\frac{f}{g}(x)$
$\frac{f}{g}(x) = \square$ (simplify your answer.)

Explanation:

Step1: Define the function quotient

$\frac{f}{g}(x) = \frac{f(x)}{g(x)}$

Step2: Substitute given functions

$\frac{f}{g}(x) = \frac{5 - x}{\frac{1}{x}}$

Step3: Simplify the fraction

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

Step4: Identify domain restrictions

Original $g(x)=\frac{1}{x}$ is undefined at $x=0$, so $x
eq 0$. The simplified polynomial has no new restrictions.

Answer:

$\frac{f}{g}(x) = 5x - x^2$
Domain: All real numbers except $x=0$, or in interval notation $(-\infty, 0) \cup (0, \infty)$