Question

let $f(x)=4 - x$ and $g(x)=\frac{5}{x}$. perform the function operation and then find the domain of the result.
$(g - f)(x)$

$(g - f)(x)=\square$ (simplify your answer.)

Explanation:

Step1: Define the function subtraction

$(g-f)(x) = g(x) - f(x)$

Step2: Substitute given functions

$(g-f)(x) = \frac{5}{x} - (4 - x)$

Step3: Simplify the expression

$(g-f)(x) = \frac{5}{x} - 4 + x$

Step4: Find domain restrictions

The term $\frac{5}{x}$ is undefined when $x=0$, so $x$ cannot equal 0.

Answer:

$(g-f)(x) = x - 4 + \frac{5}{x}$
Domain: All real numbers except $x=0$ (or in interval notation: $(-\infty,0) \cup (0,\infty)$)