Question

find f + g, f - g, fg, \frac{1}{g}. determine the domain for each function.\\( f(x) = \frac{5x}{x - 6} \\), \\( g(x) = \frac{9}{x + 7} \\)\\( (f + g)(x) = \frac{5x^2 + 44x - 54}{(x - 6)(x + 7)} \\) (simplify your answer.)what is the domain of f + g?\\( \bigcirc \\) a. the domain of f + g is \\( \square \\) (type your answer in interval notation.)\\( \bigcirc \\) b. the domain of f + g is \\( \bigcirc \\) (use a comma to separate answers as needed.)\\( \bigcirc \\) c. the domain of f + g is \\( \varnothing \\).

Explanation:

Step1: Find domain of $f(x)$

Denominator $x-6
eq0 \implies x
eq6$

Step2: Find domain of $g(x)$

Denominator $x+7
eq0 \implies x
eq-7$

Step3: Domain of $f+g$ is intersection

Combine restrictions: $x
eq-7$ and $x
eq6$

Answer:

A. The domain of $f+g$ is $(-\infty,-7)\cup(-7,6)\cup(6,\infty)$