Question

find ( f + g ), ( f - g ), ( fg ), ( \frac{f}{g} ). determine the domain for each function. ( f(x) = \frac{5x}{x - 6} ), ( g(x) = \frac{9}{x + 7} ) b. the domain of ( f - g ) is ( {} ). (use a comma to separate answers as needed.) c. the domain of ( f - g ) is ( varnothing ). ( (fg)(x) = \frac{45x}{(x - 6)(x + 7)} ) (simplify your answer.) what is the domain of ( fg )? a. the domain of ( fg ) is ( {} ). (use a comma to separate answers as needed.) b. the domain of ( fg ) is ( square ). (type your answer in interval notation.) c. the domain of ( fg ) is ( varnothing ).

Explanation:

Step1: Find domain of $f(x)$

Denominator of $f(x)$: $x-6
eq 0 \implies x
eq 6$

Step2: Find domain of $g(x)$

Denominator of $g(x)$: $x+7
eq 0 \implies x
eq -7$

Step3: Domain of $fg(x)$

Domain is all $x$ valid for both $f$ and $g$: $x
eq -7, 6$

Step4: Write in interval notation

Combine valid intervals: $(-\infty, -7) \cup (-7, 6) \cup (6, \infty)$

Answer:

For $(fg)(x)$:

$\frac{45x}{(x-6)(x+7)}$

For the domain of $fg$:

B. The domain of fg is $(-\infty, -7) \cup (-7, 6) \cup (6, \infty)$. (Type your answer in interval notation.)