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} \\)\
\\( (f - g)(x) = \frac{5x^2 + 26x + 54}{(x - 6)(x + 7)} \\) (simplify your answer.)\
what is the domain of f - g?\
a. the domain of f - g is \\( \boxed{} \\) (type your answer in interval notation.)\
b. the domain of f - g is \\( \boxed{} \\) (use a comma to separate answers as needed.)\
c. the domain of f - g is \\( \varnothing \\).\
for the domain of f + g:\
a. the domain of f + g is \\( (-\infty, -7) \cup (-7, 6) \cup (6, \infty) \\) (type your answer in interval notation.)\
b. the domain of f + g is \\( \boxed{} \\) (use a comma to separate answers as needed.)\
c. the domain of f + g is \\( \varnothing \\).

Explanation:

Step1: Find domain of $f(x)$

Denominator of $f(x)$: $x-6
eq 0 \implies x
eq 6$. Domain of $f(x)$: $(-\infty,6)\cup(6,\infty)$

Step2: Find domain of $g(x)$

Denominator of $g(x)$: $x+7
eq 0 \implies x
eq -7$. Domain of $g(x)$: $(-\infty,-7)\cup(-7,\infty)$

Step3: Domain of $f+g$

Intersection of domains of $f$ and $g$: $(-\infty,-7)\cup(-7,6)\cup(6,\infty)$

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

$$(f-g)(x)=\frac{5x}{x-6}-\frac{9}{x+7}=\frac{5x(x+7)-9(x-6)}{(x-6)(x+7)}$$
Expand numerator: $5x^2+35x-9x+54=5x^2+26x+54$
So $(f-g)(x)=\frac{5x^2+26x+54}{(x-6)(x+7)}$

Step5: Domain of $f-g$

Same as $f+g$: intersection of domains of $f$ and $g$, since numerator has no real roots (discriminant $26^2-4*5*54=676-1080=-404<0$), no new restrictions.

Answer:

For $f+g$:
A. The domain of $f+g$ is $(-\infty,-7)\cup(-7,6)\cup(6,\infty)$
For $(f-g)(x)$: $\frac{5x^2+26x+54}{(x-6)(x+7)}$
For domain of $f-g$:
A. The domain of $f-g$ is $(-\infty,-7)\cup(-7,6)\cup(6,\infty)$