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 + 44x - 54}{(x - 6)(x + 7)} \\) (simplify your answer.)\
what is 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 \\( \\{ \\} \\). (use a comma to separate answers as needed.)\
c. the domain of f + g is \\( \varnothing \\).\
\\( (f - g)(x) = \square \\) (simplify your answer.)

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: Compute $(f-g)(x)$

Combine fractions with common denominator
$$(f-g)(x) = \frac{5x}{x-6} - \frac{9}{x+7} = \frac{5x(x+7) - 9(x-6)}{(x-6)(x+7)}$$

Step5: Expand numerator

Simplify the expanded terms
$$5x(x+7)-9(x-6) = 5x^2 +35x -9x +54 = 5x^2 +26x +54$$

Step6: Final form of $(f-g)(x)$

Write simplified rational function
$$(f-g)(x) = \frac{5x^2 +26x +54}{(x-6)(x+7)}$$

Answer:

1. Domain of $f+g$: $\boldsymbol{(-\infty,-7)\cup(-7,6)\cup(6,\infty)}$ (Option A)
2. $(f-g)(x) = \boldsymbol{\frac{5x^2 +26x +54}{(x-6)(x+7)}}$