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) = \square \\) (simplify your answer.)

Explanation:

Step1: Define sum of functions

$(f+g)(x) = f(x) + g(x)$

Step2: Substitute given functions

$(f+g)(x) = \frac{5x}{x-6} + \frac{9}{x+7}$

Step3: Find common denominator

Common denominator is $(x-6)(x+7)$

Step4: Rewrite fractions with LCD

$(f+g)(x) = \frac{5x(x+7)}{(x-6)(x+7)} + \frac{9(x-6)}{(x-6)(x+7)}$

Step5: Expand numerators

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

Step6: Combine like terms

$(f+g)(x) = \frac{5x^2 + 44x - 54}{(x-6)(x+7)}$

Step7: Find domain of $f+g$

Denominator cannot be zero: $x-6
eq 0 \implies x
eq 6$; $x+7
eq 0 \implies x
eq -7$

Answer:

$(f+g)(x) = \frac{5x^2 + 44x - 54}{(x-6)(x+7)}$
Domain: All real numbers except $x = -7$ and $x = 6$