Question

find \\(f+g\\), \\(f-g\\), \\(fg\\) and \\(\frac{1}{g}\\). determine the domain for each function.

\\f(x) = \frac{5x+2}{x^2-81}, g(x) = \frac{4x-7}{x^2-81}\\

c. the domain of \\(f+g\\) is \\(\varnothing\\)

\\((f-g)(x) = \frac{1}{x-9}\\) (simplify your answer)

what is the domain of \\(f-g\\)?

a. the domain of \\(f-g\\) is \\(\\{\quad\\}\\) (use a comma to separate answers as needed)

b. the domain of \\(f-g\\) is \\((-\infty, -9) \cup (-9, 9) \cup (9, \infty)\\) (type your answer in interval notation.)

c. the domain of \\(f-g\\) is \\(\varnothing\\)

\\((fg)(x) = \square\\) (simplify your answer)

Explanation:

Calculate the product function

$$ (fg)(x) = f(x) \cdot g(x) = \frac{5x+2}{x^2-81} \cdot \frac{4x-7}{x^2-81} = \frac{(5x+2)(4x-7)}{(x^2-81)^2} $$

Expand the numerator and denominator

$$ (5x+2)(4x-7) = 20x^2 - 35x + 8x - 14 = 20x^2 - 27x - 14 $$

$$ (x^2-81)^2 = (x-9)^2(x+9)^2 $$

State the simplified product function

$$ (fg)(x) = \frac{20x^2-27x-14}{(x^2-81)^2} $$

Answer:

\((fg)(x) =\) <blank>\(\frac{20x^2-27x-14}{(x^2-81)^2}\)</blank>