Question

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

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

\\((f+g)(x) = \frac{9x-5}{x^2-81}\\) (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\\).

\\((f-g)(x) = \square\\) (simplify your answer.)

Explanation:

Find the sum function \( (f+g)(x) \)

Using the Function Operations knowledge point

$$ (f+g)(x) = f(x) + g(x) = \frac{5x+2}{x^2-81} + \frac{4x-7}{x^2-81} = \frac{(5x+2) + (4x-7)}{x^2-81} = \frac{9x-5}{x^2-81} $$

Determine the domain of \( f+g \)

Using the Domain of Rational Functions and Interval Notation knowledge points

$$
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Find the difference function \( (f-g)(x) \)

Using the Function Operations knowledge point

$$ (f-g)(x) = f(x) - g(x) = \frac{5x+2}{x^2-81} - \frac{4x-7}{x^2-81} = \frac{(5x+2) - (4x-7)}{x^2-81} = \frac{x+9}{x^2-81} $$

Simplify the difference function

Using the Domain of Rational Functions knowledge point

$$ (f-g)(x) = \frac{x+9}{(x-9)(x+9)} = \frac{1}{x-9} \quad (\text{for } x eq -9) $$

Answer:

Question 1

\( (f+g)(x) = \frac{9x-5}{x^2-81} \)

Question 2

• A. The domain of \( f+g \) is { }.
• B. The domain of \( f+g \) is \( (-\infty, -9) \cup (-9, 9) \cup (9, \infty) \) (Correct answer)
• C. The domain of \( f+g \) is \( \varnothing \)

Question 3

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