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}\

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

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}\

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

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Domain of Rational Functions", "Function Operations" ], "new_concepts": [], "current_concepts": [ "Domain of Rational Functions", "Function Operations", "Interval Notation" ] } ## Find the sum of the functions Using the *Function Operations* knowledge point $$ \begin{aligned} (f+g)(x) &= f(x) + g(x) \ &= \frac{5x+2}{x^2-81} + \frac{4x-7}{x^2-81} \ &= \frac{(5x+2) + (4x-7)}{x^2-81} \ &= \frac{9x-5}{x^2-81} \end{aligned} $$ ## Determine the domain of the sum function Using the *Domain of Rational Functions* and *Interval Notation* knowledge points $$ \begin{aligned} &x^2 - 81 eq 0 \implies (x-9)(x+9) eq 0 \implies x eq \pm 9 \ &\text{Domain: } (-\infty, -9) \cup (-9, 9) \cup (9, \infty) \end{aligned} $$ 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.) { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Function Operations" ] }

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Find the sum of the functions

Determine the domain of the sum function

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