given the rational function $f(x) = \frac{4x - 5}{x^3 - 25x}$, answer the following questions.

a) what is the domain of $f(x)$? enter your answer in interval notation.

b) find the vertical asymptote(s) of $f(x)$. enter each asymptote as an equation, separating multiple asymptotes with a comma. if there are no vertical asymptotes, enter none.

c) find the horizontal asymptote(s) of $f(x)$. enter each asymptote as an equation, separating multiple asymptotes with a comma. if there are no horizontal asymptotes, enter none.

Question

given the rational function $f(x) = \frac{4x - 5}{x^3 - 25x}$, answer the following questions.

a) what is the domain of $f(x)$? enter your answer in interval notation.

b) find the vertical asymptote(s) of $f(x)$. enter each asymptote as an equation, separating multiple asymptotes with a comma. if there are no vertical asymptotes, enter none.

c) find the horizontal asymptote(s) of $f(x)$. enter each asymptote as an equation, separating multiple asymptotes with a comma. if there are no horizontal asymptotes, enter none.

Accepted Answer

{ "quality": "clear", "question_count": 3, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Domain of Rational Functions", "Interval Notation" ], "new_concepts": [ "Vertical Asymptotes", "Horizontal Asymptotes" ], "current_concepts": [ "Domain of Rational Functions", "Interval Notation", "Vertical Asymptotes", "Horizontal Asymptotes" ] } ## Find the domain of the function $$ \begin{aligned} &x^3 - 25x = 0 \ &x(x^2 - 25) = 0 \ &x(x - 5)(x + 5) = 0 \ &x eq 0, \quad x eq 5, \quad x eq -5 \ &\text{Domain: } (-\infty, -5) \cup (-5, 0) \cup (0, 5) \cup (5, \infty) \end{aligned} $$ ## Find the vertical asymptotes $$ \begin{aligned} &f(x) = \frac{4x - 5}{x(x - 5)(x + 5)} \ &\text{Numerator } 4x - 5 eq 0 \text{ at } x = 0, \pm 5 \ &\text{Vertical Asymptotes: } x = -5, x = 0, x = 5 \end{aligned} $$ ## Find the horizontal asymptote $$ \begin{aligned} &\text{Degree of numerator } = 1 \ &\text{Degree of denominator } = 3 \ &1 < 3 \implies \lim_{x \to \pm\infty} f(x) = 0 \ &\text{Horizontal Asymptote: } y = 0 \end{aligned} $$ a) $(-\infty, -5) \cup (-5, 0) \cup (0, 5) \cup (5, \infty)$ b) $x = -5, x = 0, x = 5$ c) $y = 0$ { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Rational Functions" ] }

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QUESTION IMAGE

Question

Explanation:

Find the domain of the function

Find the vertical asymptotes

Find the horizontal asymptote

Answer: