consider the table representing a rational function.

which statement describes the graph of the function?

- the function has holes when $x = 0$ and $x = 4$.
- the function has vertical asymptotes when $x = 0$ and $x = 4$.
- the function has a vertical asymptote when $x = 0$ and a hole when $x = 4$.
- the function has a hole when $x = 0$ and a vertical asymptote when $x = 4$.

Question

consider the table representing a rational function.

which statement describes the graph of the function?

- the function has holes when $x = 0$ and $x = 4$.
- the function has vertical asymptotes when $x = 0$ and $x = 4$.
- the function has a vertical asymptote when $x = 0$ and a hole when $x = 4$.
- the function has a hole when $x = 0$ and a vertical asymptote when $x = 4$.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Vertical Asymptotes", "Holes in Rational Graphs", "Removable Discontinuities" ], "new_concepts": [], "current_concepts": [ "Vertical Asymptotes", "Holes in Rational Graphs", "Removable Discontinuities", "Rational Functions" ] } ## Analyze behavior near x = 0 Using the *Holes in Rational Graphs* and *Removable Discontinuities* knowledge points $$ \begin{aligned} &\lim_{x \to 0^-} f(x) \approx -0.25\ &\lim_{x \to 0^+} f(x) \approx -0.25\ &f(0) \text{ is undefined} \end{aligned} $$ ## Analyze behavior near x = 4 Using the *Vertical Asymptotes* knowledge point $$ \begin{aligned} &\lim_{x \to 4^-} f(x) = -\infty\ &\lim_{x \to 4^+} f(x) = +\infty\ &f(4) \text{ is undefined} \end{aligned} $$ ## Determine graph features Using the *Vertical Asymptotes* and *Holes in Rational Graphs* knowledge points $$ \begin{aligned} &\text{Since } \lim_{x \to 0} f(x) = L \text{ (finite), there is a hole at } x = 0.\ &\text{Since } \lim_{x \to 4} f(x) = \pm\infty, \text{ there is a vertical asymptote at } x = 4. \end{aligned} $$

The function has holes when $x = 0$ and $x = 4$.

The function has vertical asymptotes when $x = 0$ and $x = 4$.

The function has a vertical asymptote when $x = 0$ and a hole when $x = 4$.

The function has a hole when $x = 0$ and a vertical asymptote when $x = 4$.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Rational Functions" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze behavior near x = 0

Analyze behavior near x = 4

Determine graph features

Answer: