select all the correct answers.

which key features describe this inverse variation function?

- the domain is all real numbers.
- there is an asymptote at $x = 0$.
- the function is always decreasing.
- as $x$ approaches infinity, $y$ approaches infinity.

Question

select all the correct answers.

which key features describe this inverse variation function?

- the domain is all real numbers.
- there is an asymptote at $x = 0$.
- the function is always decreasing.
- as $x$ approaches infinity, $y$ approaches infinity.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Inverse Variation", "Domain of a Function", "Asymptotes" ], "new_concepts": [], "current_concepts": [ "Inverse Variation", "Domain of a Function", "Asymptotes", "Increasing and Decreasing Functions" ] } ## Analyze the domain of the function $$ x eq 0 \implies \text{Domain: } (-\infty, 0) \cup (0, \infty) $$ ## Identify the asymptotes $$ \begin{aligned} &\lim_{x \to 0^+} y = \infty, \quad \lim_{x \to 0^-} y = -\infty \implies \text{Vertical asymptote at } x = 0\ &\lim_{x \to \infty} y = 0, \quad \lim_{x \to -\infty} y = 0 \implies \text{Horizontal asymptote at } y = 0 \end{aligned} $$ ## Determine interval behavior and end behavior $$ \begin{aligned} &\text{For } x \in (-\infty, 0): y' < 0 \implies \text{Decreasing}\ &\text{For } x \in (0, \infty): y' < 0 \implies \text{Decreasing}\ &\text{As } x \to \infty, y \to 0 \end{aligned} $$

The domain is all real numbers.

There is an asymptote at $x = 0$.

The function is always decreasing.

As $x$ approaches infinity, $y$ approaches infinity.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Inverse Variation" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze the domain of the function

Identify the asymptotes

Determine interval behavior and end behavior

Answer: