select the correct answer.

which statement is true about the graph of function $f$?

\f(x) = \log_2 x\

- the graph has an asymptote of $y = 0$ and is increasing as $x$ approaches positive infinity.
- the graph has an asymptote of $x = 0$ and is positive over the interval $(0, 1)$.
- the graph has an asymptote of $y = 0$ and is decreasing as $x$ approaches positive infinity.
- the graph has an asymptote of $x = 0$ and is negative over the interval $(0, 1)$.

Question

select the correct answer.

which statement is true about the graph of function $f$?

\f(x) = \log_2 x\

- the graph has an asymptote of $y = 0$ and is increasing as $x$ approaches positive infinity.
- the graph has an asymptote of $x = 0$ and is positive over the interval $(0, 1)$.
- the graph has an asymptote of $y = 0$ and is decreasing as $x$ approaches positive infinity.
- the graph has an asymptote of $x = 0$ and is negative over the interval $(0, 1)$.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Logarithmic Functions" ], "new_concepts": [], "current_concepts": [ "Logarithmic Functions", "Asymptotes", "Function Intervals" ] } ## Determine the asymptote of the logarithmic function $$ \begin{aligned} &f(x) = \log_2 x \ &\text{Domain: } x > 0 \ &\lim_{x \to 0^+} \log_2 x = -\infty \implies \text{Vertical Asymptote: } x = 0 \end{aligned} $$ ## Analyze the function values on the interval (0, 1) $$ \begin{aligned} &0 < x < 1 \implies \log_2 x < \log_2 1 \ &\log_2 1 = 0 \implies f(x) < 0 \text{ for } x \in (0, 1) \end{aligned} $$ ## Evaluate the given statements $$ \begin{aligned} &\text{Asymptote: } x = 0 \ &\text{Sign on } (0, 1): \text{negative} \end{aligned} $$

(A) The graph has an asymptote of $y = 0$ and is increasing as $x$ approaches positive infinity.

(B) The graph has an asymptote of $x = 0$ and is positive over the interval $(0, 1)$.

(C) The graph has an asymptote of $y = 0$ and is decreasing as $x$ approaches positive infinity.

(D) The graph has an asymptote of $x = 0$ and is negative over the interval $(0, 1)$.

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Question

Explanation:

Determine the asymptote of the logarithmic function

Analyze the function values on the interval (0, 1)

Evaluate the given statements

Answer: