select all the correct answers.

consider the graph of the function $f(x) = \log x$.

which are features of function $g$ if $g(x) = 4 \log(x) + 4$?

range of $(-\infty, \infty)$
y-intercept of $(0, 4)$
x-intercept of

Question

select all the correct answers.

consider the graph of the function $f(x) = \log x$.

which are features of function $g$ if $g(x) = 4 \log(x) + 4$?

range of $(-\infty, \infty)$
  y-intercept of $(0, 4)$
  x-intercept of

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Logarithmic Functions", "Range of a Function", "Graph Intercepts" ], "new_concepts": [], "current_concepts": [ "Logarithmic Functions", "Range of a Function", "Graph Intercepts", "Asymptotes" ] } ## Determine the base of the logarithm from the graph of $f(x)$ $$ \begin{aligned} f(x) &= \log_b(x) \ f(1) &= 0 \ f(3) &= 0.5 \implies \log_b(3) = 0.5 \implies b^{0.5} = 3 \implies b = 9 \ f(x) &= \log_9(x) \end{aligned} $$ ## Analyze the features of $g(x) = 4\log_9(x) + 4$ $$ \begin{aligned} \text{Domain: } &x > 0 \ \text{Range: } &(-\infty, \infty) \ \text{y-intercept: } &\text{None (since } x = 0 \text{ is outside the domain)} \ \text{x-intercept: } &g(x) = 0 \implies 4\log_9(x) + 4 = 0 \implies \log_9(x) = -1 \implies x = 9^{-1} = \frac{1}{9} \approx 0.11 \end{aligned} $$ ## Evaluate the given options $$ \begin{aligned} &\text{Option 1: range of } (-\infty, \infty) \quad \text{(Correct)} \ &\text{Option 2: y-intercept of } (0, 4) \quad \text{(Incorrect, no y-intercept)} \ &\text{Option 3: x-intercept of } (0.1, 0) \quad \text{(Correct, since } \frac{1}{9} \approx 0.11 \approx 0.1\text{)} \end{aligned} $$

range of $(-\infty, \infty)$

y-intercept of $(0, 4)$

x-intercept of $(0.1, 0)$

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

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Question

Explanation:

Determine the base of the logarithm from the graph of \(f(x)\)

Analyze the features of \(g(x) = 4\log_9(x) + 4\)

Evaluate the given options

Answer: