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 ((0.1, 0))
domain of ((4, \infty))
vertical asymptote of (x = 0)

Question

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 ((0.1, 0))
  domain of ((4, \infty))
  vertical asymptote of (x = 0)

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", "Domain of a Function", "Range of a Function", "Asymptotes" ], "new_concepts": [], "current_concepts": [ "Logarithmic Functions", "Domain of a Function", "Range of a Function", "Asymptotes", "Graph Intercepts" ] } ## Analyze the domain and vertical asymptote of $g(x)$ $$ \begin{aligned} &\text{For } g(x) = 4\log(x) + 4, \text{ the argument of the logarithm must be strictly positive: } x > 0.\ &\text{Domain: } (0, \infty)\ &\text{Vertical Asymptote: } x = 0 \end{aligned} $$ ## Determine the range of $g(x)$ $$ \begin{aligned} &\text{Since } f(x) = \log(x) \text{ has a range of } (-\infty, \infty), \text{ any vertical stretch and shift preserves this range.}\ &\text{Range: } (-\infty, \infty) \end{aligned} $$ ## Evaluate the intercepts of $g(x)$ $$ \begin{aligned} &\text{y-intercept: } g(0) \text{ is undefined, so there is no y-intercept.}\ &\text{x-intercept: } 4\log(x) + 4 = 0 \implies \log(x) = -1 \implies x = 10^{-1} = 0.1\ &\text{x-intercept: } (0.1, 0) \end{aligned} $$

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

y-intercept of $(0, 4)$

x-intercept of $(0.1, 0)$

domain of $(4, \infty)$

vertical asymptote of $x = 0$

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

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Question

Explanation:

Analyze the domain and vertical asymptote of \(g(x)\)

Determine the range of \(g(x)\)

Evaluate the intercepts of \(g(x)\)

Answer: