which of the following explains why $f(x) = \log_{4}x$ does not have a $y$-intercept? check all that apply.

there is no power of 4 that is equal to 0.
there is no power of 4 that is equal to 1.
its inverse does not have any $x$-intercepts.
its inverse does not have any $y$-intercepts.

Question

which of the following explains why $f(x) = \log_{4}x$ does not have a $y$-intercept? check all that apply.

there is no power of 4 that is equal to 0.
there is no power of 4 that is equal to 1.
its inverse does not have any $x$-intercepts.
its inverse does not have any $y$-intercepts.

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": [ "Logarithmic Functions", "Logarithmic and Exponential Inverses" ], "new_concepts": [], "current_concepts": [ "Logarithmic Functions", "Logarithmic and Exponential Inverses", "y-intercept" ] } ## Define the y-intercept condition Using the *Logarithmic Functions* knowledge point $$ \begin{aligned} &\text{To find the } y\text{-intercept of } f(x) = \log_4(x), \text{ we evaluate } f(0).\ &f(0) = \log_4(0) \end{aligned} $$ ## Analyze the logarithmic expression Using the *Logarithmic Functions* knowledge point $$ \begin{aligned} &y = \log_4(0) \iff 4^y = 0\ &\text{Since } 4^y > 0 \text{ for all real numbers } y, \text{ there is no power of } 4 \text{ that is equal to } 0.\ &\text{Thus, } f(0) \text{ is undefined, and } f(x) \text{ has no } y\text{-intercept.} \end{aligned} $$ ## Analyze the inverse function relationship Using the *Logarithmic and Exponential Inverses* knowledge point $$ \begin{aligned} &f^{-1}(x) = 4^x\ &\text{The } y\text{-intercept of } f(x) \text{ corresponds to the } x\text{-intercept of its inverse } f^{-1}(x).\ &\text{Since } 4^x = 0 \text{ has no real solution, } f^{-1}(x) \text{ has no } x\text{-intercepts.}\ &\text{Therefore, } f(x) \text{ has no } y\text{-intercept.} \end{aligned} $$

There is no power of 4 that is equal to 0.

There is no power of 4 that is equal to 1.

Its inverse does not have any x-intercepts.

Its inverse does not have any y-intercepts.

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

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Question

Explanation:

Define the y-intercept condition

Analyze the logarithmic expression

Analyze the inverse function relationship

Answer: