which best describes the range of the function $f(x) = \frac{2}{3}(6)^x$ after it has been reflected over the $x$-axis?

all real numbers
all real numbers less than 0
all real numbers greater than 0
all real numbers less than or equal to 0

Question

which best describes the range of the function $f(x) = \frac{2}{3}(6)^x$ after it has been reflected over the $x$-axis?

all real numbers
all real numbers less than 0
all real numbers greater than 0
all real numbers less than or equal to 0

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Exponential Functions", "Function Transformations", "Exponential Range" ], "new_concepts": [], "current_concepts": [ "Exponential Functions", "Function Transformations", "Exponential Range" ] } ## Determine the range of the original function $$ \begin{aligned} &f(x) = \frac{2}{3}(6)^x \ &\text{Since } 6^x > 0 \text{ for all real } x, \ &\text{Range of } f(x): y > 0 \end{aligned} $$ ## Apply reflection over the x-axis $$ \begin{aligned} &g(x) = -f(x) = -\frac{2}{3}(6)^x \ &\text{Since } f(x) > 0, \ &g(x) < 0 \end{aligned} $$ ## Identify the matching description $$ \text{Range of } g(x): \{y \in \mathbb{R} \mid y < 0\} \quad (\text{all real numbers less than 0}) $$

all real numbers

all real numbers less than 0

all real numbers greater than 0

all real numbers less than or equal to 0

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Exponential Range" ] }

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QUESTION IMAGE

Question

Explanation:

Determine the range of the original function

Apply reflection over the x-axis

Identify the matching description

Answer: