which function has a range of $y

Question

which function has a range of $y < 3$?

$y = 3(2)^x$
$y = 2(3)^x$
$y = -(2)^x + 3$
$y = (2)^x - 3$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Range of Exponential Functions", "Exponential Transformations" ], "new_concepts": [], "current_concepts": [ "Range of Exponential Functions", "Exponential Transformations" ] } ## Analyze the range of standard exponential terms $$ (2)^x > 0 \quad \text{and} \quad (3)^x > 0 \quad \text{for all } x \in \mathbb{R} $$ ## Determine the range of each option $$ \begin{aligned} &y = 3(2)^x \implies y > 0\ &y = 2(3)^x \implies y > 0\ &y = -(2)^x + 3 \implies -(2)^x < 0 \implies y < 3\ &y = (2)^x - 3 \implies y > -3 \end{aligned} $$ ## Identify the matching function $$ y = -(2)^x + 3 \implies \text{Range: } y < 3 $$

(A) $y = 3(2)^x$

(B) $y = 2(3)^x$

(C) $y = -(2)^x + 3$

(D) $y = (2)^x - 3$

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

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Question

Explanation:

Analyze the range of standard exponential terms

Determine the range of each option

Identify the matching function

Answer: