the function $y = \sqrt3{-x} - 3$ is graphed only over the domain of $\{x \mid -8 \le x \le 8\}$. what is the range of the graph?

$\{y \mid -5 \le y \le 5\}$
$\{y \mid -5 \le y \le -1\}$
$\{y \mid 1 \le y \le 5\}$
$\{y \mid 1 \le y \le -1\}$

Question

the function $y = \sqrt3{-x} - 3$ is graphed only over the domain of $\{x \mid -8 \le x \le 8\}$. what is the range of the graph?

$\{y \mid -5 \le y \le 5\}$
$\{y \mid -5 \le y \le -1\}$
$\{y \mid 1 \le y \le 5\}$
$\{y \mid 1 \le y \le -1\}$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Range of Radical Functions", "Cube Root Transformations" ], "current_concepts": [ "Range of Radical Functions", "Cube Root Transformations", "Monotonic Functions" ] } ## Evaluate the function at the domain boundaries $$ \begin{aligned} y(-8) &= \sqrt[3]{-(-8)} - 3 = \sqrt[3]{8} - 3 = 2 - 3 = -1 \ y(8) &= \sqrt[3]{-(8)} - 3 = \sqrt[3]{-8} - 3 = -2 - 3 = -5 \end{aligned} $$ ## Determine the range using monotonicity $$ \begin{aligned} &\text{Since } y = \sqrt[3]{-x} - 3 \text{ is strictly decreasing on } [-8, 8]: \ &\min(y) = y(8) = -5 \ &\max(y) = y(-8) = -1 \ &\text{Range: } \{y \mid -5 \le y \le -1\} \end{aligned} $$

{y| -5 \le y \le 5}

{y| -5 \le y \le -1}

{y| 1 \le y \le 5}

{y| 1 \le y \le -1}

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

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QUESTION IMAGE

Question

Explanation:

Evaluate the function at the domain boundaries

Determine the range using monotonicity

Answer: