which of the following graphs are identical?

$y = \sqrt{x}$

$y = \sqrt3{x}$

$y = \sqrt{-x}$

$y = \sqrt3{-x}$

$y = -\sqrt{x}$

$y = -\sqrt3{x}$

Question

which of the following graphs are identical?

$y = \sqrt{x}$

$y = \sqrt3{x}$

$y = \sqrt{-x}$

$y = \sqrt3{-x}$

$y = -\sqrt{x}$

$y = -\sqrt3{x}$

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": [ "Cube Root Transformations", "Function Reflections" ], "new_concepts": [ "Odd Function Symmetry", "Identical Graphs" ], "current_concepts": [ "Square Root Function Domain", "Cube Root Transformations", "Function Reflections", "Odd Function Symmetry", "Identical Graphs" ] } ## Analyze square root functions Using the *Square Root Function Domain* and *Function Reflections* knowledge points $$ \begin{aligned} &y = \sqrt{x} \quad (\text{Domain: } x \ge 0) \ &y = \sqrt{-x} \quad (\text{Domain: } x \le 0, \text{ reflection of } \sqrt{x} \text{ across the y-axis}) \ &y = -\sqrt{x} \quad (\text{Domain: } x \ge 0, \text{ reflection of } \sqrt{x} \text{ across the x-axis}) \end{aligned} $$ None of these square root graphs are identical because they have different domains or ranges. ## Analyze cube root functions Using the *Cube Root Transformations* and *Function Reflections* knowledge points $$ \begin{aligned} &y = \sqrt[3]{x} \quad (\text{Domain: } \mathbb{R}) \ &y = \sqrt[3]{-x} \quad (\text{Domain: } \mathbb{R}, \text{ reflection across the y-axis}) \ &y = -\sqrt[3]{x} \quad (\text{Domain: } \mathbb{R}, \text{ reflection across the x-axis}) \end{aligned} $$ ## Compare the cube root transformations Using the *Cube Root Transformations* and *Function Reflections* knowledge points $$ \begin{aligned} &\sqrt[3]{-x} = \sqrt[3]{-1 \cdot x} = \sqrt[3]{-1} \cdot \sqrt[3]{x} = -\sqrt[3]{x} \ &\text{Therefore, } y = \sqrt[3]{-x} \text{ and } y = -\sqrt[3]{x} \text{ represent the exact same function.} \end{aligned} $$ Thus, their graphs are identical.

$y = \sqrt{x}$

$y = \sqrt[3]{x}$

$y = \sqrt{-x}$

$y = \sqrt[3]{-x}$

$y = -\sqrt{x}$

$y = -\sqrt[3]{x}$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Cube Root Transformations" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze square root functions

Analyze cube root functions

Compare the cube root transformations

Answer: