LEARN · EXPLAIN · REVISE

Interpret x^(2/3) as the square of the real cube root. Find its domain, range and parity.

Read the idea, work independently, then explain what changed.

高一必修第一册(A版).pdf · 3.3 · PDF 96 / printed page 89

TOPIC 01

Power functions

Build understanding of power functions through definitions, contrasting cases and justified applications.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Transfer#Your turn

f(x)=(\sqrt[3]x)^2

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

The cube root accepts every real input.

Hint 2

Squaring removes the sign of the cube root.

Worked solution

The cube root accepts every real input.
Calculate or simplify this relation.
$f(-x)=(-\sqrt[3]x)^2=f(x);\quad y\ge0\Rightarrow x=y^{3/2},\quad f(x)=y$
The displayed preimage attains each nonnegative output; specify the real-root interpretation for negative inputs.

Domain R; range [0,∞); even.

Checks and common pitfalls: The displayed preimage attains each nonnegative output; specify the real-root interpretation for negative inputs.

Reasoning checklist · self / teacher assessment

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

Interpret and use power functions with domain checks.
Fixed exponent: A power function has the form x to a fixed exponent with its real domain.
$y=x^a$
Domains and order: Negative, fractional and integer exponents have different restrictions and behaviours.
Close with: conditions → representation → reasoning → check.

Expected reasoning: A power function has the form x to a fixed exponent with its real domain.
Expected reasoning: Negative, fractional and integer exponents have different restrictions and behaviours.
Expected correction: A negative exponent does not mean a negative output.