LEARN · EXPLAIN · REVISE

Find the real domain and range of the square-root power function.

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 / Foundation#Your turn

f(x)=x^{1/2}

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the principal real square root.

Hint 2

Show that every nonnegative output has a preimage.

Worked solution

Use the principal real square root.
Calculate or simplify this relation.
$x\ge0;\quad y=\sqrt x\ge0;\quad x=y^2$
A nonnegative output bound alone does not establish the complete range without attainability.

Both domain and range are [0,∞).

Checks and common pitfalls: A nonnegative output bound alone does not establish the complete range without attainability.

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.