LEARN · EXPLAIN · REVISE

Give the real domain and range.

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 / Standard#Worked example

f(x)=x^{-2}

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Identify the exponent, its real domain and how it changes order.

Hint 2

Rewrite as a reciprocal.

Worked solution

Identify the exponent, its real domain and how it changes order.
Calculate or simplify this relation.
$x^{-2}=1/x^2>0\quad(x\ne0)$
Calculate or simplify this relation.
$y>0\Rightarrow x=1/\sqrt y\ne0,\quad f(x)=y$
Every positive output is attained by the displayed nonzero input; zero is excluded as both input and output.

Domain: x≠0; range: y>0.

Checks and common pitfalls: Every positive output is attained by the displayed nonzero input; zero is excluded as both input and output.

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.