LEARN · EXPLAIN · REVISE

Is f(x)=1/x odd on its natural domain? Justify the domain condition.

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

高一必修第一册(A版).pdf · 3.2 · PDF 83 / printed page 76

TOPIC 01

Basic properties of functions

Build understanding of basic properties of 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#Your turn

f(x)=1/x,\quad x\ne0

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Check both symmetry of the domain and the algebraic identity.

Hint 2

An odd function need not be defined at zero.

Worked solution

Check both symmetry of the domain and the algebraic identity.
Calculate or simplify this relation.
$f(-x)=1/(-x)=-f(x);\quad x\ne0\Rightarrow -x\ne0$
The punctured real line remains symmetric about zero.

Yes.

Checks and common pitfalls: The punctured real line remains symmetric about zero.

Reasoning checklist · self / teacher assessment

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

Teacher preparation and assessment

Expected reasoning: Compare outputs at any ordered pair of domain inputs.
Expected reasoning: The domain must be symmetric before testing f(−x)=±f(x).
Expected correction: A vertex outside the allowed interval cannot give its minimum.