← Senior Mathematics Studio

LEARN · EXPLAIN · REVISE

Classify parity on the real line.

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

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

Revisit first: Function concept and representations

TOPIC 01

Basic properties of functions

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

What you will be able to explain

Interpret and use basic properties of functions with domain checks.
Connect representations and justify the steps, including boundary cases.
Explain a solution and apply the idea to a changed situation.

Try it. Leave your reasoning visible.

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

01 / Foundation#Your turn

Classify parity on the real line.

f(x)=x^3+3x

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the definition of the property and keep the domain in view.

Hint 2

Replace every x by −x.

Worked solution

Use the definition of the property and keep the domain in view.
Calculate or simplify this relation.
$f(-x)=-x^3-3x=-f(x)$
The domain is symmetric about zero.

Odd.

Checks and common pitfalls: The domain is symmetric about zero.

Reasoning checklist · self / teacher assessment

State the relevant definition, condition or model.
Show a valid calculation, proof or counterexample.
Interpret the conclusion with its restrictions.

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

Teacher preparation and assessment

Question sequence

What must be true before using the main rule for basic properties of functions?
Which representation makes this task easier, and why?
Change one assumption. Does the conclusion survive?

Board plan

Interpret and use basic properties of functions with domain checks.
Monotonicity: Compare outputs at any ordered pair of domain inputs.
Parity: The domain must be symmetric before testing f(−x)=±f(x).
Close with: conditions → representation → reasoning → check.

Anticipated thinking

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.

Assessment checklist

1: identify the givens and required quantity.
1: choose a valid definition, representation or method.
1: present connected, correct reasoning.
1: check conditions and explain the result.