Kenneth ChengMathematical modeling
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Basic properties of functions

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

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.

Monotonicity

Compare outputs at any ordered pair of domain inputs.

Parity

The domain must be symmetric before testing f(−x)=±f(x).

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Before calculating, predict how the conclusion changes when one defining condition in basic properties of functions changes. Record a reason.

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

Quadratic graph: horizontal shift 0, vertical shift 0. Input is replaced by x−0.

Quadratic graph: horizontal shift 0, vertical shift 0. Input is replaced by x−0.

Explain: Compare two admissible cases and one boundary or invalid case. Explain the observed difference using the stated definition.

Transfer: Construct a new example and a tempting incorrect solution. Repair the solution by naming the missing condition.

Try it. Leave your reasoning visible.

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

01 / Foundation#Worked example

State the monotonicity on the real line and justify it.

f(x)=−2x+2f(x)=-2x+2
  • 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
The slope is negative.
Worked solution

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

  2. Calculate or simplify this relation.

    x1<x2⇒f(x2)−f(x1)=−2(x2−x1)<0x_1<x_2\Rightarrow f(x_2)-f(x_1)=-2(x_2-x_1)<0

  3. The definition compares the outputs at ordered inputs.

Strictly decreasing.

Checks and common pitfalls: The definition compares the outputs at ordered inputs.

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.

02 / Standard#Worked example

Classify parity on the real line.

f(x)=x3+2xf(x)=x^3+2x
  • 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

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

  2. Calculate or simplify this relation.

    f(−x)=−x3−2x=−f(x)f(-x)=-x^3-2x=-f(x)

  3. 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.

03 / Transfer#Worked example

Is the function even on its stated domain? Explain.

f(x)=x2;x∈[0,2]f(x)=x^2;\quad x\in[0,2]
  • 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
A parity claim requires both x and −x in the domain.
Worked solution

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

  2. Calculate or simplify this relation.

    1∈[0,2],−1∉[0,2]1\in[0,2],\quad -1\notin[0,2]

  3. The algebraic expression alone does not determine parity.

No: the domain is not symmetric about zero.

Checks and common pitfalls: The algebraic expression alone does not determine parity.

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.

04 / Foundation#Your turn

State the monotonicity on the real line and justify it.

f(x)=−3x+2f(x)=-3x+2
  • 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
The slope is negative.
Worked solution

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

  2. Calculate or simplify this relation.

    x1<x2⇒f(x2)−f(x1)=−3(x2−x1)<0x_1<x_2\Rightarrow f(x_2)-f(x_1)=-3(x_2-x_1)<0

  3. The definition compares the outputs at ordered inputs.

Strictly decreasing.

Checks and common pitfalls: The definition compares the outputs at ordered inputs.

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.

05 / Foundation#Your turn

Classify parity on the real line.

f(x)=x3+3xf(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

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

  2. Calculate or simplify this relation.

    f(−x)=−x3−3x=−f(x)f(-x)=-x^3-3x=-f(x)

  3. 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.

06 / Foundation#Your turn

An even function has the stated value. Find the reflected value.

f(3)=8;f(−3)f(3)=8;\quad f(-3)
  • 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
Use f(−x)=f(x).
Worked solution

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

  2. Calculate or simplify this relation.

    f(−3)=f(3)=8f(-3)=f(3)=8

  3. Even symmetry reflects the graph across the vertical axis.

The requested value is 8.

Checks and common pitfalls: Even symmetry reflects the graph across the vertical axis.

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

07 / Foundation#Your turn

Find the minimum on the stated interval.

f(x)=(x−3)2+2;x∈[4,6]f(x)=(x-3)^2+2;\quad x\in[4,6]
  • 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
The vertex lies outside this interval.
Worked solution

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

  2. Calculate or simplify this relation.

    x−3∈[1,3];f(x)≥12+2=3x-3\in[1,3];\quad f(x)\ge1^2+2=3

  3. The unrestricted vertex value cannot be used as the constrained minimum.

The requested value is 3.

Checks and common pitfalls: The unrestricted vertex value cannot be used as the constrained minimum.

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

08 / Standard#Your turn

How does adding a constant affect monotonicity? Prove your answer.

g(x)=f(x)+3g(x)=f(x)+3
  • 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
Compare two outputs by subtraction.
Worked solution

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

  2. Calculate or simplify this relation.

    g(x2)−g(x1)=f(x2)−f(x1)g(x_2)-g(x_1)=f(x_2)-f(x_1)

  3. The constant cancels from every output difference.

It preserves increasing or decreasing behaviour.

Checks and common pitfalls: The constant cancels from every output difference.

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.

09 / Standard#Your turn

Can a nonzero constant function be odd? Explain.

f(x)=3,x∈Rf(x)=3,\quad x\in\mathbb R
  • 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
Oddness would require c=−c.
Worked solution

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

  2. Calculate or simplify this relation.

    f(−x)=3=f(x)≠−f(x)f(-x)=3=f(x)\ne-f(x)

  3. Only the zero constant function is both even and odd on a symmetric domain.

No; it is even but not odd.

Checks and common pitfalls: Only the zero constant function is both even and odd on a symmetric domain.

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.

10 / Standard#Your turn

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

f(x)=1/x,x≠0f(x)=1/x,\quad x\ne0
  • Use the domain, units and sampling assumptions stated in the question.
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

  1. Check both symmetry of the domain and the algebraic identity.

  2. Calculate or simplify this relation.

    f(−x)=1/(−x)=−f(x);x≠0⇒−x≠0f(-x)=1/(-x)=-f(x);\quad x\ne0\Rightarrow -x\ne0

  3. 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
  • 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.

11 / Standard#Your turn

State the monotonicity on the real line and justify it.

f(x)=−4x+2f(x)=-4x+2
  • 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
The slope is negative.
Worked solution

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

  2. Calculate or simplify this relation.

    x1<x2⇒f(x2)−f(x1)=−4(x2−x1)<0x_1<x_2\Rightarrow f(x_2)-f(x_1)=-4(x_2-x_1)<0

  3. The definition compares the outputs at ordered inputs.

Strictly decreasing.

Checks and common pitfalls: The definition compares the outputs at ordered inputs.

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.

12 / Transfer#Your turn

Classify parity on the real line.

f(x)=x3+4xf(x)=x^3+4x
  • 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

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

  2. Calculate or simplify this relation.

    f(−x)=−x3−4x=−f(x)f(-x)=-x^3-4x=-f(x)

  3. 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.

13 / Transfer#Your turn

An even function has the stated value. Find the reflected value.

f(4)=9;f(−4)f(4)=9;\quad f(-4)
  • 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
Use f(−x)=f(x).
Worked solution

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

  2. Calculate or simplify this relation.

    f(−4)=f(4)=9f(-4)=f(4)=9

  3. Even symmetry reflects the graph across the vertical axis.

The requested value is 9.

Checks and common pitfalls: Even symmetry reflects the graph across the vertical axis.

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

Focus on one question

End-of-lesson check and correction

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    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.

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