Kenneth ChengMathematical modeling
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Power functions

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

Revisit first: Basic properties of functions

TOPIC 01

Power functions

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

What you will be able to explain

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

Fixed exponent

A power function has the form x to a fixed exponent with its real domain.

y=xay=x^a

Domains and order

Negative, fractional and integer exponents have different restrictions and behaviours.

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 power 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.

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

Cubic 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

Evaluate the power.

232^3
  • Use the domain, units and sampling assumptions stated in the question.
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
A positive integer exponent means repeated multiplication.
Worked solution

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

  2. Calculate or simplify this relation.

    23=2×2×2=82^3=2\times2\times2=8

  3. Do not replace a power by base times exponent.

The requested value is 8.

Checks and common pitfalls: Do not replace a power by base times exponent.

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

02 / Standard#Worked example

Give the real domain and range.

f(x)=x−2f(x)=x^{-2}
  • Use the domain, units and sampling assumptions stated in the question.
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

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

  2. Calculate or simplify this relation.

    x−2=1/x2>0(x≠0)x^{-2}=1/x^2>0\quad(x\ne0)

  3. Calculate or simplify this relation.

    y>0⇒x=1/y≠0,f(x)=yy>0\Rightarrow x=1/\sqrt y\ne0,\quad f(x)=y

  4. 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
  • 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

Solve the power comparison for positive x.

x−1>2x^{-1}>2
  • Use the domain, units and sampling assumptions stated in the question.
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
Multiply by x only after noting x>0.
Worked solution

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

  2. Calculate or simplify this relation.

    1>2x⇒0<x<121>2x\Rightarrow0<x<\frac1{2}

  3. A reciprocal power decreases on the positive half-line.

0<x<1/2.

Checks and common pitfalls: A reciprocal power decreases on the positive half-line.

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

Evaluate the power.

333^3
  • Use the domain, units and sampling assumptions stated in the question.
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
A positive integer exponent means repeated multiplication.
Worked solution

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

  2. Calculate or simplify this relation.

    33=3×3×3=273^3=3\times3\times3=27

  3. Do not replace a power by base times exponent.

The requested value is 27.

Checks and common pitfalls: Do not replace a power by base times exponent.

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

05 / Foundation#Your turn

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

f(x)=x1/2f(x)=x^{1/2}
  • 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 principal real square root.
Hint 2
Show that every nonnegative output has a preimage.
Worked solution

  1. Use the principal real square root.

  2. Calculate or simplify this relation.

    x≥0;y=x≥0;x=y2x\ge0;\quad y=\sqrt x\ge0;\quad x=y^2

  3. 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
  • 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

Evaluate the real cube root.

(−27)1/3(-27)^{1/3}
  • Use the domain, units and sampling assumptions stated in the question.
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
An odd root preserves the sign.
Worked solution

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

  2. Calculate or simplify this relation.

    (−3)3=−27(-3)^3=-27

  3. Unlike an even root, a real odd root allows negative inputs.

The requested value is -3.

Checks and common pitfalls: Unlike an even root, a real odd root allows negative inputs.

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

07 / Foundation#Your turn

For 0<x<1, compare x² and x³ and prove the comparison.

0<x<10<x<1
  • Use the domain, units and sampling assumptions stated in the question.
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
Factor their difference.
Worked solution

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

  2. Calculate or simplify this relation.

    x2−x3=x2(1−x)>0x^2-x^3=x^2(1-x)>0

  3. For bases between zero and one, larger positive exponents give smaller values.

x³<x².

Checks and common pitfalls: For bases between zero and one, larger positive exponents give smaller values.

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.

08 / Standard#Your turn

Classify the parity of this power function on the reals.

f(x)=x6f(x)=x^{6}
  • Use the domain, units and sampling assumptions stated in the question.
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
An even number of negative factors gives a positive product.
Worked solution

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

  2. Calculate or simplify this relation.

    f(−x)=(−x)6=x6=f(x)f(-x)=(-x)^{6}=x^{6}=f(x)

  3. The exponent is an even integer and the domain is symmetric.

Even.

Checks and common pitfalls: The exponent is an even integer and the domain is symmetric.

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

Find m so that this is a power function with coefficient one.

f(x)=(m−3)x3f(x)=(m-3)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
Identify the exponent, its real domain and how it changes order.
Hint 2
The standard form is x raised to a fixed real power.
Worked solution

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

  2. Calculate or simplify this relation.

    m−3=1⇒m=4m-3=1\Rightarrow m=4

  3. A scalar multiple is not the stated coefficient-one power-function form.

The requested value is 4.

Checks and common pitfalls: A scalar multiple is not the stated coefficient-one power-function form.

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

10 / Standard#Your turn

Solve the power comparison for positive x.

x−1>3x^{-1}>3
  • Use the domain, units and sampling assumptions stated in the question.
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
Multiply by x only after noting x>0.
Worked solution

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

  2. Calculate or simplify this relation.

    1>3x⇒0<x<131>3x\Rightarrow0<x<\frac1{3}

  3. A reciprocal power decreases on the positive half-line.

0<x<1/3.

Checks and common pitfalls: A reciprocal power decreases on the positive half-line.

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

Evaluate the power.

434^3
  • Use the domain, units and sampling assumptions stated in the question.
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
A positive integer exponent means repeated multiplication.
Worked solution

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

  2. Calculate or simplify this relation.

    43=4×4×4=644^3=4\times4\times4=64

  3. Do not replace a power by base times exponent.

The requested value is 64.

Checks and common pitfalls: Do not replace a power by base times exponent.

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

12 / Transfer#Your turn

Interpret x^(2/3) as the square of the real cube root. Find its domain, range and parity.

f(x)=(x3)2f(x)=(\sqrt[3]x)^2
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
The cube root accepts every real input.
Hint 2
Squaring removes the sign of the cube root.
Worked solution

  1. The cube root accepts every real input.

  2. Calculate or simplify this relation.

    f(−x)=(−x3)2=f(x);y≥0⇒x=y3/2,f(x)=yf(-x)=(-\sqrt[3]x)^2=f(x);\quad y\ge0\Rightarrow x=y^{3/2},\quad f(x)=y

  3. The displayed preimage attains each nonnegative output; specify the real-root interpretation for negative inputs.

Domain R; range [0,∞); even.

Checks and common pitfalls: The displayed preimage attains each nonnegative output; specify the real-root interpretation for negative 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.

13 / Transfer#Your turn

Evaluate the real cube root.

(−64)1/3(-64)^{1/3}
  • Use the domain, units and sampling assumptions stated in the question.
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
An odd root preserves the sign.
Worked solution

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

  2. Calculate or simplify this relation.

    (−4)3=−64(-4)^3=-64

  3. Unlike an even root, a real odd root allows negative inputs.

The requested value is -4.

Checks and common pitfalls: Unlike an even root, a real odd root allows negative inputs.

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 power functions?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

    • 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=xay=x^a
    • Domains and order: Negative, fractional and integer exponents have different restrictions and behaviours.
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • 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.

    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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    Curriculum and source notes ↗