Kenneth ChengMathematical modeling
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Inverse functions and domain restrictions

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

2027 JM02 考試大綱 · 1. Functions · PDF 2 / printed page 2

Revisit first: Function concept and representationsBasic properties of functions

TOPIC 01

Inverse functions and domain restrictions

Build understanding of inverse functions and domain restrictions through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Choose an equivalent method and explain exclusions before calculating.
  • Connect representations and justify the steps, including boundary cases.
  • Explain a solution and apply the idea to a changed situation.

Undoing a rule

An inverse exchanges the original domain and range and requires a one-to-one rule.

f−1(f(x))=xf^{-1}(f(x))=x

Choosing a branch

Restricting a quadratic to either side of its vertex creates an invertible rule.

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 inverse functions and domain restrictions 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.

Restrict y=(x−0)²+0 to x≥0. Its inverse is y=0+√(x−0), x≥0. The graphs reflect in y=x; their domain and range exchange.

Restrict y=(x−0)²+0 to x≥0. Its inverse is y=0+√(x−0), x≥0. The graphs reflect in y=x; their domain and range exchange.

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

Find the inverse of the linear function.

f(x)=2x+3f(x)=2x+3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Specify a one-to-one domain, solve for the input, then exchange the variables.
Hint 2
Undo addition before dividing by the nonzero slope.
Worked solution

  1. Specify a one-to-one domain, solve for the input, then exchange the variables.

  2. Calculate or simplify this relation.

    y=2x+3  ⟺  x=(y−3)/2y=2x+3\iff x=(y-3)/2

  3. An inverse function is not the reciprocal 1/f(x).

f⁻¹(x)=(x−3)/2.

Checks and common pitfalls: An inverse function is not the reciprocal 1/f(x).

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

Find the inverse and its domain.

f(x)=1x−2f(x)=\frac{1}{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
Specify a one-to-one domain, solve for the input, then exchange the variables.
Hint 2
The original function never outputs zero.
Worked solution

  1. Specify a one-to-one domain, solve for the input, then exchange the variables.

  2. Calculate or simplify this relation.

    y(x−2)=1;x=2+1/y;y≠0y(x-2)=1;\quad x=2+1/y;\quad y\ne0

  3. The original range becomes the inverse domain.

f⁻¹(x)=2+1/x, x≠0.

Checks and common pitfalls: The original range becomes the inverse 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.

03 / Transfer#Worked example

Find the inverse of the fractional linear function with all exclusions.

f(x)=x+2x−1f(x)=\frac{x+2}{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
Specify a one-to-one domain, solve for the input, then exchange the variables.
Hint 2
Solve the equation before noticing that this rule is its own inverse.
Worked solution

  1. Specify a one-to-one domain, solve for the input, then exchange the variables.

  2. Calculate or simplify this relation.

    yx−y=x+2;x(y−1)=y+2;x=y+2y−1yx-y=x+2;\quad x(y-1)=y+2;\quad x=\frac{y+2}{y-1}

  3. The output 1 would require 2=−1, which is false.

f⁻¹(x)=(x+2)/(x−1), x≠1.

Checks and common pitfalls: The output 1 would require 2=−1, which is false.

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

Find the inverse of the linear function.

f(x)=3x+3f(x)=3x+3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Specify a one-to-one domain, solve for the input, then exchange the variables.
Hint 2
Undo addition before dividing by the nonzero slope.
Worked solution

  1. Specify a one-to-one domain, solve for the input, then exchange the variables.

  2. Calculate or simplify this relation.

    y=3x+3  ⟺  x=(y−3)/3y=3x+3\iff x=(y-3)/3

  3. An inverse function is not the reciprocal 1/f(x).

f⁻¹(x)=(x−3)/3.

Checks and common pitfalls: An inverse function is not the reciprocal 1/f(x).

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

Find the inverse and its domain.

f(x)=1x−3f(x)=\frac{1}{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
Specify a one-to-one domain, solve for the input, then exchange the variables.
Hint 2
The original function never outputs zero.
Worked solution

  1. Specify a one-to-one domain, solve for the input, then exchange the variables.

  2. Calculate or simplify this relation.

    y(x−3)=1;x=3+1/y;y≠0y(x-3)=1;\quad x=3+1/y;\quad y\ne0

  3. The original range becomes the inverse domain.

f⁻¹(x)=3+1/x, x≠0.

Checks and common pitfalls: The original range becomes the inverse 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.

06 / Foundation#Your turn

Find the inverse on the stated restricted domain.

f(x)=(x−3)2;x≥3f(x)=(x-3)^2;\quad x\ge3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Specify a one-to-one domain, solve for the input, then exchange the variables.
Hint 2
The domain selects the positive square-root branch.
Worked solution

  1. Specify a one-to-one domain, solve for the input, then exchange the variables.

  2. Calculate or simplify this relation.

    x−3≥0;x=3+yx-3\ge0;\quad x=3+\sqrt y

  3. The unrestricted quadratic does not have an inverse function on the whole real line.

f⁻¹(x)=3+√x, x≥0.

Checks and common pitfalls: The unrestricted quadratic does not have an inverse function on the whole real 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.

07 / Foundation#Your turn

Evaluate the inverse composition at 5.

f(x)=3x+3;f−1(f(5))f(x)=3x+3;\quad f^{-1}(f(5))
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Specify a one-to-one domain, solve for the input, then exchange the variables.
Hint 2
An inverse undoes the original rule on its domain.
Worked solution

  1. Specify a one-to-one domain, solve for the input, then exchange the variables.

  2. Calculate or simplify this relation.

    f(5)=18;f−1(18)=5f(5)=18;\quad f^{-1}(18)=5

  3. Composition order matters outside the relevant domain and range.

The requested value is 5.

Checks and common pitfalls: Composition order matters outside the relevant domain and range.

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

08 / Standard#Your turn

Can f(x)=x² on the real line have an inverse function? Give a counterexample.

f:R→[0,∞);f(x)=x2f:\mathbb R\to[0,\infty);\quad f(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
Specify a one-to-one domain, solve for the input, then exchange the variables.
Hint 2
Find distinct inputs with the same output.
Worked solution

  1. Specify a one-to-one domain, solve for the input, then exchange the variables.

  2. Calculate or simplify this relation.

    1≠−1;f(1)=f(−1)1\ne-1;\quad f(1)=f(-1)

  3. Surjectivity onto the stated range alone does not give injectivity.

No: f(1)=f(−1)=1.

Checks and common pitfalls: Surjectivity onto the stated range alone does not give injectivity.

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 both inverse branches by choosing x≤3 or x≥3.

y=(x−3)2y=(x-3)^2
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Specify a one-to-one domain, solve for the input, then exchange the variables.
Hint 2
A single restricted function uses one branch, not both.
Worked solution

  1. Specify a one-to-one domain, solve for the input, then exchange the variables.

  2. Calculate or simplify this relation.

    x−3=±yx-3=\pm\sqrt y

  3. Writing ± without a domain does not define an inverse function.

x≤3: f⁻¹(y)=3−√y; x≥3: f⁻¹(y)=3+√y, y≥0.

Checks and common pitfalls: Writing ± without a domain does not define an inverse function.

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

Find the inverse of the fractional linear function with all exclusions.

f(x)=x+3x−1f(x)=\frac{x+3}{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
Specify a one-to-one domain, solve for the input, then exchange the variables.
Hint 2
Solve the equation before noticing that this rule is its own inverse.
Worked solution

  1. Specify a one-to-one domain, solve for the input, then exchange the variables.

  2. Calculate or simplify this relation.

    yx−y=x+3;x(y−1)=y+3;x=y+3y−1yx-y=x+3;\quad x(y-1)=y+3;\quad x=\frac{y+3}{y-1}

  3. The output 1 would require 3=−1, which is false.

f⁻¹(x)=(x+3)/(x−1), x≠1.

Checks and common pitfalls: The output 1 would require 3=−1, which is false.

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

Find the inverse of the linear function.

f(x)=4x+3f(x)=4x+3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Specify a one-to-one domain, solve for the input, then exchange the variables.
Hint 2
Undo addition before dividing by the nonzero slope.
Worked solution

  1. Specify a one-to-one domain, solve for the input, then exchange the variables.

  2. Calculate or simplify this relation.

    y=4x+3  ⟺  x=(y−3)/4y=4x+3\iff x=(y-3)/4

  3. An inverse function is not the reciprocal 1/f(x).

f⁻¹(x)=(x−3)/4.

Checks and common pitfalls: An inverse function is not the reciprocal 1/f(x).

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

Find the inverse and its domain.

f(x)=1x−4f(x)=\frac{1}{x-4}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Specify a one-to-one domain, solve for the input, then exchange the variables.
Hint 2
The original function never outputs zero.
Worked solution

  1. Specify a one-to-one domain, solve for the input, then exchange the variables.

  2. Calculate or simplify this relation.

    y(x−4)=1;x=4+1/y;y≠0y(x-4)=1;\quad x=4+1/y;\quad y\ne0

  3. The original range becomes the inverse domain.

f⁻¹(x)=4+1/x, x≠0.

Checks and common pitfalls: The original range becomes the inverse 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.

13 / Transfer#Your turn

Find the inverse on the stated restricted domain.

f(x)=(x−4)2;x≥4f(x)=(x-4)^2;\quad x\ge4
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Specify a one-to-one domain, solve for the input, then exchange the variables.
Hint 2
The domain selects the positive square-root branch.
Worked solution

  1. Specify a one-to-one domain, solve for the input, then exchange the variables.

  2. Calculate or simplify this relation.

    x−4≥0;x=4+yx-4\ge0;\quad x=4+\sqrt y

  3. The unrestricted quadratic does not have an inverse function on the whole real line.

f⁻¹(x)=4+√x, x≥0.

Checks and common pitfalls: The unrestricted quadratic does not have an inverse function on the whole real 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.

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

    Board plan

    • Choose an equivalent method and explain exclusions before calculating.
    • Undoing a rule: An inverse exchanges the original domain and range and requires a one-to-one rule.
      f−1(f(x))=xf^{-1}(f(x))=x
    • Choosing a branch: Restricting a quadratic to either side of its vertex creates an invertible rule.
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: An inverse exchanges the original domain and range and requires a one-to-one rule.
    • Expected reasoning: Restricting a quadratic to either side of its vertex creates an invertible rule.
    • Expected correction: A candidate produced by algebra may violate the original domain or sign condition.

    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 ↗