Unit 01 · Lesson 05 · Open-Age

Inverse Functions & One-to-One Relationships

An inverse reverses a one-to-one function. Swap input and output, solve for the new output, and verify by composition. A restricted domain may be needed before an inverse is a function.

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Learning objectives

Explain the central structure of inverse functions & one-to-one relationships using precise mathematical language.
Apply the governing relationship and verify results with an alternate representation or substitution.
State domain, unit, precision, and context restrictions whenever they affect the result.
Analyze a misconception and communicate a correction with evidence.

Essential question: How do notation, transformations, composition, and inverse thinking reveal the structure and behavior of functions?

Governing relationship

f(f⁻¹(x))=x on the appropriate domain.

An inverse reverses a one-to-one function. Swap input and output, solve for the new output, and verify by composition. A restricted domain may be needed before an inverse is a function.

Worked examples

Read each example, close it, and reconstruct the reasoning independently. Then verify by substitution, graph, table, or alternate method.

Find the inverse of f(x)=3x−7.Let y=3x−7. Swap: x=3y−7. Solve: y=(x+7)/3, so f⁻¹(x)=(x+7)/3.

Why does f(x)=x² need restriction?Without restriction, positive outputs have two inputs. Restrict x≥0 or x≤0 before inverting.

Guided and independent practice

Find the inverse of (x−4)/5.
Verify a pair of inverse functions by composition.
Restrict and invert y=(x+2)².
Create one original example involving inverse functions & one-to-one relationships and solve it completely.
Use the Khaemenes Scientific Calculator to verify one result, then document the setup and reasonableness check.
Write a one-paragraph explanation connecting symbolic, graphical, numerical, and contextual representations.

Differentiated pathways

Foundation

Review prerequisite vocabulary, annotate each worked step, complete the first four prompts, and use a table or graph to support symbolic work.

Core

Complete every prompt, explain domain and verification, correct errors, and submit the applied task with clear mathematical communication.

Extended

Generalize the relationship, compare methods, use technology strategically, analyze sensitivity, and create a proof, counterexample, or advanced model.

Application laboratory

Create a conversion function and its inverse, then explain how units confirm the reversal.

Required evidence: defined quantities and units; at least two representations; transparent calculations; verification; one limitation; and a defensible conclusion.

Misconception and correction

Common misconception: Writing 1/f(x) instead of f⁻¹(x). Reciprocal and inverse function are different ideas.

Write the first invalid step, the governing rule that corrects it, and one counterexample or verification that demonstrates the correction.

Mastery evidence

Accurate symbolic work with preserved equivalence.
Correct domain, interval, unit, and precision statements.
At least one independent verification.
A correction explaining why an incorrect approach fails.
A clear application conclusion that distinguishes mathematical result from recommendation.

Reflection

What structure did you recognize? Which representation was most useful? What assumption or restriction mattered most? What will you do differently on the next problem?