BetterGrades Precalculus · Unit 4 · Lesson

One-to-one functions

Determine whether a function is one-to-one and explain why one-to-one behavior is required for an inverse function.

Opening

Start with the situation

A one-to-one function gives each output at most one input, making reverse assignment possible.

Many real models are built in stages—a conversion followed by a cost rule, or a measurement followed by a calibration. Composition records that order, while inverse reasoning asks whether the stages can be undone.

Before you begin

Prerequisite check

Evaluate function notation.
Determine domains from formulas.
Solve equations for a selected variable.

Core explanation

Explanation

Check repeated outputs, use the horizontal-line test, or prove $f(a)=f(b)$ implies $a=b$ .

A function that fails globally may be restricted to a monotonic interval.

A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through horizontal-line test, reverse mapping failure, or another equivalent representation.

Conceptual reading

What the idea is really doing

Composition and inversion are about information flow. A composite sends an input through stages in a fixed order; an inverse reverses that flow only when each output identifies a unique input.

This lesson narrows that lens to one goal: determine whether a function is one-to-one and explain why one-to-one behavior is required for an inverse function. The point is not to memorize an isolated trick; it is to know what evidence makes the conclusion valid and how a second representation can check it.

Reusable method

A reliable route through the problem

Check repeated outputs.
Use the horizontal-line test.
Or prove $f(a)=f(b)$ implies $a=b$ .

Verification: Name the intermediate quantity, enforce its domain, and verify an inverse with composition. Units are especially useful because the output unit of one stage must match the input unit of the next.

Foundation walkthrough

Plan before calculating

Problem

Is $x^2$ one-to-one on all real numbers?

Plan: Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Check repeated outputs, use the horizontal-line test, or prove $f(a)=f(b)$ implies $a=b$ .
Conclusion: No; $x$ and -x share outputs. Restrict to $x\ge 0$ or $x\le 0$ .
Why the check works: A branch restriction restores reverse uniqueness.

Worked examples

See the idea in three forms

foundation example

Is $x^2$ one-to-one on all real numbers?

SolutionNo; $x$ and -x share outputs. Restrict to $x\ge 0$ or $x\le 0$ .

A branch restriction restores reverse uniqueness.

representation example

Is $x^3$ one-to-one?

SolutionYes.

This example expresses one-to-one functions in a second form.

transfer example

Horizontal-line test.

SolutionEvery horizontal line meets at most once.

A function that fails globally may be restricted to a monotonic interval.

Horizontal-line test. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: A branch restriction restores reverse uniqueness. — **Anchor figure · Horizontal-line test**
Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: A branch restriction restores reverse uniqueness.

Reverse mapping failure. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for one-to-one functions. — **Mechanism figure · Reverse mapping failure**
Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for one-to-one functions.

Quadratic branch restrictions. Compare the valid path with the tempting shortcut. The figure shows why confusing the ordinary function condition with one-to-one behavior leads to a false conclusion. — **Comparison and error figure · Quadratic branch restrictions**
Compare the valid path with the tempting shortcut. The figure shows why confusing the ordinary function condition with one-to-one behavior leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is confusing the ordinary function condition with one-to-one behavior.

Check yourself

Is $2x+1$ one-to-one?

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice

Ten concrete questions

Practice 101

Is $2x+1$ one-to-one?

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 202

Is $x^3$ one-to-one?

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 303

Horizontal-line test.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 404

Can a constant function be one-to-one on multiple inputs?

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 505

Explain why this conclusion is valid: No; $x$ and -x share outputs. Restrict to $x\ge 0$ or $x\le 0$ . Use the foundation problem as evidence: Is $x^2$ one-to-one on all real numbers?

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 606

Solve the representation example, then name the feature of one-to-one functions that it illustrates: Is $x^3$ one-to-one?

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 707

Correct this reasoning and identify the first unsafe assumption: A frequent error is confusing the ordinary function condition with one-to-one behavior.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 808

Connect two representations for this example: Is $x^2$ one-to-one on all real numbers? Describe what a graph, table, mapping, or algebraic form would have to show.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 909

Create a nearby example by changing one number or condition in this prompt: Horizontal-line test. Predict the effect, solve your new example, and compare it with the original.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 1010

Write a short verification checklist for one-to-one functions, then apply it to one worked example from this lesson.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Lesson close

Connect forward

The next lesson, Constructing inverse functions, uses this result as part of a larger structure.

Source record

Original BetterGrades manuscript, rights-separated references.

Lippman and Rasmussen, Precalculus Volume 1
Utah College Algebra
AP Precalculus framework

No long source passage is reproduced.

Is x2x^2x2 one-to-one on all real numbers?

Is x2x^2x2 one-to-one on all real numbers?

Is x3x^3x3 one-to-one?

Horizontal-line test.

Original BetterGrades manuscript, rights-separated references.

Is $x^2$ one-to-one on all real numbers?

Is $x^2$ one-to-one on all real numbers?

Is $x^3$ one-to-one?