BetterGrades Precalculus · Unit 2 · Lesson

Relations and functions

Determine whether a relation is a function by applying the rule that each allowed input has exactly one output.

Opening

Start with the situation

A relation is a function when every allowed input has exactly one output. Several inputs may share an output.

The same dependency may arrive as a story, table, graph, or formula. Learning to preserve the inputs, outputs, units, and domain while moving among those forms is the central language of the course.

Before you begin

Prerequisite check

Use function notation from the algebra and function readiness unit.
Read ordered pairs and interval notation.
Attach units to contextual quantities.

Core explanation

Explanation

Check repeated inputs in a table, outgoing arrows in a mapping, vertical intersections in a graph, or the rule stated by a context.

The function condition applies only to inputs in the domain. One-to-one behavior is a separate stronger property.

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

Conceptual reading

What the idea is really doing

A function is a dependency, not merely an equation. Inputs, outputs, units, and domain must agree in words, tables, graphs, and formulas; each representation should tell the same mathematical story.

This lesson narrows that lens to one goal: determine whether a relation is a function by applying the rule that each allowed input has exactly one output. 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 inputs in a table.
Outgoing arrows in a mapping.
Vertical intersections in a graph.
Or the rule stated by a context.

Verification: Choose two representations and make them verify one another. A table can test a formula, a graph can expose a domain or range claim, and units can reveal a model that is algebraically neat but conceptually wrong.

Foundation walkthrough

Plan before calculating

Problem

Does $(0,3),(1,5),(2,5),(3,8)$ define a function?

Plan: Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Check repeated inputs in a table, outgoing arrows in a mapping, vertical intersections in a graph, or the rule stated by a context.
Conclusion: Yes. Every input has one output.
Why the check works: Repeated output $5$ is allowed.

Worked examples

See the idea in three forms

foundation example

Does $(0,3),(1,5),(2,5),(3,8)$ define a function?

SolutionYes. Every input has one output.

Repeated output $5$ is allowed.

representation example

Is $y=x^2$ a function of $x$ ?

SolutionYes.

This example expresses relations and functions in a second form.

transfer example

Give a function not one-to-one.

SolutionFor example $y=x^2$ .

The function condition applies only to inputs in the domain. One-to-one behavior is a separate stronger property.

Mapping comparison. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: Repeated output 5 is allowed. — **Anchor figure · Mapping comparison**
Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: Repeated output $5$ is allowed.

Vertical-line derivation. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for relations and functions. — **Mechanism figure · Vertical-line derivation**
Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for relations and functions.

Discrete and continuous functions. Compare the valid path with the tempting shortcut. The figure shows why rejecting a many-to-one function because an output repeats leads to a false conclusion. — **Comparison and error figure · Discrete and continuous functions**
Compare the valid path with the tempting shortcut. The figure shows why rejecting a many-to-one function because an output repeats leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is rejecting a many-to-one function because an output repeats.

Check yourself

Is $(1,2),(2,3),(1,4)$ a function?

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 $(1,2),(2,3),(1,4)$ a function?

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 $y=x^2$ a function of $x$ ?

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

Give a function not 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 404

Why does a vertical line fail as a function of $x$ ?

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: Yes. Every input has one output. Use the foundation problem as evidence: Does $(0,3),(1,5),(2,5),(3,8)$ define a function?

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 relations and functions that it illustrates: Is $y=x^2$ a function of $x$ ?

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 rejecting a many-to-one function because an output repeats.

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: Does $(0,3),(1,5),(2,5),(3,8)$ define a function? 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: Give a function not one-to-one. 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 relations and 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, Function notation, evaluation, and solving, uses this result as part of a larger structure.

Source record

Original BetterGrades manuscript, rights-separated references.

Yoshiwara, Modeling, Functions, and Graphs
Lippman and Rasmussen, Precalculus Volume 1
Stitz and Zeager, Precalculus

No long source passage is reproduced.

Does (0,3),(1,5),(2,5),(3,8)(0,3),(1,5),(2,5),(3,8)(0,3),(1,5),(2,5),(3,8) define a function?

Does (0,3),(1,5),(2,5),(3,8)(0,3),(1,5),(2,5),(3,8)(0,3),(1,5),(2,5),(3,8) define a function?

Is y=x2y=x^2y=x2 a function of xxx?

Give a function not one-to-one.

Original BetterGrades manuscript, rights-separated references.

Does $(0,3),(1,5),(2,5),(3,8)$ define a function?

Does $(0,3),(1,5),(2,5),(3,8)$ define a function?

Is $y=x^2$ a function of $x$ ?