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Introduction
Relations and Functions helps you connect a mathematical rule to a result you can explain and check. Check the one-output-per-input requirement in a relation. Apply the same definition to words, tables, graphs, and equations. This lesson belongs to coordinate geometry and graphing and builds toward independent Algebra I problem solving.
Alignment uses 19 TAC §111.39(c). Review lessons sample their listed standards. Readiness and supporting labels describe STAAR emphasis, not whether a topic is required.
Learning objectives
- Identify the quantities and representations used in relations and functions.
- Apply the lesson's rule to solve a numerical or contextual problem.
- Explain each mathematical step using the correct vocabulary.
- Verify an answer and correct a likely error.
Prerequisite knowledge
Read ordered pairs, tables, and graphs.
Previous lesson: Tables of Values
Key mathematical vocabulary
- equation
- A statement that two expressions are equal.
- function
- A rule assigning one output to each permitted input.
- domain
- The set of permitted inputs.
- input
- A value supplied to a rule.
- output
- The result of a rule at an allowed input.
Formulas and rules
- a(b+c)=ab+ac
- Equivalent expressions agree for all allowed inputs.
- A solution satisfies every original condition.
Recognize a function in every representation
Check the one-output-per-input requirement in a relation. Apply the same definition to words, tables, graphs, and equations.
A function assigns exactly one output to each input in its domain. Different inputs may share an output. A relation fails to be a function only when at least one allowed input has more than one output.
In a table or set of pairs, inspect repeated inputs and compare their outputs. On a graph, the vertical-line test checks whether one x-coordinate corresponds to more than one y-coordinate. In symbols, y=x² defines a function of x, but x=y² generally does not without a restriction on y. A verbal rule must also give a unique output.
A method you can explain
- Read the question and identify the known information, unknown quantities, and any domain restrictions. Select a suitable representation and tool.
- Apply the relationship explained above. Look for repeated first coordinates, not repeated second coordinates.
- Write each calculation or logical inference on a separate line. Keep units and signs visible. When two methods are possible, compare which one uses the given information most directly.
- Check the result in the ORIGINAL conditions, not merely the last simplified line. The two outputs differ by 5, so they are genuinely different.
- Interpret what the answer means. Decide whether its size, sign, and precision make sense; identify a restriction or assumption that would change the conclusion.
Five worked examples
Example 1 · Beginner
Does the relation {(12, 20), (12, 25), (13, 20)} define y as a function of x?
- Look for repeated first coordinates, not repeated second coordinates.
- Input 12 is paired with 20 and 25, two different outputs.
- A function assigns exactly one output to each allowed input, so this relation fails the function definition.
Answer: No; one input has two outputs
Verify: The two outputs differ by 5, so they are genuinely different.
Avoid this mistake: Rejecting repeated outputs: repeated inputs with different outputs are the issue.
Example 2 · Basic
Does the relation {(13, 20), (13, 25), (14, 20)} define y as a function of x?
- Look for repeated first coordinates, not repeated second coordinates.
- Input 13 is paired with 20 and 25, two different outputs.
- A function assigns exactly one output to each allowed input, so this relation fails the function definition.
Answer: No; one input has two outputs
Verify: The two outputs differ by 5, so they are genuinely different.
Avoid this mistake: Using a horizontal-line test to decide whether a graph is a function.
Example 3 · Intermediate
Does the relation {(14, 20), (14, 25), (15, 20)} define y as a function of x?
- Look for repeated first coordinates, not repeated second coordinates.
- Input 14 is paired with 20 and 25, two different outputs.
- A function assigns exactly one output to each allowed input, so this relation fails the function definition.
Answer: No; one input has two outputs
Verify: The two outputs differ by 5, so they are genuinely different.
Avoid this mistake: Ignoring a stated domain restriction.
Example 4 · Advanced
Does the relation {(2, 21), (2, 26), (3, 21)} define y as a function of x?
- Look for repeated first coordinates, not repeated second coordinates.
- Input 2 is paired with 21 and 26, two different outputs.
- A function assigns exactly one output to each allowed input, so this relation fails the function definition.
Answer: No; one input has two outputs
Verify: The two outputs differ by 5, so they are genuinely different.
Avoid this mistake: Rejecting repeated outputs: repeated inputs with different outputs are the issue.
Example 5 · Challenge
Does the relation {(3, 21), (3, 26), (4, 21)} define y as a function of x?
- Look for repeated first coordinates, not repeated second coordinates.
- Input 3 is paired with 21 and 26, two different outputs.
- A function assigns exactly one output to each allowed input, so this relation fails the function definition.
Answer: No; one input has two outputs
Verify: The two outputs differ by 5, so they are genuinely different.
Avoid this mistake: Using a horizontal-line test to decide whether a graph is a function.
Common mistakes and corrections
- Rejecting repeated outputs: repeated inputs with different outputs are the issue.
- Using a horizontal-line test to decide whether a graph is a function.
- Ignoring a stated domain restriction.
Real-world applications
- Assign a single shipping price to each permitted package weight. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Check whether a student ID identifies one account. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Check the one-output-per-input requirement in a relation. Apply the same definition to words, tables, graphs, and equations.
A function assigns exactly one output to each input in its domain. Different inputs may share an output. A relation fails to be a function only when at least one allowed input has more than one output.
Frequently asked questions
What should I know before relations and functions?
Read ordered pairs, tables, and graphs.
How can I check my work?
The two outputs differ by 5, so they are genuinely different. A numerical check can detect an error; when claiming an identity for all inputs, also explain the algebraic property that makes it true.
Which mistakes should I watch for?
Rejecting repeated outputs: repeated inputs with different outputs are the issue. Using a horizontal-line test to decide whether a graph is a function. Ignoring a stated domain restriction.
Can I use another valid method?
Yes. Explain your assumptions, show enough intermediate work to justify the result, and verify it in the original problem. Equivalent expressions and valid alternative methods receive credit.
Mathematical process practice · A.1A–A.1G
- A.1A: Assign a single shipping price to each permitted package weight.
- A.1B: Plan, solve, check, and interpret the written response.
- A.1C: Select paper, a calculator, a table, or a graph; explain the choice.
- A.1D: Connect symbols with a verbal explanation and another representation.
- A.1E: Organize the solution with a table, diagram, or labeled equations.
- A.1F: Explain how the representation and the algebraic rule express the same relationship.
- A.1G: Justify each operation and the reasonableness of the conclusion.
Quick revision notes
- Check the one-output-per-input requirement in a relation. Apply the same definition to words, tables, graphs, and equations.
- Rejecting repeated outputs: repeated inputs with different outputs are the issue.
- Using a horizontal-line test to decide whether a graph is a function.
- Ignoring a stated domain restriction.
Practice and explain
Try all four multiple-choice questions, then show your reasoning in the written response. Objective scores and AI feedback assess different skills.
Continue learning
Original Cambridge Veritas practice. These are not official STAAR items. Cambridge Veritas is not affiliated with or endorsed by TEA.
Official Algebra I TEKS (§111.39) · TEA assessed curriculum · STAAR blueprint · TEA instructional materials