Chapter 3 · Lesson 2 of 10 · Grade 8 · Functions · MAT-08-FN-002
Determine Whether a Relation Is a Function
Represent and solve determine whether a relation is a function problems using precise mathematical notation, multiple representations, and justified reasoning.
Math V4 · Course chapter
Chapter 3: Functions and Linear Models
Functions
Essential question: How can functions connect inputs, outputs, rates of change, graphs, tables, and equations?
Where this lesson fits
Lesson 2 of 10. Each lesson builds on earlier chapter representations and ideas rather than resetting the topic.
Chapter destination
Build and defend a multi-representation model using functions and linear models, including interpretation, verification, and error analysis.
Learning objectives
What you should be able to do
- Represent and solve determine whether a relation is a function problems using precise mathematical notation, multiple representations, and justified reasoning.
- Interpret and verify determine whether a relation is a function results in context, including units, constraints, and reasonableness.
Prerequisite check
Make sure the foundation is ready.
Grade 8 uses this prerequisite as active knowledge in a more formal representation.
Learn
Build the idea from meaning, not memorization.
Chapter 3: Functions and Linear Models
Functions. Lesson 2 of 10 extends earlier chapter reasoning.
Connection to the course
Use earlier Grade 8 chapters, Grade 7 prerequisite knowledge as active knowledge. Connect today's representation to at least one earlier representation and explain the relationship.
Meaning and structure
Determine Whether a Relation Is a Function belongs to functions. Identify the governing property or relationship before choosing a procedure.
Multiple representations
Represent determine whether a relation is a function numerically and symbolically, then connect it to the Grade 8 Math Box. Tables, graphs, equations, coordinate models, diagrams, and verbal descriptions must agree.
Verification and interpretation
Check with substitution, an inverse operation, an equivalent representation, geometric invariants, estimation, or context. State whether the result is exact or approximate and preserve relevant units and constraints.
Math Box · Interactive lesson
Touch the math. Change it. See what stays true.
Use the interactive model before and after the worked examples. Change the inputs, make a prediction, then use the model to test whether your reasoning holds.
Function Machine
Every input passes through one rule and produces one output.
- Model determine whether a relation is a function in the Math Box.
- Change one input and predict which quantities or invariants should change.
- Explain what the visual, equation, and numerical result say about the same relationship.
Worked examples
Twenty different ways to see the concept work.
Case 1: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
noInput 1 has two different outputs.
Represent the situation with a table, graph, diagram, number line, or equation before solving: Case 2: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
yesA representation makes the mathematical structure visible before calculation. Each input has exactly one output.
Explain why the method works, not only what steps to perform: Case 3: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
noJustification connects each step to a definition, property, or relationship. Input 1 has two different outputs.
Estimate or predict the result first, then calculate and judge whether the answer is reasonable: Case 4: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
yesAn estimate provides an independent reasonableness check. Each input has exactly one output.
Verify the result with substitution, an inverse operation, or a second representation: Case 5: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
noIndependent verification should support the result without merely repeating the same procedure. Input 1 has two different outputs.
Interpret the result in context, including units, direction, rate, or geometric meaning: Case 6: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
yesInterpretation connects a computed value to the mathematical situation. Each input has exactly one output.
Error analysis: a student says, "Rejecting a function because outputs repeat." Explain the error, then solve this related task correctly: Case 7: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
Outputs may repeat; one input cannot have two different outputs. Correct solution: noError analysis identifies the broken idea and replaces it with valid reasoning. Input 1 has two different outputs.
Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: Case 8: does the relation {(1,2),(2,4),(3,5)} define y as a function of x? Case B: Case 19: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
Case A: yes Case B: noComparison should address structure and representation, not only numerical answers. Each input has exactly one output.
Case 9: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
noInput 1 has two different outputs.
Represent the situation with a table, graph, diagram, number line, or equation before solving: Case 10: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
yesA representation makes the mathematical structure visible before calculation. Each input has exactly one output.
Explain why the method works, not only what steps to perform: Case 11: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
noJustification connects each step to a definition, property, or relationship. Input 1 has two different outputs.
Estimate or predict the result first, then calculate and judge whether the answer is reasonable: Case 12: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
yesAn estimate provides an independent reasonableness check. Each input has exactly one output.
Verify the result with substitution, an inverse operation, or a second representation: Case 13: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
noIndependent verification should support the result without merely repeating the same procedure. Input 1 has two different outputs.
Interpret the result in context, including units, direction, rate, or geometric meaning: Case 14: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
yesInterpretation connects a computed value to the mathematical situation. Each input has exactly one output.
Error analysis: a student says, "Rejecting a function because outputs repeat." Explain the error, then solve this related task correctly: Case 15: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
Outputs may repeat; one input cannot have two different outputs. Correct solution: noError analysis identifies the broken idea and replaces it with valid reasoning. Input 1 has two different outputs.
Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: Case 16: does the relation {(1,2),(2,4),(3,5)} define y as a function of x? Case B: Case 27: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
Case A: yes Case B: noComparison should address structure and representation, not only numerical answers. Each input has exactly one output.
Case 17: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
noInput 1 has two different outputs.
Represent the situation with a table, graph, diagram, number line, or equation before solving: Case 18: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
yesA representation makes the mathematical structure visible before calculation. Each input has exactly one output.
Explain why the method works, not only what steps to perform: Case 19: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
noJustification connects each step to a definition, property, or relationship. Input 1 has two different outputs.
Estimate or predict the result first, then calculate and judge whether the answer is reasonable: Case 20: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
yesAn estimate provides an independent reasonableness check. Each input has exactly one output.
Practice
Guided practice with hints
01Verify the result with substitution, an inverse operation, or a second representation: Case 21: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: no
02Interpret the result in context, including units, direction, rate, or geometric meaning: Case 22: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: yes
03Error analysis: a student says, "Rejecting a function because outputs repeat." Explain the error, then solve this related task correctly: Case 23: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: Outputs may repeat; one input cannot have two different outputs. Correct solution: no
04Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: Case 24: does the relation {(1,2),(2,4),(3,5)} define y as a function of x? Case B: Case 35: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: Case A: yes Case B: no
05Case 25: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: no
06Represent the situation with a table, graph, diagram, number line, or equation before solving: Case 26: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: yes
Practice
Independent practice
01Explain why the method works, not only what steps to perform: Case 27: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
Answer: no
02Estimate or predict the result first, then calculate and judge whether the answer is reasonable: Case 28: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
Answer: yes
03Verify the result with substitution, an inverse operation, or a second representation: Case 29: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
Answer: no
04Interpret the result in context, including units, direction, rate, or geometric meaning: Case 30: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
Answer: yes
05Error analysis: a student says, "Rejecting a function because outputs repeat." Explain the error, then solve this related task correctly: Case 31: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
Answer: Outputs may repeat; one input cannot have two different outputs. Correct solution: no
06Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: Case 32: does the relation {(1,2),(2,4),(3,5)} define y as a function of x? Case B: Case 43: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
Answer: Case A: yes Case B: no
07Case 33: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
Answer: no
08Represent the situation with a table, graph, diagram, number line, or equation before solving: Case 34: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
Answer: yes
Common mistakes
Learn to catch the error, not just the answer.
Outputs may repeat; one input cannot have two different outputs.
Slope is change per unit; intercept is output at x=0.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Model constant rates and starting values with tables, graphs, and equations.
- Compare functions by rate of change and initial value.
Challenge problems
Explain why the method works, not only what steps to perform: Case 35: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
no
Estimate or predict the result first, then calculate and judge whether the answer is reasonable: Case 36: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
yes
Verify the result with substitution, an inverse operation, or a second representation: Case 37: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
no
Interpret the result in context, including units, direction, rate, or geometric meaning: Case 38: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
yes
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 8 Algebra & Modeling Sprint
Solve determine whether a relation is a function problems and justify each result with a valid verification method.
Practice
Mastery check
01Error analysis: a student says, "Rejecting a function because outputs repeat." Explain the error, then solve this related task correctly: Case 39: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
Answer: Outputs may repeat; one input cannot have two different outputs. Correct solution: no
02Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: Case 40: does the relation {(1,2),(2,4),(3,5)} define y as a function of x? Case B: Case 51: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
Answer: Case A: yes Case B: no
03Case 41: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
Answer: no
04Represent the situation with a table, graph, diagram, number line, or equation before solving: Case 42: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
Answer: yes
05Explain why the method works, not only what steps to perform: Case 43: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
Answer: no
06Estimate or predict the result first, then calculate and judge whether the answer is reasonable: Case 44: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
Answer: yes
07Verify the result with substitution, an inverse operation, or a second representation: Case 45: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
Answer: no
08Interpret the result in context, including units, direction, rate, or geometric meaning: Case 46: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
Answer: yes
Terminology
Words to know
- function
- A relation assigning exactly one output to each input.
- slope
- Vertical change divided by horizontal change for a nonvertical line.
- y-intercept
- The output where a graph crosses the y-axis.
- rate of change
- Output change per unit input change.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
Grade 8 scope, sequence, and public task structure
Coherent sequencing, representations, proof-oriented reasoning, and tasks.Eureka Math² Grade 8 program structure
Module coherence, cumulative practice, and algebraic fluency.enVision Mathematics Grade 8 instructional model
Problem-based entry points, visual learning, modeling, and application.Common Core State Standards for Mathematics — Grade 8
Grade-level content and mathematical-practice expectations.