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.
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.
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.
Case 2: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
yesEach input has exactly one output.
Case 3: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
noInput 1 has two different outputs.
Case 4: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
yesEach input has exactly one output.
Case 5: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
noInput 1 has two different outputs.
Case 6: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
yesEach input has exactly one output.
Case 7: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
noInput 1 has two different outputs.
Case 8: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
yesEach 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.
Case 10: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
yesEach input has exactly one output.
Case 11: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
noInput 1 has two different outputs.
Case 12: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
yesEach input has exactly one output.
Case 13: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
noInput 1 has two different outputs.
Case 14: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
yesEach input has exactly one output.
Case 15: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
noInput 1 has two different outputs.
Case 16: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
yesEach 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.
Case 18: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
yesEach input has exactly one output.
Case 19: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
noInput 1 has two different outputs.
Case 20: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
yesEach input has exactly one output.
Practice
Guided practice with hints
01Case 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
02Case 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
03Case 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: no
04Case 24: 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
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
06Case 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
01Case 27: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
Answer: no
02Case 28: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
Answer: yes
03Case 29: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
Answer: no
04Case 30: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
Answer: yes
05Case 31: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
Answer: no
06Case 32: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
Answer: yes
07Case 33: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
Answer: no
08Case 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
Case 35: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
no
Case 36: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
yes
Case 37: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
no
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
01Case 39: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
Answer: no
02Case 40: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
Answer: yes
03Case 41: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
Answer: no
04Case 42: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
Answer: yes
05Case 43: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
Answer: no
06Case 44: does the relation {(1,2),(2,4),(3,5)} define y as a function of x?
Answer: yes
07Case 45: does the relation {(1,2),(1,4),(2,5)} define y as a function of x?
Answer: no
08Case 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
California Mathematics Framework (2023)
Grade 8 instructional guidance, mathematical practices, modeling, and progression toward high-school mathematics.Common Core State Standards for Mathematics — Grade 8
Grade-level expectations for real numbers, equations, functions, geometry, and statistics.Principles to Actions: Ensuring Mathematical Success for All
Research-informed emphasis on reasoning, representations, discourse, productive struggle, and conceptual understanding.