Relations and Functions
A function is a rule where every input gets exactly one output. This one idea runs through every algebra course after grade 8.
{(−1, 2), (0, 2), (1, 2)}
Know these first
- Master List skill 6.EE.9
This unlocks
- More lessons that build on this are coming.
What you'll be able to do
- Decide whether a relation is a function from pairs, a table, a mapping diagram or a real situation.
- Name the input that breaks a relation that isn't a function.
- Decide which direction makes a pairing a function.
- Find the values that keep a relation a function.
- List the domain and range of a relation, and evaluate a function written as f(x).
- List the domain and range of a relation, and evaluate a function written as f(x).
The big idea
A relation is any set of input-output pairs. A function is a relation with one rule: each input has exactly one output.
Think of a vending machine. Press B4 and you get the same snack every time. If B4 sometimes gave chips and sometimes candy, the machine would be broken. Two buttons giving the same snack is fine.
Direction matters. Turn every pair around and the answer can change, because the old outputs become the new inputs.
Words you need
- Input
- The value you start with, usually x.
- Output
- The value you get, usually y.
- Domain
- The set of all inputs.
- Range
- The set of all outputs.
Worked examples
Example 1
Is {(−1, 2), (0, 2), (1, 2)} a function?
- List the inputs: −1, 0, 1. None repeat.
- Every output is 2. A repeated output is allowed.
Answer: Yes, it is a function
{(−1, 2), (0, 2), (1, 2)}
Example 2
Does this table show y as a function of x?
- The input 2 appears twice.
- It goes with 1 the first time and 5 the second time, so input 2 has two outputs.
Answer: No - the input 2 breaks it
| x | 2 | 4 | 2 | 6 |
|---|---|---|---|---|
| y | 1 | 3 | 5 | 7 |
Example 3
For which values of k is {(1, 2), (3, 4), (k, 9)} a function?
- The inputs 1 and 3 are already used with outputs 2 and 4.
- If k is 1 or 3, that input gets a second output, 9.
- Any other k is a new input, so the relation stays a function.
Answer: Every value except k = 1 or 3
Common mistakes
"An output repeats, so it's not a function." Only a repeated input with different outputs breaks the rule. {(1, 4), (2, 4)} is a function.
Forgetting the direction. "Is the price a function of the item?" and "is the item a function of the price?" are different questions. Decide which is the input first.
Counting arrows that arrive. In a mapping diagram, count only the arrows that leave each input.
Practice: 20 questions
11 of 20 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- Is this relation a function?
{(3, 6), (4, 6), (5, 6), (6, 6)}
- Does this table show y as a function of x?
x 5 5 6 7 y 0 1 2 3 - Lockers are assigned in pairs: Ravi and Mei share locker 112, Sofia has 118, and Jonah has 121. Is the locker number a function of the student? Word problem
- A pizza shop charges by toppings: 0 toppings $10.00, 1 topping $11.50, 2 toppings $13.00, 3 toppings $14.50. Is the price a function of the number of toppings? Word problem
- A vending machine: A1 gives pretzels, A2 gives chips, A3 gives pretzels, B1 gives a granola bar. Is the snack a function of the button? Word problem
Tier 2 Intermediate One more step to think through
- Is this relation a function? If not, name the input that breaks it.
{(−2, 5), (0, 1), (−2, −5), (4, 3)}
- Does this table show y as a function of x?
x 0 1 2 1 y 4 6 8 6 - Is the relation in this mapping diagram a function?
- Find the range of this relation.
{(−2, 3), (0, 3), (4, −1), (6, 8)}
- This week's highs were Monday 61°F, Tuesday 64°F, Wednesday 61°F and Thursday 70°F. Is the day a function of the temperature? Word problem
- Hana's phone saves two numbers for her aunt (mobile and work) and one each for Omar and Priya. Is the phone number a function of the contact? Word problem
- A taxi charges f(x) = 2x + 3 dollars for a ride of x miles. What is f(7)? Word problem
Tier 3 Advanced Several steps - find the hidden structure
- For which values of p is {(1, 7), (p, 3), (4, 9)} NOT a function?
- A table has x-values 2, 3, 2 and y-values 8, 5, n. For which value of n is y a function of x?
- A parking garage prints each ticket number once and records the car's plate. Ticket 501: KJX-204. Ticket 502: PLM-118. Ticket 503: KJX-204 (the car came back). Is the ticket number a function of the plate? Word problem
- A tutoring center logs sessions per day: Monday 3, Wednesday 5, Friday 2. Someone adds a second Monday entry with s sessions. For which s is the log still a function of the day? Word problem
- A teacher posts quiz scores by student ID: 2041 scored 18, 2042 scored 20, 2043 scored 18, 2044 scored 15. Is the ID a function of the score? Word problem
Tier 4 Challenge Stretch problems, some above grade level
- For which values of a is {(a, 3), (2, a), (5, 1)} NOT a function?
- A train fare chart: Zone 1 costs $2.75, Zone 2 $3.50, Zone 3 $4.25. Is the zone a function of the fare on this chart? Word problem
- Mateo claims you can spot a function by checking that no output repeats. Which relation proves him wrong? Word problem
- A. {(2, 3), (2, 4)}
- B. {(5, 1), (6, 1), (7, 2)}
- C. {(0, 0), (0, 1), (1, 1)}
Answers
Tier 1 Foundations
- Yes, it is a function. No input has two different outputs. A repeated output is allowed.
- No - the input 5 breaks it. The input 5 goes with 0 and 1.
- Yes, it is a function. No input has two different outputs. A repeated output is allowed.
- Yes, it is a function. No input has two different outputs.
- Yes, it is a function. No input has two different outputs. A repeated output is allowed.
Tier 2 Intermediate
- No - the input −2 breaks it. The input −2 goes with −5 and 5.
- Yes, it is a function. No input has two different outputs. A repeated output is allowed.
- No - the input 3 breaks it. The input 3 goes with 4 and 5.
- {−1, 3, 8}. List the outputs (second numbers), each value once: {−1, 3, 8}.
- No - the input 61 breaks it. The input 61 goes with Monday and Wednesday.
- No - the input the aunt breaks it. The input the aunt goes with mobile and work.
- 17. Replace x with 7: 2(7) + 3 = 17.
Tier 3 Advanced
- Every value except p = 1 or 4. Those values reuse an existing input with a new output. Any other value is a fresh input.
- n = 8. Any other value gives one input two different outputs.
- No - the input KJX204 breaks it. The input KJX204 goes with 501 and 503.
- s = 3. Any other value gives one input two different outputs.
- No - the input 18 breaks it. The input 18 goes with 2041 and 2043.
Tier 4 Challenge
- Every value except a = 2 or 5. Those values reuse an existing input with a new output. Any other value is a fresh input.
- Yes, it is a function. No input has two different outputs.
- B. B repeats an output and is still a function. The others fail because an input repeats.
Stuck on this topic?
Trouble here often traces back to an earlier skill. Our free check-in finds the exact gap. A tutor then works on it with your child.
Find gaps free More practice