Relations and Functions
A function is a rule that gives every input exactly one output. You will use this idea in every algebra class after grade 8.
{(1, 8), (2, 8), (3, 9)}
What you'll be able to do
- Decide if a set of pairs, a table or a story is a function.
- Read a mapping diagram.
- Find the domain and range.
- Find the output of a rule.
- Spot common mistakes with functions.
The big idea
A relation is any set of input-output pairs, like (2, 5). A function is a relation where each input has exactly one output.
The test. Look at each input. If any input has two different outputs, it is not a function. So {(2, 5), (2, 7)} is not a function.
Repeats are fine. Two inputs can share an output. So {(1, 4), (2, 4)} is still a function.
The same pair twice is fine too. If (7, 2) is listed twice, the input 7 still has just one output.
Reverse it. Swap each pair to get the reverse. The reverse of a function is not always a function. Swap {(1, 3), (2, 3)}, and the input 3 gets two outputs.
Domain and range. The domain is the list of inputs. The range is the list of outputs. Write each value once.
Rules. A rule like y = 3x − 5 is a function. Each x gives one y. For x = 4, y = 3(4) − 5 = 7.
Words you need
- Relation
- Any set of input-output pairs.
- Function
- A relation where each input has exactly one output.
- Input and output
- In the pair (2, 5), 2 is the input and 5 is the output.
- Domain
- All the inputs, each listed once.
- Range
- All the outputs, each listed once.
- Mapping diagram
- A picture with arrows from each input to its output.
Function or not?
| Pairs | Function? | Why |
|---|---|---|
| (1, 4), (2, 4) | Yes | Each input has one output. The repeated 4 is fine. |
| (2, 5), (2, 7) | No | The input 2 has two outputs. |
| (7, 2), (7, 2) | Yes | The same pair twice still gives 7 one output. |
Worked examples
Example 1
Is {(1, 8), (2, 8), (3, 9)} a function?
- Look at each input: 1, 2 and 3.
- Each one has just one output.
- The 8 repeats, but that is fine. So yes, it is a function.
Answer: Yes, it is a function
{(1, 8), (2, 8), (3, 9)}
Example 2
Is {(5, 1), (6, 2), (5, 3)} a function?
- Look at each input. The input 5 appears twice.
- It goes with 1 and with 3.
- One input has two outputs, so it is not a function.
Answer: No - the input 5 breaks it
{(5, 1), (6, 2), (5, 3)}
Example 3
What are the domain and range of {(2, 7), (4, 7), (9, 0)}?
- The domain is the inputs: {2, 4, 9}.
- The range is the outputs: 7, 7 and 0.
- Write each value once. The range is {0, 7}.
Answer: {0, 7}
{(2, 7), (4, 7), (9, 0)}
Common mistakes
Saying a repeated output breaks a function. Only an input with two different outputs breaks it. Outputs may repeat.
Mixing up the domain and the range. The domain is the inputs. The range is the outputs.
Thinking the reverse of a function is always a function. Swap the pairs and test again. A shared output becomes an input with two outputs.
Forgetting that a missing value can match an input already used. Try the inputs you already have. Those are the values that break it.
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
- A game gives 6 stars for finishing level 3, 4, 5 or 6. The level is the input and the stars are the output: (3, 6), (4, 6), (5, 6), (6, 6). Is this a function? Word problem
- Age is the input and height in cm is the output. Four students give the pairs (13, 152), (13, 158), (14, 160) and (15, 165). Is height a function of age? Word problem
- Ravi and Mei share locker 112. Sofia has locker 118, and Jonah has locker 121. Is 'student → locker' a function? Word problem
- A taxi ride costs y = 3x + 2 dollars for x miles. How much does a 4-mile ride cost? Word problem
- Kai says {(1, 4), (2, 4), (3, 4)} is not a function because 4 repeats. Find his mistake. Is it a function?
Tier 2 Intermediate One more step to think through
- Is the relation in the mapping diagram a function? If not, which input breaks it?
- A bus sensor counts riders. The input is the time in minutes. The output is the number of riders: (0, 12), (10, 15), (20, 12) and (30, 9). What is the domain? Word problem
- Maya records her bowling score for each game. The input is the game number, and the output is the score. She scores 120 in game 1, 135 in game 2, 120 in game 3 and 150 in game 4. What is the range? Word problem
- At lunch, Ana picks an apple, Ben picks a pear and Cho picks an apple. The fruit is the input and the student is the output. Is this relation a function? Word problem
- Relation A is {(1, 5), (1, 6), (2, 7)}. Relation B is {(1, 5), (2, 5), (3, 6)}. Which one repeats an output but is still a function?
Tier 3 Advanced Several steps - find the hidden structure
- A theater pairs seat numbers with rows: (1, 7), (k, 3) and (4, 9). For which values of k is the row not a function of the seat number? Word problem
- For which value of k is {(2, 5), (k, 5), (2, k)} a function?
- A function uses the rule y = x² − 4x. Find y when x = −3.
- A function uses the rule y = 3x − 2 with domain {−2, 0, 4}. What is its range?
- Lena says the reverse of a function is always a function. Test her claim on {(1, 3), (2, 3), (5, 8)}. Is the reverse a function?
Tier 4 Challenge Stretch problems, some above grade level
- A club has 4 members and 3 shirt colors. Each member picks one color, and all 3 colors get picked. Is 'color → member' a function? Explain. Word problem
- A quiz has 3 true-or-false questions. Each way to answer it is a function from {Question 1, Question 2, Question 3} to {True, False}. How many different functions are possible? Word problem
- Raj says only k = 2 breaks {(1, k), (k, 4), (2, 6)}, because 2 is already an input with output 6. Find his mistake. Which values of k break it?
- A coach lists each player's number and goals as pairs: (7, 2), (10, 5) and (k, 2). Between k = 7 and k = 10, which value stops goals being a function of player number? Word problem
- f(x) means the output of f for the input x. If f(x) = 2x + 1, find f(f(3)). Above grade level
Answers
Tier 1 Foundations
- Yes, it is a function. No input has two different outputs. A repeated output is allowed.
- No - the input 13 breaks it. The input 13 goes with 152 and 158.
- Yes, it is a function. No input has two different outputs. A repeated output is allowed.
- 14. Replace x with 4: 3(4) + 2 = 14.
- Yes, it is a function. Kai thought a repeated output breaks a function. Only an input with two outputs does. 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.
- {0, 10, 20, 30}. List the inputs (first numbers), each value once: {0, 10, 20, 30}.
- {120, 135, 150}. List the outputs (second numbers), each value once: {120, 135, 150}.
- No - the input apple breaks it. The input apple goes with Ana and Cho.
- B. B repeats an output and is still a function. The other fails because an input repeats.
Tier 3 Advanced
- k = 1 or k = 4 break it. Those values reuse an input that already has a different output. Any other value works.
- k = 5. The input 2 goes with 5 and with k. For one output, k must be 5. Then (5, 5) is fine too.
- 21. Replace x with −3: (−3)² − 4(−3) = 9 + 12 = 21.
- {−8, −2, 10}. Put each input into the rule: x = −2 gives −8, x = 0 gives −2, x = 4 gives 10. The range is the set of outputs.
- No - the input 3 breaks it. Lena is wrong. Swapping gives (3, 1) and (3, 2), so 3 has two outputs. The input 3 goes with 1 and 2.
Tier 4 Challenge
- No. 4 members share 3 colors, so at least one color is picked by two members. That color would have two outputs.
- 8. Each question can be true or false, so 2 choices each. For 3 questions that is 2 × 2 × 2 = 8 functions.
- k = 1 or k = 2 break it. Raj missed k = 1. If k = 1, the pairs (1, 1) and (1, 4) give the input 1 two outputs. If k = 2, the input 2 goes with 4 and 6. Any other k works.
- k = 10 breaks it. If k = 7, the pair (7, 2) is just listed twice, so player 7 still has one goal count. If k = 10, player 10 has 5 goals and 2 goals. So only k = 10 breaks it.
- 15. Replace x with 3: f(f(3)) = f(7) = 2(7) + 1 = 15.
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