8.F.A.1: Understanding Functions as Input-Output Rules
In plain English: 8.F.A.1 is the Common Core grade 8 math standard that asks students to understand that a function is a rule that gives each input exactly one output, and that the graph of a function is the set of its (input, output) ordered pairs. Students test tables, mapping diagrams, lists of pairs and graphs. It is the first function standard in Grade 8 Math.
Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
Official note: Function notation is not required in Grade 8.
Common Core State Standards for Mathematics · Domain: Functions (F) · Cluster: Define, evaluate, and compare functions. Also written as 8.F.1 · Official standard
Students meet the idea of a function for the first time as a named concept. A function is a rule that assigns to each input exactly one output. Students test tables, lists of ordered pairs, mapping diagrams (drawings with an arrow from each input to its output), graphs and everyday sentences, and they learn that the one thing that breaks a function is an input with two different outputs. Repeated outputs are fine.
The second idea is that the graph of a function is the set of its ordered pairs (input, output). Students make tables from rules, plot the pairs, and decide whether a point belongs to a graph. The official standard carries this note: "Function notation is not required in Grade 8." So every rule on this page is written in words or as an equation such as y = 3x - 1, and no quiz or homework item uses f(x).
Learning Objectives
By the end of this lesson, students will be able to:
Explain that a function assigns to each input exactly one output
Decide whether a table, list of ordered pairs, mapping diagram, graph or sentence shows a function, and give the reason
Explain why repeated outputs are allowed in a function but repeated inputs with different outputs are not
Write the ordered pairs (input, output) of a rule and plot them as the graph of the function
Decide whether a given point is on the graph of a function
Prior Knowledge Required
Students should already be comfortable with:
Plotting ordered pairs in the coordinate plane 5.G.A.1
Graphing points with negative coordinates in all four quadrants 6.NS.C.8
Writing an equation that shows how one quantity depends on another 6.EE.C.9
Substituting a number into an expression such as 3x - 1 and working with integers
Show a picture of a school vending machine with buttons labeled A1 to C4. Read the prompt aloud and give pairs two minutes to talk.
Warm-Up Prompt
"You press B3 and get a bag of pretzels. Your friend presses B3 and gets pretzels too. Then you press B3 again and get a granola bar. Is the machine working the way it should? Now suppose A1 and A2 both give pretzels. Is that a problem?"
Collect ideas. Students usually say the first machine is broken, because one button should always give the same snack. The second machine is fine: two buttons can sell the same snack. Write two words on the board. An input is what you put in (the button). An output is what comes out (the snack). Tell students that today they will name the "working machine" idea: it is called a function.
Direct Instruction20 minutes
Define the key words, one at a time, and write each on an anchor chart:
Relation: any pairing of inputs with outputs. It can be a table, a list of ordered pairs, a mapping diagram, a graph or a sentence.
Ordered pair: two numbers written as (input, output). The order matters: (2, 5) and (5, 2) are different pairs.
Function: a rule that assigns to each input exactly one output. Each input gets one output, never two.
Graph of a function: the set of all its ordered pairs, drawn as points. The input is the x-coordinate and the output is the y-coordinate.
How to test a relation: look for an input that appears twice. If the same input has two different outputs, it is not a function. Repeated outputs are fine.
Work through the four examples below. For each one, ask students to name the inputs first, then the outputs.
Table to ordered pairs
Movie tickets cost $9 each. Input: the number of tickets, 1 to 4. Output: the total cost in dollars.
Equation: Pairs (1, 9), (2, 18), (3, 27), (4, 36). Each number of tickets has one cost, so this is a function.
Testing a list of pairs
Is {(2, 5), (4, 7), (2, 9), (6, 11)} a function?
Equation: No. The input 2 is paired with two outputs, 5 and 9.
Repeated outputs are allowed
Rule: square the input. Inputs -2, -1, 0, 1, 2.
Equation: Pairs (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4). A function: -2 and 2 share the output 4, but each input still has only one output.
Is a point on the graph?
Rule: y = 3x - 1, where any number can be an input. Make a table for inputs 0, 1, 2, 3, then test the points (4, 11) and (2, 6).
Equation: Pairs (0, -1), (1, 2), (2, 5), (3, 8). (4, 11) is on the graph because 3(4) - 1 = 11. (2, 6) is not, because input 2 gives 5.
Use Diagram 1 to show the same idea with arrows: in a function, every input has exactly one arrow leaving it. Then use Diagram 2 to connect the rule to its graph. The graph is not a separate picture: it is the list of ordered pairs, drawn as points. Point out the difference between Example 1 and Example 4. You cannot buy 2.5 tickets, so the ticket graph is only four dots. For y = 3x - 1, every number is an allowed input, so the dots fill in to make a line. The notation f(x) is not needed in grade 8: write rules in words or as equations such as y = 3x - 1.
Guided Practice15 minutes
Pairs work on four items, one at a time. After each one, a pair explains its answer at the board.
{(3, 1), (5, 1), (7, 2)}. (A function: the inputs 3, 5 and 7 each appear once.)
A table with inputs 0, 1, 1, 2 and outputs 4, 5, 6, 7. (Not a function: the input 1 has outputs 5 and 6.)
Input: a student in the class. Output: that student's height in centimeters. (A function: each student has one height today. Ask: is height to student also a function? Not always, since two students can be the same height.)
Rule y = x + 4 for inputs -2, 0 and 3. List the ordered pairs, then decide whether (1, 6) is on the graph. ((-2, 2), (0, 4), (3, 7). (1, 6) is not on the graph, because input 1 gives 5.)
Listen for students who say "not a function" because an output repeats, and for students who write pairs as (output, input).
Independent Practice15 minutes
Students work alone on five items, then compare with a partner:
A mapping diagram with arrows 10 to 2, 20 to 2 and 30 to 3. (A function.)
{(-1, 0), (0, -1), (-1, 2)}. (Not a function: -1 has two outputs.)
Make a table for y = 5x with inputs 0, 1, 2, 3 and plot the points on grid paper. ((0, 0), (1, 5), (2, 10), (3, 15).)
Is (6, 30) on the graph of y = 5x? (Yes, 5 times 6 is 30.)
Write your own list of four ordered pairs that is not a function, and circle the problem input.
Closure5-10 minutes
Exit ticket: (1) Is {(1, 8), (2, 8), (3, 9)} a function? Explain in one sentence. (Yes: each input has one output, and sharing the output 8 is allowed.) (2) The graph of a function contains the point (4, 6). Name a point that can never be on the same graph, and explain why. (Any point with input 4 and an output other than 6, for example (4, 1).) (3) Finish this sentence: "The graph of a function is ..."
Differentiation Strategies
For Struggling Students
Give a two-column input-output table template and have students highlight every input that appears more than once before deciding
Start with mapping diagrams, where a second arrow from the same input is easy to see, before moving to lists of pairs
Keep a card with the sentence "Same input, two outputs: not a function" on the desk during practice
For Advanced Students
Ask students to write a sentence rule from their own life that is a function, then reverse the input and output and decide whether the reverse is still a function
Ask: can a function have only one output for all of its inputs? Can a relation with only one input fail to be a function? Have them give an example for each answer
Preview (beyond this standard): show how high school courses write y = 3x - 1 as f(x) = 3x - 1, and have students read f(2) = 5 aloud as "the output for input 2 is 5"
Assessment Guidance
What to Look For
Listen for the reason, not only the answer. A strong answer names the input that breaks the rule, for example "the input 2 has two outputs." Watch for three errors: calling a relation "not a function" because an output repeats, writing ordered pairs in the order (output, input), and thinking a function must have a formula. When students test a point, check that they substitute the first number of the pair as the input.
02
Classroom Activities
3 Activities
1
Function or Not? Card Sort
20 minPairs
Pairs sort 12 cards into two piles, "Function" and "Not a function." The cards show the same idea in five ways: tables, lists of ordered pairs, mapping diagrams, graphs and sentences. Each pair must write the reason on the back of every card.
Card 10: Graph: the points (2, 1), (2, 3), (3, 2) (not a function (2 has outputs 1 and 3))
Card 11: Rule: the output is the input plus 7 (function)
Card 12: Sentence: input a whole number, output any whole number less than it (not a function (the input 5 could give 4, 3, 2, 1 or 0))
Procedure
Pairs read each card aloud and name its inputs before sorting it
For every "Not a function" card, circle the input that has two outputs, or explain in words why one input can give several outputs
When all 12 cards are sorted, pairs check with another pair and settle any disagreement using the definition
Discussion Questions
Six cards are functions and six are not. Which "Not a function" card was hardest to spot, and why?
Cards 3 and 5 repeat an output. Why are they still functions?
Cards 7, 8 and 12 have no numbers in a table. How did you decide?
Modification for Distance Learning
Put the cards on a shared slide with two labeled boxes. Pairs drag each card into a box and type their reason in a speech bubble next to it.
2
Human Function Machine
20 minGroups of 4
One student is the "machine" and holds a secret rule card. The others call out whole-number inputs from 0 to 10, the machine answers with outputs, and a recorder writes each (input, output) pair in a table. The group then plots the pairs on grid paper and guesses the rule.
Rule Cards
Card A: multiply the input by 2, then add 3
Card B: subtract the input from 10
Card C: the output is always 4
Card D, the "broken machine": ignore the input, roll the number cube and say the number you roll
Sample Record
With Card A, a group called out 1, 4, 0 and 7 and recorded (1, 5), (4, 11), (0, 3) and (7, 17). Their plotted points lie on a straight line. With Card D, the input 4 was called twice and gave 2 and then 6.
Procedure
Rotate the machine role so each student uses one card; Card D is always used last
Each group must call at least one input twice, to test whether the machine is consistent
Groups plot each set of pairs on its own grid and label it "function" or "not a function"
Discussion Questions
Card C gave the same output every time. Is it a function? What does its graph look like?
Why is Card D not a function, even though every output was a real number?
Which cards did you guess from the graph alone?
3
Measure and Map: Hand Spans
20 minGroups of 8 (or the whole class)
Students measure their hand span, from the tip of the thumb to the tip of the little finger with the hand stretched flat, to the nearest centimeter. They build a function from real data, graph it as ordered pairs, and then test whether the reverse pairing is still a function.
Sample Data (invented)
An invented group of 8 students, numbered 1 to 8, measured hand spans of 17, 19, 18, 19, 20, 16, 18 and 21 cm.
Procedure
Each student measures twice with a centimeter ruler and records one value, so that every input has exactly one output
Make a table: input = student number, output = hand span in cm. Plot the pairs (student number, hand span) on grid paper
Now reverse the table: input = hand span, output = student number. Plot these pairs on a second grid
Decide which of the two graphs shows a function and mark any input with two points above it
Discussion Questions
In the sample data, the spans 18 cm and 19 cm each belong to two students. What does that do to the reversed graph?
Why did we record one value per student instead of both measurements?
Could the reversed pairing ever be a function? What would the data need to look like?
Challenge Variation
Groups add the pair (student number, height in cm) for each student and decide whether height could be used as the input of a function whose output is student number in their group.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Mapping Diagrams, Function and Not a Function
A mapping diagram draws an arrow from each input to its output. On the left, the rule "square the input" sends -1 and 1 to the same output, 1, which is allowed. On the right, the input 2 has two arrows, so the relation is not a function.
Diagram 2: The Graph Is the Set of Ordered Pairs
Left: the table for y = 3x - 1 gives the dots (0, -1), (1, 2), (2, 5) and (3, 8), drawn to scale. Because every number is an allowed input, the whole graph is the line through them; the window shows inputs from 0 to 4. Right: two points share the input 2, so this set of points is not the graph of a function.
04
Homework Assignment
~30 min
8.F.A.1 Homework: Functions and Their Graphs
Directions: Show your thinking. When you decide whether something is a function, name the input that proves your answer or explain why every input has one output. Use grid paper for the graphs.
Part 1: Is It a Function? (Problems 1-3)
Decide whether each set of ordered pairs is a function, and explain. (a) {(1, 4), (3, 4), (5, 6), (7, 8)} (b) {(0, 2), (1, 3), (0, -2), (4, 6)}
A parking garage charges by the hour. Input: hours parked. Output: fee. The table shows 1 hour: $4, 2 hours: $7, 3 hours: $10, 4 hours: $12, 5 hours: $12. (a) Is the fee a function of the hours parked? (b) If you reverse the table, so that the fee is the input and the hours are the output, is it a function? Explain both answers.
Decide whether each sentence describes a function. (a) Input: a student in your class. Output: the month of that student's birthday. (b) Input: a month. Output: a student in your class born in that month. (c) Input: a number. Output: the number that is 5 more than it.
Part 2: Graphs as Ordered Pairs (Problems 4-6)
Make a table for the rule y = 4x - 7 with inputs -1, 0, 1, 2 and 3. Write the ordered pairs and plot them on grid paper.
Which of these points are on the graph of y = 2x + 5? Show the substitution for each: (3, 11), (-2, 1), (6, 16), (0, 5).
A bike rental costs $5 per hour plus a $3 helmet fee, for 1 to 4 whole hours. Input: hours. Output: total cost in dollars. (a) List the four ordered pairs and plot them. (b) Explain why this graph is the graph of a function. (c) Your friend says the point (2, 15) could also be on this graph. Explain why that is wrong.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Function Decisions
Every decision is correct
One or two decisions are wrong
Most decisions are wrong
Reasons
Names the input with two outputs, or explains why each input has one output
Reasons given but vague
No reasons
Ordered Pairs and Tables
All pairs correct and written as (input, output)
One arithmetic error or one pair reversed
Many pairs wrong or reversed
Graphs
Points plotted accurately and labeled
Most points plotted correctly
Graph missing or incorrect
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
Jordan says, "A relation is a function only if all of its outputs are different." Which statement corrects Jordan?
Answer: A
A function gives each input exactly one output, and two inputs may share an output. Choice B swaps the roles of input and output. Choice C accepts Jordan's error: the rule "square the input" repeats outputs and is still a function. Choice D describes linear functions only (8.F.A.3); many functions do not graph as lines.
Question 2 of 20 · Multiple Choice
Which set of ordered pairs is NOT a function?
Answer: C
In choice C the input 8 has two outputs, 2 and 4. Choices B and D repeat an output, which is allowed; a student who checks the outputs instead of the inputs picks one of them by mistake.
Question 3 of 20 · Multiple Choice
A table has inputs 1, 2, 3, 4 and outputs 7, 7, 9, 9. Is the relation a function?
Answer: D
The inputs 1, 2, 3 and 4 each appear once, so each input has one output. Choice A treats a repeated output as a problem, but it is not. Choice C gives a false reason: the output 7 has two inputs, 1 and 2.
Question 4 of 20 · Multiple Choice
In a mapping diagram, arrows go from the input 3 to the output 6 and from the input 3 to the output 9. Every other input has one arrow. Is the relation a function?
Answer: B
Two arrows leave the input 3, so 3 has two outputs and the relation is not a function. Choice C looks at the arrows arriving at outputs instead of the arrows leaving inputs. Choice D is false: Diagram 1 shows a mapping diagram of a function.
Question 5 of 20 · Multiple Choice
Which sentence describes a function?
Answer: D
Each state has exactly one capital, so choice D is a function. Choice A fails because the input 9 gives both 3 and -3. Choices B and C fail because many students share a first name and many people share a birth year.
Question 6 of 20 · Multiple Choice
Which value of k makes {(1, 3), (2, 5), (k, 8), (4, 9)} NOT a function?
Answer: C
If k = 4, the input 4 has two outputs, 8 and 9. Choices A, B and D use numbers that appear as outputs (3, 8 and 5); none of them is already an input, so with any of them each input still has one output.
Question 7 of 20 · Multiple Choice
Which ordered pair is on the graph of y = 2x + 1?
Answer: B
Input 6 gives 2(6) + 1 = 13, so (6, 13) is on the graph. Choice A writes the pair as (output, input): input 13 would give 27. Choice C forgets to add 1. Choice D takes the multiplier 2 as the output for input 0, but 2(0) + 1 = 1.
Question 8 of 20 · Multiple Choice
The graph of a function has the points (0, 2), (1, 4), (2, 6) and (3, 8). What does the point (2, 6) tell you?
Answer: A
An ordered pair is written (input, output). Choice B reads the pair backward. Choice C fits (2, 6) alone but fails for (0, 2), since 0 + 4 is not 2; the rule here is "multiply by 2, then add 2."
Question 9 of 20 · Multiple Choice
Which of these could NOT be the graph of a function?
Answer: C
Every point on a vertical line through (3, 0) has input 3, so the input 3 would have endlessly many outputs. Choice A is a function: every input has the output 4. Students who think repeated outputs break a function pick A by mistake.
Question 10 of 20 · Multiple Choice
The graph of a function contains the point (5, 12). Which point can NOT also be on this graph?
Answer: B
The input 5 already has the output 12, so it cannot also have the output 10. Choice A has the input 12, a different input, so it is allowed. Choices C and D share the output 12, which is also allowed.
Question 11 of 20 · Multiple Choice
A thermometer recorded these temperatures on a spring day: 8 am, 55°F; 10 am, 61°F; noon, 66°F; 2 pm, 66°F; 4 pm, 61°F. Which statement is true?
Answer: A
Each time has one reading, so temperature is a function of time. Reversed, the input 66°F has two outputs (noon and 2 pm), and 61°F also has two (10 am and 4 pm). Choice D reverses the two roles. Choice C treats the repeated temperatures as a problem for both directions.
Question 12 of 20 · Multiple Choice
Nina lists the graph of y = 2x + 3 for the inputs 0, 1, 2, 3 as (3, 0), (5, 1), (7, 2), (9, 3). What should she fix?
Answer: D
Her outputs 3, 5, 7, 9 are right, but the input must come first. For example, input 3 gives 9, not 0, so (3, 0) is not on the graph. Choice B forgets to add 3. Choice C changes nothing important: the order of the list does not matter, only the order inside each pair.
Question 13 of 20 · Multiple Choice
The inputs are the positive whole numbers. Which rule does NOT describe a function?
Answer: B
The input 6 has the factors 1, 2, 3 and 6, so "a factor of the input" can give several outputs. Choice C is a function because rounding gives one answer. Choice D is a function even though all outputs are the same.
Question 14 of 20 · Multiple Choice
The complete graph of a function is exactly the four points (1, 4), (2, 7), (3, 10) and (4, 13). Is (5, 16) on this graph?
Answer: D
The graph of a function is exactly the set of its ordered pairs. Here the inputs are 1, 2, 3 and 4 only, so no point with input 5 belongs to it. Choices A and B continue the pattern past the listed inputs. Choice C gives a wrong reason: a large output does not keep a point off a graph. What matters is that 5 is not one of the inputs.
Question 15 of 20 · Short Answer
Is {(-3, 5), (0, 1), (2, 5), (-3, 7)} a function? If not, name one ordered pair you could remove to make it a function.
Not a function. The input -3 has two outputs, 5 and 7. Remove either (-3, 5) or (-3, 7). The pairs (-3, 5) and (2, 5) sharing the output 5 is not a problem.
Question 16 of 20 · Short Answer
Make a table for y = 3x - 2 with the inputs -1, 0, 2 and 4. Write the points of the graph as ordered pairs.
3(-1) - 2 = -5, 3(0) - 2 = -2, 3(2) - 2 = 4 and 3(4) - 2 = 10, so the pairs are (-1, -5), (0, -2), (2, 4), (4, 10).
Question 17 of 20 · Short Answer
A movie theater charges by age: age 8 pays $9, age 12 pays $9, age 15 pays $12, age 40 pays $12 and age 70 pays $8. Is price a function of age? Is age a function of price? Explain.
Price is a function of age: each age has one price. Age is not a function of price: the price $9 has two ages, 8 and 12 (and $12 has 15 and 40).
Question 18 of 20 · Short Answer
The points (2, 7) and (2, k) are both on the graph of the same function. What must k be? Explain.
k = 7. Both points have the input 2. A function gives each input exactly one output, so the second output must also be 7.
Question 19 of 20 · Short Answer
Leo has $20 and saves $5 each week. Input: the number of weeks. Output: his total savings in dollars. Write the ordered pairs for weeks 0, 1, 2 and 3, and explain what the last pair means.
The pairs are (0, 20), (1, 25), (2, 30), (3, 35). The last pair means that after 3 weeks Leo has $35 in total.
Question 20 of 20 · Short Answer
A graph is made of the points (1, 4), (3, 2), (1, -4) and (5, 0). Is it the graph of a function? Name a vertical line that shows your answer.
Not a function. The vertical line through the input 1 passes through both (1, 4) and (1, -4), so the input 1 has two outputs, 4 and -4. No other vertical line meets two of the points.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 8.F.A.1 mean?
8.F.A.1 means students understand what a function is: a rule that gives each input exactly one output. It also asks them to see the graph of a function as the set of its (input, output) ordered pairs, plotted as points. Students do not need a formula to recognize a function; a table, a list of pairs or a sentence can show one.
Is 8.F.A.1 taught in grade 8 or in Algebra I?
8.F.A.1 is a grade 8 standard, and it is usually the first lesson of the functions unit in Grade 8 Math. Some schools teach Algebra I in grade 8, and those courses cover the same idea early in the year. High school revisits it in HSF.IF.A.1 with new words (domain and range) and function notation.
What is the difference between a relation and a function?
Every function is a relation, but not every relation is a function. A relation is any pairing of inputs with outputs. It becomes a function only when no input is paired with two different outputs. For example, "a teacher and the students in their class" is a relation but not a function, because one teacher has many students.
Can two different inputs have the same output?
Yes. Sharing an output is allowed. The rule "square the input" sends both 4 and -4 to 16, and it is still a function. What is not allowed is one input with two different outputs. Many students mix these up at first, so ask them to always check the inputs column.
What is the vertical line test, and do grade 8 students need it?
It is a quick way to read the definition from a graph. A vertical line joins points that have the same input, so if any vertical line passes through two points of a graph, one input has two outputs and the graph is not a function. The standard does not name the test, but it is a helpful shortcut once students understand why it works.
Do students need f(x) notation for 8.F.A.1?
No. The official standard has a footnote that says "Function notation is not required in Grade 8." Students can write rules in words or as equations such as y = 2x + 7. Function notation is introduced in high school in HSF.IF.A.2.
What does "the graph of a function is the set of ordered pairs" mean?
It means the graph is not a separate drawing: it is every (input, output) pair of the function, plotted as a point. The input is the x-coordinate and the output is the y-coordinate. So a point is on the graph exactly when its second number is the output the rule gives for its first number.
Why do some function graphs show separate dots and others show a line?
It depends on which inputs make sense. If the input counts whole things, such as tickets or people, the graph is a set of separate dots. If any number can be an input, as in y = 3x - 1, there are infinitely many ordered pairs, and the dots fill in to make a line or a curve.
What mistakes do students make with functions?
A common mistake is saying a relation is not a function because an output repeats. Another is writing ordered pairs backward, as (output, input), which moves every point on the graph. A third is thinking a function needs a formula. Everyday rules, such as "a book and its number of pages," are functions too.
How can parents help their child practice functions at home?
Ask "Is it a function?" about everyday pairings. A person and their birthday: yes, each person has one birthday. A birthday and a person: no, many people share one. A price tag and an item in a store, a house and its street address, a song and its length: have your child name the input and say whether any input could have two outputs.
07
Related Standards
6 standards
These standards connect to 8.F.A.1: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
5.G.A.1Prerequisite
Use a coordinate system and locate points with ordered pairs of numbers
Lesson coming soon
6.EE.C.9Prerequisite
Use variables for two related quantities and relate the equation to graphs and tables
