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 wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
A function is a rule that takes an input and gives back exactly one output. A vending machine is a good mental picture: press B4 and you always get the same snack, never two different ones. Grade 8 is the first time students meet the word function formally, and they apply that one-output test to tables, sets of ordered pairs, mapping diagrams and graphs.
The test only restricts inputs, not outputs. Two different inputs may share an output (both 2 and -2 square to 4) and the relation is still a function, but one input with two different outputs breaks the rule. Students also learn that the graph of a function is the collection of all its (input, output) pairs plotted as points, which is why a vertical line crossing a graph twice reveals a relation that is not a function. Formal function notation such as f(x) is not required yet.
Students think {(1, 5), (2, 5)} is not a function because 5 appears twice. Repeated outputs are fine; only a repeated input with different outputs breaks the rule.
A table of daily high temperatures is a function of the date even without an equation. A function is any rule that gives one output per input.
A horizontal line crossing a graph twice does not disqualify it. Only a vertical line hitting two points shows one input with two outputs.
Curves such as y = x² are functions too. Linear functions are just one family studied in grade 8.
Decide whether each relation is a function. A: {(3, 7), (4, 9), (5, 7), (6, 11)}. B: {(2, 1), (2, 5), (3, 8)}.
Answer: A is a function; B is not, because the input 2 is paired with both 1 and 5.
Input-output machines drawn on the board, with students acting as the rule, make the idea concrete. Then give a mix of real relationships (person to birthday, birthday to person, ZIP code to city, city to ZIP code) and ask which are functions and why. The reverse pairs are especially good for discussion.
Tests often present several tables or graphs and ask which is not a function. Teach students to scan for repeated inputs in tables and to slide a ruler across a graph vertically. Later, 8.F.A.2 and 8.F.A.3 build on this definition to compare and classify functions.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: A) {(4, 1), (4, 3), (5, 2)}
The input 4 is paired with both 1 and 3, so that relation gives two outputs for one input.
Answer: D) Yes, each student has exactly one height
Each input (a student) gives exactly one output (their height). Two students sharing a height does not matter.
Answer: yes
Every input has exactly one output. All outputs being 9 is allowed; it is a constant function.
Answer: B) The graph is not a function
Two points on the same vertical line share the input x = 2 but have different outputs, so the graph is not a function.
Answer: 14
3 × 6 = 18, and 18 - 4 = 14.
Answer: D) Each input gives exactly one output
The defining rule is one output for each input. Outputs may repeat, and functions need not be linear or have formulas.
A full lesson with slides, activities and an exit ticket on what a function is, pitched to grade 8 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 8.F.A.1 with an answer key, ready in about a minute.
Make a worksheet →Turn what a function is into a quiz students answer online that marks itself, with a class summary for you.
Build a test →Function notation is introduced in high school. In grade 8 students describe functions with words, tables, graphs and equations like y = 3x + 2.
No. A vertical line through the middle of a circle hits it twice, so one input would have two outputs.