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Understand What Makes a Relation a Function

Students define a function as a rule that assigns to each input exactly one output, and identify functions and non-functions from tables, graphs (vertical line test), sets of ordered pairs, and mapping diagrams. They give real-world examples of functional and non-functional relationships.

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Understand What Makes a Relation a Function

Grade 8 · Functions · question 1 of 3

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20 rungs. A star at the top.

Which is a function: {(3, 1), (3, 3), (5, 5)}, {(3, 1), (4, 3), (4, 5)}, {(3, 1), (4, 3), (5, 5)} or {(3, 1), (4, 3), (3, 5)}?

How it is taught

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A function gives one output for each input.

A vertical line that hits twice means two outputs for one input.

Read it: What makes a function

A function gives each input exactly one output. (1, 4), (2, 6), (3, 8) is a function: each input has one output. (1, 4), (1, 6), (3, 8) is not: the input 1 has two outputs, 4 and 6.

  1. List the inputs, the first number of each pair.
  2. Look for an input that comes twice.
  3. If it comes with two different outputs, the set is not a function.

An input is what goes in; its output is what comes out. A function always gives the same output for the same input.

In (1, 4), (1, 6), (3, 8), the input 1 gives 4 once and 6 once: two outputs for one input.

Two inputs may share an output: (1, 4) and (2, 4) can both belong to a function.

A relation is a function when no input has two different outputs.

Calling a set not a function because an output repeats, as in (1, 4), (2, 4), (3, 8).

Look at the inputs, not the outputs. 1, 2 and 3 each come once, so each has one output: it is a function.

A function holds (3, 7), (7, 8) and (3, ___). — 7. The input 3 already has the output 7, and a function gives each input one output.

Comparing two relations

Think of a vending machine. You press one button, and you get exactly one snack — never two snacks for the same button.

That's the rule for a function: every input is paired with exactly one output.

Let's check Relation A. The inputs are one, two, three, and four. The outputs are two, four, six, and eight.

Each input is connected to one output: one to two, two to four, three to six, four to eight.

Look at every input. Does each one have exactly one arrow leaving it? Yes!

When is a relation a function?Press a button → get one snack.Same button, same snack — every time.A function:each input → exactly one outputRelation AINPUTOUTPUT12342468✓ FunctionRelation BINPUTOUTPUT1232456two outputs!✗ Not a functionThe function ruleone input → exactly one outputREMEMBER1 → 1input to output
Grade 8 · Every input gets exactly one output

Relation A is a function.

Now Relation B. The inputs are one, two, and three. The outputs are two, four, five, and six.

Here are the pairings: one goes to two, two goes to four, two also goes to five, and three goes to six.

Wait — look at input two. It has two arrows leaving it, going to two different outputs.

That breaks the rule. Relation B is not a function.

Remember: one input, exactly one output. That's what makes a relation a function.

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics
Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal
1 star — one skill mastered, every rung of its ladder climbed
Climbing now
Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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