What is a function: one output for each input

Discover

A vending machine: you press a button (input) and get a snack (output). Pressing B4 sometimes gives chips and sometimes gives a cookie. What is wrong?

One input gives two different outputs
Two buttons might give the same snack

Key idea

A function assigns exactly one output to each input.

A function is a rule that assigns to each input exactly one output. Different inputs may share an output. On a graph, no vertical line meets a function's graph twice. A function can be given by a table, a graph, an equation or words.

Worked example

$(1, 5), (2, 5), (3, 7)$ and $(1, 5), (1, 6), (2, 7)$ .

First: a function

Each input once.

Second: not a function

Input 1 has outputs 5 and 6.

Your turn, guided

Your turn: the rule “triple the input, then subtract 4”. The output for input 5.

11

15 − 4.

Practice ★

A table pairs inputs $x$ with outputs $y$ : $(x, y) = (5, 1), (5, 2), (9, 3)$ . Is $y$ a function of $x$ ?
1. No: the input 5 has two different outputs
2. Yes: every input has an output
3. Yes: the outputs are all different
A table pairs inputs $x$ with outputs $y$ : $(x, y) = (4, 1), (4, 2), (6, 3)$ . Is $y$ a function of $x$ ?
1. No: the input 4 has two different outputs
2. Yes: every input has an output
3. Yes: the outputs are all different

A table pairs inputs $x$ with outputs $y$ : $(x, y) = (1, 5), (5, 5), (9, 5)$ . Is $y$ a function of $x$ ?
1. Yes: each input has exactly one output
2. No: the output 5 repeats
3. No: a function needs a formula

Watch out!

Practice ★★

A function's rule: multiply the input by 6, then add 6. Which input gives the output 24?
A function's rule: multiply the input by 4, then add 9. Which input gives the output 21?

Exit ticket

Three questions to check you have got it.

A table pairs inputs $x$ with outputs $y$ : $(x, y) = (5, 1), (5, 2), (9, 3)$ . Is $y$ a function of $x$ ?
1. No: the input 5 has two different outputs
2. Yes: every input has an output
3. Yes: the outputs are all different
A table pairs inputs $x$ with outputs $y$ : $(x, y) = (4, 1), (4, 2), (6, 3)$ . Is $y$ a function of $x$ ?
1. No: the input 4 has two different outputs
2. Yes: every input has an output
3. Yes: the outputs are all different
A table pairs inputs $x$ with outputs $y$ : $(x, y) = (2, 1), (2, 2), (8, 3)$ . Is $y$ a function of $x$ ?
1. No: the input 2 has two different outputs
2. Yes: every input has an output
3. Yes: the outputs are all different

Summary card

Functions

Exactly one output for each input.

Watch out: shared outputs are fine; two outputs for one input are not.

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