Math Core

Press B4 on a vending machine and you get pretzels. Press B4 again tomorrow and you get pretzels again. You never press one button and get two different snacks. A function works the same way: you put in an input, and you get back exactly one output.

Inputs and outputs

A function is often given as a rule. Here is one: "multiply the input by 2, then add 1." Feed it some inputs and record what comes out.

input, $x$	0	1	2	3	4
output, $y$	1	3	5	7	9

You can also write the rule as an equation, $y = 2x + 1$ , or as a list of ordered pairs (input, output):

(0, 1),\ (1, 3),\ (2, 5),\ (3, 7),\ (4, 9)

Every input goes in and one definite output comes out. That is the whole idea.

Definition

Function

A function is a rule that assigns to each input exactly one output. We often call the input $x$ and the output $y$ , and write the pairs as ordered pairs $(x, y)$ .

Checking a table or a list of pairs

Sometimes you are handed a table with no rule at all. To decide whether it is a function, look at the inputs.

How to spot a function

If some input appears twice with different outputs, it is not a function.
If every input has just one output, it is a function.
Different inputs sharing the same output is fine.

Worked example: Pairs of numbers

Which of these sets of ordered pairs is a function?

Set A: $(1, 4),\ (2, 5),\ (3, 4),\ (6, 0)$
Set B: $(2, 1),\ (3, 6),\ (2, 8),\ (5, 5)$

Set A is a function. The inputs 1, 2, 3 and 6 each appear once. The output 4 shows up twice (for inputs 1 and 3), but that's allowed. Two buttons on a vending machine can both give pretzels.

Set B is not a function. The input 2 is paired with 1 and with 8. One input, two outputs.

Common mistake

Repeated outputs do not break a function. Only a repeated input with a different output does. Always look down the input column for repeats.

Functions on a graph

When you graph the pairs, each input is a spot on the $x$ -axis. Here are the points from Set A:

Set A. No two points sit on the same vertical line.Open in grapher →

If two points were stacked one above the other, they would share an input but have different outputs. That gives a quick visual check.

The vertical line test

A graph shows a function if no vertical line crosses it in more than one point.

A circle fails the test. The vertical line $x = 2$ crosses this circle twice, near $(2, 3.5)$ and $(2, -3.5)$ . The input 2 would have two outputs.

The circle is not the graph of a function: the line x = 2 hits it twice.Open in grapher →

A non-vertical straight line always passes. Any vertical line crosses $y = 2x + 1$ exactly once.

y = 2x + 1 is a function. Every vertical line meets it once.Open in grapher →

Functions in everyday situations

Worked example: Is it a function?

Input: a student in your class. Output: that student's birthday.
Input: a birthday month. Output: a student born in that month.

Solutions.

Function. Each student has exactly one birthday. (Two students might share a birthday; that's two inputs with the same output, which is fine.)
Not a function. The input "March" could give several different students.

Worked example: Using a rule

A function has the rule $y = 3x - 4$ .

Find the output when the input is 5.
Which input gives an output of 17?

Solutions.

$y = 3(5) - 4 = 15 - 4 = 11$ .
Solve $3x - 4 = 17$ . Add 4: $3x = 21$ . Divide by 3: $x = 7$ . Check: $3(7) - 4 = 17$ . ✓

Tip

To check a rule, plug your answer back in. If it gives the output you expected, you're done.

Practice

Practice 1

Which set of ordered pairs is a function?

Practice 2

A function has the rule $y = 4x + 3$ . What is the output when the input is 6?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

A function has the rule $y = 5x - 2$ . Which input gives an output of 33?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

Is this table a function?

$x$	2	4	4	6
$y$	1	3	5	7

Practice 5

The graph shows four points. It is not a function. Which input causes the problem?

(1, 2)(2, 4)(2, -1)(4, 3)Open in grapher →

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

Which of these is not a function?

Practice 7

The set $(1, 4),\ (3, 6),\ (a, 9),\ (5, 2)$ is a function. Which value can $a$ not be?