22 min read

What Is a Function?

A function is a special relation where every input has exactly one output.

The One-Output Rule

The key test for a function: no input is allowed two different outputs. One input, one output — always. A relation that gives some input two different outputs is not a function.

Example 1 — a function

Is {(1, 2), (2, 4), (3, 6)} a function? 1 Check every input: 1, 2, 3 — none repeats. 2 Each input has exactly one output. Answer: Yes, this is a function.

Example 2 — not a function

Is {(1, 2), (1, 5), (2, 4)} a function? 1 Check every input: 1 appears twice, connected to both 2 and 5 . 2 Input 1 has two different outputs. Answer: No, this is not a function.

A function is usually named f, g, or h, and written f(x), read "f of x ". f(x) is the output when the input is x — it does not mean f multiplied by x . f(x) = 2 x + 1 is the rule "double the input, then add 1 ".

Function machine · f(x) = 2x + 1

Inputf(x) = 2x + 1

→ 3

Change a value and watch:

Every input produces exactly one output. That is what makes this a function.

f(x) = 2 x + 1 doubles the input and adds 1 . Try tracing each input through the machine.

Practice

Which of these relations is NOT a function?