Functions

What Is a Function?

Understanding a function as a rule assigning exactly one output to each input.

Intermediate15 min lesson3 min readUpdated August 12, 2026Author not yet attributed

A Rule with Exactly One Output per Input

A vending machine assigns exactly one snack to each button pressed — press B4 today, and you always get the same snack. Before reading on, think about this: would a machine still make sense if pressing B4 sometimes gave chips and sometimes gave a candy bar, unpredictably? What's the one property that makes the vending machine reliable?

Definition — Function

A rule that assigns exactly one output to each input. A relationship is not a function if any single input could produce more than one possible output.

The vending machine works precisely because it's a function — every input (button) has exactly one guaranteed output (snack). A relationship where one input could produce two different outputs, like 'name a fruit that starts with the letter A' (which could mean apple or apricot), fails the function test, since the input alone doesn't pin down a single answer.

Worked Example — Testing a Table for the Function Rule

A table pairs inputs 1, 2, 3, 2 with outputs 5, 8, 11, 9. Is this a function? The input 2 appears twice, paired with two different outputs, 8 and 9. Since one input has two different outputs, this is not a function.

Worked Example — Testing a Graph for the Function Rule

A graph shows a curve that a single vertical line could cross at two different points. Does this represent a function? A vertical line at a fixed x-value represents all the outputs for that one input — crossing the curve twice means that one input has two outputs, so this graph does not represent a function.

Function Explorer

f(x) =

sin(x)

Transform: g(x) = a·f(b(x − h)) + k

a — vertical stretch/reflect1

b — horizontal stretch/reflect1

h — horizontal shift0

k — vertical shift0

Composition

Compose with another function

Trace mode

Show inverse view

Analysis (of the transformed function, in view)

y-intercept: (0, 0)
x-intercepts: (-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Extrema: local min at (-7.85, -1); local max at (-4.71, 1); local min at (-1.57, -1); local max at (1.57, 1); local min at (4.71, -1); local max at (7.85, 1)
Inflection points: (-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Vertical asymptotes: none found in view
Horizontal asymptotes: none found
Domain: all real numbers in view
Range (estimated): approximately [-1, 1]

Tip

The vertical line test is a fast visual check: if any vertical line drawn on a graph would cross the graph more than once, the graph isn't a function — some input is producing more than one output.

Common Mistakes

Checking whether outputs repeat instead of checking whether any single input repeats with different outputs.
A function only requires each input to have one output — the same output value showing up for two different inputs is completely fine and doesn't break the function rule.
Applying the vertical line test using a horizontal line instead, checking the wrong direction.
The vertical line test specifically checks for repeated x-values (inputs) — a vertical line represents a single fixed input value, so use a vertical line, not a horizontal one.

Key Takeaways

A function assigns exactly one output to every input.
A relationship fails to be a function if any single input could produce more than one output.
The vertical line test checks a graph for the function rule visually — more than one crossing means it's not a function.

Summary

Understanding what makes a rule a function sets up comparing functions given in different forms — the focus of the next lesson.

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