Function Formula

Q: What do the symbols mean in the Function formula?

$f(x)$ denotes the output of function $f$ at input $x$. Also written $f\colon X \to Y$.

Q: What do students get wrong about Function?

A circle is not a function (fails vertical line test)—one $x$ gives two $y$ values.

The Formula

y = f(x)

When to use: A machine: put something in, get exactly one thing out. Same input always gives same output.

Quick Example

f(x) = x^2 Input 3, output 9. Input -3, output 9. Never two different outputs for one input.

Notation

f(x) denotes the output of function f at input x. Also written f\colon X \to Y.

What This Formula Means

A function is a rule that assigns to each input in the domain exactly one output in the codomain — every input maps to precisely one output, never two.

A machine: put something in, get exactly one thing out. Same input always gives same output.

Formal View

f\colon X \to Y is a function \iff \forall x \in X,\; \exists!\, y \in Y: (x, y) \in f

Worked Examples

Example 1

easy

Determine whether the relation \{(1, 3), (2, 5), (3, 7), (4, 9)\} is a function.

Solution

1
A relation is a function if every input (first element) maps to exactly one output (second element).
2
List the inputs: 1, 2, 3, 4. Each input appears exactly once, so each has a unique output.
3
Since no input is repeated with a different output, this relation is a function.

Answer

\text{Yes, it is a function.}

A function assigns exactly one output to each input. The vertical line test for graphs and the uniqueness of first elements in ordered pairs are equivalent ways to check this.

Example 2

medium

Does the equation x^2 + y^2 = 25 define y as a function of x?

Example 3

medium

Given f(x) = 2x^2 - 3x + 1, find f(4) and determine if g = \{(1,2), (3,4), (1,5)\} is a function.

Common Mistakes

Thinking every equation is a function — x^2 + y^2 = 1 (a circle) is NOT a function because one x gives two y values
Confusing 'undefined' with 'zero' — f(x) = \frac{1}{x} at x = 0 is undefined, not f(0) = 0
Believing a function must have a formula — functions can be defined by tables, graphs, or verbal rules

Why This Formula Matters

Functions are the central objects of mathematics—they describe relationships.

Frequently Asked Questions

What is the Function formula?

A function is a rule that assigns to each input in the domain exactly one output in the codomain — every input maps to precisely one output, never two.