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What is High School / Foundations

a Function?

The fundamental building block of all mathematics — a rule that assigns exactly one output to every input.

Simple Definition

A function is a rule that takes an input, does something to it, and produces exactly one output. Think of it as a machine: you put something in, you get something out — and every time you put the same thing in, you get the same thing out.

The critical word is "exactly one." An input cannot produce two different outputs — that would break the rule. But two different inputs can produce the same output (that's fine).

Functions are everywhere: the square root of a number, the price of a product as a function of quantity, the position of a car as a function of time.

Input

→

Function

f(x) = x²

→

Output

x²

Intuition

🏧

The vending machine model

Press button A3 → you always get the same snack. That's a function. Now imagine a broken machine where pressing A3 sometimes gives chips, sometimes gives nothing, sometimes gives two things. That's not a function. Consistency and uniqueness of output are everything.

✅

The vertical line test

Draw the graph of any relation. If any vertical line crosses the graph more than once, it's not a function — because one x-value would map to multiple y-values. If every vertical line hits the graph at most once, it passes the test and is a function.

Formal Definition

A function $f$ from a set $A$ to a set $B$, written $f: A \to B$, is a relation such that:

Set-theoretic definition

$$\forall\, x \in A,\quad \exists!\, y \in B \text{ such that } f(x) = y$$

The symbol $\exists!$ means "there exists exactly one." Set $A$ is the domain (all valid inputs) and set $B$ is the codomain (where outputs live). The set of actual outputs is called the range.

Types of Functions

Linear

$f(x) = mx + b$

Constant rate of change. Produces a straight line. Slope is $m$, y-intercept is $b$.

Quadratic

$f(x) = ax^2 + bx + c$

Degree 2. Produces a parabola. Opens up if $a > 0$, down if $a < 0$.

Exponential

$f(x) = a^x$

Grows (or decays) multiplicatively. Models population, interest, radioactive decay.

Trigonometric

$f(x) = \sin(x),\,\cos(x)$

Periodic oscillating functions. Period $2\pi$. Used in waves, circles, physics.

Piecewise

Different rules per interval

Uses different formulas depending on the value of $x$. Common in real-world modelling.

Inverse

$f^{-1}(x)$

Undoes what $f$ does. Exists only if $f$ is one-to-one. Reflects the graph across $y = x$.

Key Notation & Properties

Notation	$f(x)$ — "f of x", the output when input is $x$
Composition	$(f \circ g)(x) = f(g(x))$ — apply $g$ first, then $f$
Inverse	$f(f^{-1}(x)) = x$ — inverse undoes the function
Even function	$f(-x) = f(x)$ — symmetric about the y-axis
Odd function	$f(-x) = -f(x)$ — symmetric about the origin
One-to-one	Different inputs always give different outputs — passes horizontal line test

Quick Examples

Example 1 — Evaluating

$f(x) = 2x^2 - 3$

Find $f(4)$:
$f(4) = 2(16) - 3 = 29$

Example 2 — Composition

$f(x)=x^2,\ g(x)=x+1$

$(f \circ g)(x) = f(g(x))$
$= (x+1)^2 = x^2+2x+1$

Example 3 — Inverse

$f(x) = 3x + 6$

Swap $x$ and $y$, solve for $y$:
$f^{-1}(x) = \dfrac{x-6}{3}$

Example 4 — Not a function

$x^2 + y^2 = 1$

A circle. For $x=0$, $y=\pm 1$ — two outputs for one input. Fails the vertical line test.

Common Mistakes

✗ Confusing f(x) with f times x

$f(x)$ means "f evaluated at x" — it is not multiplication. Writing $f(x+2) = f(x) + 2$ is incorrect; you must substitute $x+2$ into the entire expression for $f$.

✗ Thinking every curve is a function

A circle, an ellipse, or any curve that loops back on itself is not a function of $x$. Always apply the vertical line test when in doubt.

✗ Inverting without checking one-to-one

$f(x) = x^2$ has no inverse on all of $\mathbb{R}$ because $f(2) = f(-2) = 4$. You must restrict the domain (e.g. $x \geq 0$) before the inverse $f^{-1}(x) = \sqrt{x}$ is well-defined.

Related Concepts

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Notation	\(f(x)\) — "f of x", the output when input is \(x\)
Composition	\((f \circ g)(x) = f(g(x))\) — apply \(g\) first, then \(f\)
Inverse	\(f(f^{-1}(x)) = x\) — inverse undoes the function
Even function	\(f(-x) = f(x)\) — symmetric about the y-axis
Odd function	\(f(-x) = -f(x)\) — symmetric about the origin
One-to-one	Different inputs always give different outputs — passes horizontal line test