Functions – Rules That Connect Inputs to Outputs

A function is a rule that assigns exactly one output to every input. Functions are the heart of mathematics — they describe how one quantity depends on another, from temperature-time relationships to population growth.

The function concept and the notation f(x) were introduced by the Swiss mathematician Leonhard Euler in the 18th century, who was also responsible for popularising much of the mathematical notation still used today. Euler's original definition was broader than a formula — a function is any well-defined rule connecting inputs to outputs — which is exactly why the concept works equally well for a simple line f(x)=2x+1, a lookup table, or a piece of code that returns a value: all are functions as long as every input produces exactly one output.

What Is a Function?

A function takes an input value, applies a rule, and produces a single output. Every input gives exactly one output. Functions are often written as f(x), read as 'f of x', where x is the input.

Notation and Language

Notation	Meaning
f(x) = 2x + 1	The rule is: double the input and add 1
f(3)	The output when x = 3
Domain	The set of all valid inputs
Range	The set of all possible outputs

Evaluating Functions

f(x) = 3x - 2. Find f(4).

f(4) = 3(4) - 2 = 12 - 2 = 10.

g(x) = x squared + 1. Find g(-3).

g(-3) = (-3) squared + 1 = 9 + 1 = 10.

h(x) = 2x squared - 5x + 3. Find h(0) and h(2).

h(0) = 0 - 0 + 3 = 3. h(2) = 8 - 10 + 3 = 1.

Composite Functions

A composite function applies one function and then another. Written as fg(x) or f(g(x)), it means: first apply g to x, then apply f to the result.

f(x) = x + 2 and g(x) = 3x. Find fg(x) and gf(x).

fg(x) = f(g(x)) = f(3x) = 3x + 2. gf(x) = g(f(x)) = g(x+2) = 3(x+2) = 3x + 6. Note: fg and gf are not the same!

Inverse Functions

The inverse function undoes what the original function did. If f(x) = y then f inverse(y) = x.

Find the inverse of f(x) = 2x - 5.

Let y = 2x - 5. Rearrange for x: 2x = y + 5, so x = (y+5)/2. The inverse is f inverse(x) = (x + 5) / 2.

Real-Life Application

A taxi costs f(x) = 2x + 3 where x is the distance in kilometres. The inverse function finds how far you can travel for a given cost.

Key Takeaways

A function gives exactly one output for each input: f(x) maps x to a unique value.
Evaluate by substituting the input for x and calculating.
Composite functions apply two rules in sequence; order matters.
Inverse functions reverse the mapping: they undo the original function.

Practice: Functions

Evaluate the Function