Functions

What is a Function?

A function is a rule that assigns exactly one output to every input. Think of it as a machine: feed in a value, get out a single value.

Notation: f(x) = 2x + 1

• f is the function name
• x is the input (independent variable)
• f(x) is the output (dependent variable), read as "f of x"

The key rule: every input must give exactly one output. An input cannot produce two different outputs.

Function Notation

Different letters can name functions: f, g, h are common.

• f(x) = 3x - 2
• g(t) = t² + 1
• h(n) = 5/n

f(4) means "evaluate f when x = 4": f(4) = 3(4) - 2 = 10

Evaluating Functions

Substitute the input value for the variable and calculate:

f(x) = 2x + 1:

• f(3) = 2(3) + 1 = 6 + 1 = 7
• f(-2) = 2(-2) + 1 = -4 + 1 = -3
• f(0) = 2(0) + 1 = 1

g(x) = x² - 3x + 1:

• g(-2) = (-2)² - 3(-2) + 1 = 4 + 6 + 1 = 11

Always put negative inputs in brackets to avoid sign errors.

Domain and Range

Domain: the set of all allowed input values Range: the set of all possible output values

For f(x) = x²:

• Domain: all real numbers (you can square any number)
• Range: x² ≥ 0, so range is all values ≥ 0

For f(x) = √x:

• Domain: x ≥ 0 (you cannot take the square root of a negative real number)
• Range: f(x) ≥ 0

For f(x) = 1/x:

• Domain: all real numbers except x = 0 (division by zero is undefined)
• Range: all real numbers except 0

Composite Functions

A composite function applies one function inside another. f(g(x)) means: apply g first, then apply f to the result.

Example: f(x) = x², g(x) = 3x

• f(g(2)): first find g(2) = 3(2) = 6, then f(6) = 6² = 36

Example: f(x) = 2x + 3, g(x) = x - 1

• f(g(4)): first g(4) = 4 - 1 = 3, then f(3) = 2(3) + 3 = 9

Note: f(g(x)) and g(f(x)) usually give different results — order matters.

Types of Functions

Type	Example	Graph shape
Linear	f(x) = 2x + 1	Straight line
Quadratic	f(x) = x²	Parabola (U-shape)
Cubic	f(x) = x³	S-shaped curve
Exponential	f(x) = 2ˣ	Rapid growth curve
Reciprocal	f(x) = 1/x	Hyperbola
Square root	f(x) = √x	Half-parabola

The Vertical Line Test

A graph represents a function if every vertical line drawn crosses the graph at most once.

• A straight line y = 2x + 1: passes the test ✓ (it is a function)
• A circle x² + y² = 9: fails the test ✗ (a vertical line through the centre crosses at two points)
• A vertical line x = 3: fails the test ✗ (every point on it has the same x but different y)

Mappings and Arrow Diagrams

A function can also be shown as an arrow diagram (mapping diagram) with inputs on the left and outputs on the right.

A valid function: every input arrow points to exactly one output. Not a function: one input arrow splits to point at two outputs.

Allowed: two different inputs can point to the same output (e.g., f(x) = x² maps both 3 and -3 to 9).

Common Mistakes

• Confusing f(x) with f × x: f(x) is notation for a function output, not multiplication.
• Wrong order in composite functions: f(g(x)) applies g first. Do not apply f first.
• Ignoring domain restrictions: always check for division by zero or square roots of negatives.
• Assuming the range equals the domain: for f(x) = x², the domain is all reals but the range is only non-negative values.

Tips and Tricks

• For composite functions, work from the inside out: in f(g(x)), evaluate g first.
• When sketching a function type, identify key features: y-intercept (set x = 0), x-intercept (set f(x) = 0).
• Linear functions have the form f(x) = mx + c, where m is the gradient and c is the y-intercept.
• If you need to find an input given an output, you are solving an equation: if f(x) = 7 and f(x) = 2x + 1, solve 2x + 1 = 7 → x = 3.