Functions, Domains & Function Notation

What Is a Function?

A function is a relation between two sets — the domain (set of allowable inputs) and the codomain (set of possible outputs) — where every element of the domain is mapped to exactly one element of the codomain. This is the defining property: one input, one output. No x-value may be paired with two different y-values.

We write f : A → B, read "f is a function from A to B", where A is the domain and B is the codomain. The rule of the function tells us what to do to each input. For example, f : ℝ → ℝ, f(x) = x² maps every real number to its square.

Distinguishing functions from non-functions: The vertical line test is the graphical check — a relation is a function if and only if every vertical line drawn through the graph intersects it at most once. For instance, the circle x² + y² = 4 is a relation but not a function, because the vertical line x = 0 intersects it at (0, 2) and (0, −2).

Worked Example: Is y² = x a function?
Choose x = 4. Then y² = 4, so y = 2 or y = −2. The input x = 4 produces two outputs, so this is not a function. ✓

By contrast, y = x² is a function: for each x, x² produces exactly one value. The vertical line test confirms: every vertical line hits the parabola at most once. ✓

Note: in Methods, functions are typically real-valued and defined on real-number domains. The notation f(x) denotes the output (or image) of x under f, and is read "f of x".

Domain, Codomain and Range

The domain of a function is the complete set of input values for which the rule is defined and produces a real output. The range (also called the image) is the actual set of output values produced — it is always a subset of the codomain.

Maximal domain: when no domain is stated, we assume the maximal (natural) domain — the largest subset of ℝ for which the rule yields a real number. Three key restrictions apply:

• You cannot divide by zero: exclude any x that makes a denominator equal zero.
• You cannot take the square root (or any even root) of a negative number (in the reals): exclude any x that makes the expression under the radical negative.
• You cannot take the logarithm of zero or a negative number: for log_a(g(x)), require g(x) > 0.

Worked Example 1 — Rational function: Find the maximal domain of f(x) = 3/(x − 2).
Denominator: x − 2 ≠ 0 ⟹ x ≠ 2.
Maximal domain: ℝ \ {2} = (−∞, 2) ∪ (2, +∞). ✓

Worked Example 2 — Square root function: Find the maximal domain and range of f(x) = √(9 − x²).
Require 9 − x² ≥ 0 ⟹ x² ≤ 9 ⟹ −3 ≤ x ≤ 3.
Maximal domain: [−3, 3].
Now find the range: √(9 − x²) ≥ 0 always, and the maximum occurs at x = 0: √9 = 3. As x → ±3, √(9 − x²) → 0.
Range: [0, 3]. ✓ (Geometrically, this is the upper semicircle of radius 3.)

Worked Example 3 — Logarithmic function: Find the maximal domain of f(x) = log_e(2x − 6).
Require 2x − 6 > 0 ⟹ 2x > 6 ⟹ x > 3.
Maximal domain: (3, +∞). ✓
Range: log_e of a positive quantity ranges over all of ℝ, so range = ℝ.

When a domain is specified, the range must be determined by examining the function's behaviour (including turning points, endpoints, and asymptotes) over that restricted domain — not the maximal domain.