| symbol | latex code | explanation | |
|---|---|---|---|
f(x) | f(x) | The value of function f at input x — read 'f of x' (function notation); the parentheses do not indicate multiplication | |
y = f(x) | y = f(x) | Output variable y defined by the rule f applied to x | |
f: A → B | f\colon A \to B | Function f mapping the set A (domain) into the set B (codomain) | |
x ↦ f(x) | x \mapsto f(x) | 'x maps to f(x)' — defines a function without naming its output variable | |
g(t), P(n) | g(t), \; P(n) | Any letters may name a function and its input; the choice often reflects context (t for time, n for integers) | |
f(a) | f(a) | The output of f at a specific input a — e.g. f(3) is the value at x = 3 | |
Dom(f) | \operatorname{Dom}(f) | The domain of f — the set of allowed inputs | |
Ran(f) | \operatorname{Ran}(f) | The range of f — the set of outputs actually produced | |
{x | x ≥ 0} | \{x \mid x \geq 0\} | Set-builder notation: 'the set of all x such that x ≥ 0' (expressing domain) | |
ℝ | \mathbb{R} | The set of all real numbers — the default universe for domains and ranges | |
x ∈ A | x \in A | x is an element of the set A | |
[a, b] | [a, b] | Closed interval — both endpoints included | |
(a, b) | (a, b) | Open interval — both endpoints excluded | |
[a, b) | [a, b) | Half-open interval — a included, b excluded | |
(−∞, 3] | (-\infty, 3] | All numbers up to and including 3; infinity always takes a parenthesis | |
(−∞, −1) ∪ (1, ∞) | (-\infty, -1) \cup (1, \infty) | Union of intervals — a disconnected domain | |
∞ | \infty | Infinity — an unbounded direction, not a reachable number | |
(f ∘ g)(x) | (f \circ g)(x) | Composition: apply g first, then f — equal to f(g(x)) | |
∘ | \circ | The composition symbol — read 'composed with'; in f ∘ g the right-hand function acts first | |
f(g(x)) | f(g(x)) | Nested form of composition — evaluate from the inside out | |
f⁻¹(x) | f^{-1}(x) | The inverse function of f — the superscript −1 is a label, not an exponent | |
(f(x))⁻¹ | (f(x))^{-1} | The reciprocal 1/f(x) — parentheses distinguish it from the inverse function | |
f⁻¹(f(x)) = x | f^{-1}(f(x)) = x | The defining property: the inverse undoes the function | |
(f + g)(x) | (f + g)(x) | Sum of functions: equals f(x) + g(x) (function arithmetic) | |
(f − g)(x) | (f - g)(x) | Difference of functions: equals f(x) − g(x) | |
(f · g)(x) | (f \cdot g)(x) | Product of functions: equals f(x) · g(x) | |
(f / g)(x) | (f / g)(x) | Quotient of functions: equals f(x) / g(x), defined where g(x) ≠ 0 | |
f(x) = { … cases … | f(x) = \begin{cases} x^2 & x < 0 \\ x + 1 & x \geq 0 \end{cases} | Brace notation: each line pairs a formula with the condition under which it applies | |
1 ≤ x < 4 | 1 \leq x < 4 | A typical piece condition — the conditions must be mutually exclusive | |
f(−x) = f(x) | f(-x) = f(x) | Even function — graph symmetric about the y-axis | |
f(−x) = −f(x) | f(-x) = -f(x) | Odd function — graph symmetric about the origin | |
f(x + T) = f(x) | f(x + T) = f(x) | Periodic function with period T |