Functions formalise the concept of mathematical relationships. Understanding domain, range, composition, inverses, and transformations is essential for the entire A Level course.
Function: a mapping where each input has exactly one output. Domain: set of allowed input values. Range: set of possible output values. One-to-one (1:1): each output from exactly one input — has an inverse. Many-to-one: different inputs can give same output — must restrict domain for inverse. Notation: f(x) = 2x + 3 or f: x ↦ 2x + 3.
Composite function fg(x) = f(g(x)): apply g first, then f. Order matters: fg ≠ gf generally. Domain of fg: values where g(x) is in domain of f. Inverse function f⁻¹: reverses f. To find: let y = f(x), rearrange for x in terms of y, swap x and y. Graph of f⁻¹ is reflection of f in y = x line. ff⁻¹(x) = f⁻¹f(x) = x. Only 1:1 functions have inverses.
f(x) + a: translate up by a. f(x + a): translate left by a. af(x): vertical stretch factor a. f(ax): horizontal stretch factor 1/a. -f(x): reflect in x-axis. f(-x): reflect in y-axis. |f(x)|: reflect negative y-values in x-axis. f(|x|): keep x ≥ 0 part, reflect in y-axis. Combination: apply inside bracket first, then outside.
An inverse function must be one-to-one — each output maps back to exactly one input. If the original function is many-to-one (like f(x) = x²), then two different inputs give the same output (e.g., f(2) = f(-2) = 4). If we tried to find the inverse, we would not know whether 4 should map back to 2 or -2. By restricting the domain (e.g., x ≥ 0 for x²), the function becomes one-to-one, and the inverse is well-defined (f⁻¹(x) = √x). The restricted domain of f becomes the range of f⁻¹.
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