Key Idea: A function is a rule that takes every input and produces exactly one output. Topic 2.2 introduces the language and notation of functions — f(x), domain, range, composition, and inverses — which underpins everything in Units 2 and 5. These are not just definitions: they describe how inputs and outputs behave, which matters every time you use a model or interpret a graph.
✅ Core vocabulary
Example: Find f(g(x)) where f(x) = 3x − 1 and g(x) = x²: g(x) = x² → f(g(x)) = f(x²) = 3x² − 1 Find f(2): f(2) = 3(4) − 1 = 11 Find f⁻¹(x) for f(x) = 2x + 5: Step 1: write y = 2x + 5 Step 2: swap x and y → x = 2y + 5 Step 3: solve for y → y = (x − 5)/2 So f⁻¹(x) = (x − 5)/2
f(g(x)) means 'plug g(x) into f' — substitute the entire expression for g(x) in place of x in f. The domain of the composite f(g(x)) is restricted to inputs where g(x) is defined AND the output of g is in the domain of f. For inverse functions: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. These two cancel each other out.
Paper 1 (GDC allowed): Evaluate composites by careful substitution. For inverse functions, show the 'swap and solve' steps clearly. Paper 2 (GDC allowed): Use the GDC to verify function values. Graphing f and f⁻¹ together shows them as reflections in y = x — a useful visual check.