Functions and Graphs
Core ideas
A function is a rule assigning each input in its domain exactly one output. Candidates must handle function notation, implied (maximal) domains, composite functions and inverse functions. Know that an inverse exists only when the function is one-to-one on its stated domain; restricting the domain of a parabola is the standard fix. Be fluent transforming graphs: y = a f(b(x − h)) + k applies dilations then translations, and the order matters when describing transformations.
Traps
- Forgetting that the implied domain of f(x) = √(x − 3) is x ≥ 3, and of log expressions is the argument being positive.
- For composite f(g(x)), the domain must exclude x where g(x) falls outside the domain of f.
- Confusing the graph of f⁻¹ with 1/f; the inverse is the reflection in y = x.
- Sign errors when finding x-intercepts of factorised quartics.
- Ignoring asymptotes and behaviour near excluded points when sketching.
Worked example
Let f(x) = x² − 4x + 1 for x ≥ 2. Find f⁻¹. Complete the square: f(x) = (x − 2)² − 3. Since x ≥ 2, (x − 2)² = y + 3 and x = 2 + √(y + 3). Swap variables: f⁻¹(x) = 2 + √(x + 3), with domain x ≥ −3 and range y ≥ 2. Check endpoints: f(2) = −3 maps to f⁻¹(−3) = 2. Rule of thumb: always state domain and range of an inverse, and verify with one test point.