Advanced Gaokao Mathematics: Functions, Sequences, and Probability

Functions constitute one of the largest and highest-difficulty topic areas in Gaokao Mathematics (both 理科 and 文科). Advanced function problems typically involve: domain and range determination for complex expressions, composite function evaluation f(g(x)) and decomposition, inverse function calculation (replace f(x) with y, solve for x, swap x and y), and piecewise function behaviour analysis. Key technique for composite functions: always start from the innermost function and work outward. For inverse functions, a critical Gaokao pattern is asking for the domain of f⁻¹(x) — remember that the domain of the inverse is the range of the original function. Piecewise functions require testing which piece applies for each input value before substituting. Common 难点 (difficulty point): problems that ask you to find a piecewise function's parameters (a, b) given that the function is continuous or monotonic — set the pieces equal at the boundary point for continuity, and compare derivatives at the boundary for monotonicity. For 理科 students, logarithmic and exponential function transformations (translations, reflections, scaling) are high-frequency and require both algebraic manipulation and graphical reasoning.