Function Analysis: Domain, Monotonicity, and Parity
Gaokao Mathematics Section I (选择题 and 填空题) and Section II (解答题) both heavily feature function analysis. Function questions test: domain and range, odd/even (parity), monotonicity, symmetry, and periodicity. These properties form the foundation of most Gaokao function questions — including questions about composite functions, inverse functions, and functions defined piecewise.
Domain and range: domain is the set of x-values for which the function is defined. Common domain restrictions: for f(x) = √(g(x)), require g(x) ≥ 0; for f(x) = log_a(g(x)), require g(x) > 0 and a > 0, a ≠ 1; for rational functions f(x) = p(x)/q(x), require q(x) ≠ 0. Range determination requires identifying the output set — typically done by solving for x in terms of y and identifying which y-values are achievable, or by completing the square for quadratic functions (range is [minimum value, ∞) for upward-opening parabola).
Monotonicity: f(x) is monotonically increasing on interval I if x₁ < x₂ implies f(x₁) < f(x₂). For differentiable functions: f is increasing where f'(x) > 0 and decreasing where f'(x) < 0. For Gaokao problems without calculus (required at some difficulty levels): use algebraic comparison — for f(x₁) − f(x₂), factor and determine sign based on whether x₁ − x₂ > 0 or < 0. The formal proof of monotonicity (设x₁, x₂ ∈ I, 且x₁ < x₂, 则f(x₁) − f(x₂) = ..., 因此..., 故f在I上单调递增) is required for full marks in 解答题.
Parity (奇偶性): a function is even if f(−x) = f(x) for all x in the domain (symmetric about y-axis). A function is odd if f(−x) = −f(x) for all x (symmetric about origin). Precondition: the domain must be symmetric about the origin for a function to be odd or even. Gaokao questions frequently test: is f(x) = x·ln(|x|/x + 1) odd or even? — requiring students to check domain symmetry first, then test the parity condition algebraically.