Mathematics - Algebra and Functions
Mathematics - Algebra and Functions checks manipulation, domain and range, quadratic and polynomial behavior, exponential and logarithmic equations, and sequences. The exam favors multi-step algebra where one illegal step ruins the answer.
Core rules
- A function’s domain excludes values making a denominator zero or a logarithm’s argument non-positive.
- Complete the square: ax² + bx + c = a(x + b/2a)² + (c − b²/4a).
- Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a); discriminant b² − 4ac gives the number of real roots.
- Logs: log(ab) = log a + log b; log(a/b) = log a − log b; log a^n = n log a.
- Exponential growth/decay: y = A e^{kt}; half-life uses k = −ln2/t½.
Worked example
Find the minimum of f(x) = x² − 4x + 7 and state its range. Complete the square: x² − 4x + 7 = (x − 2)² − 4 + 7 = (x − 2)² + 3. Since (x − 2)² ≥ 0, the minimum value is 3, attained at x = 2. The domain is all real x, and the range is [3, ∞). Check with the vertex formula x = −b/2a = 4/2 = 2; f(2) = 4 − 8 + 7 = 3.
Traps
- Squaring both sides and keeping extraneous roots without checking.
- Forgetting to restrict the domain after taking a logarithm.
- Misplacing signs when completing the square.
- Confusing f(x) + k (vertical shift) with f(x + k) (horizontal shift, opposite direction).
- Using the discriminant’s sign as the value of the roots.