Algebraic Reasoning and Functions
What the Domain Covers
This domain tests rewriting expressions into useful equivalent forms, solving equations and inequalities with attention to domain, and reasoning about function behavior - notation, domain and range, transformations, inverses, growth, and sequences.
Equivalent Forms and Solving
Solving is disciplined rewriting. Factoring, expanding, completing the square, and rationalizing each expose different structure. The quadratic formula x = [−b ± √(b² − 4ac)]/(2a) comes from completing the square, and the discriminant b² − 4ac decides the outcome: positive gives two real roots, zero gives one repeated real root, negative gives a conjugate pair of complex roots.
Functions
A function assigns exactly one output to each input. Domain restrictions come from denominators (nonzero), even roots (radicand ≥ 0), and logarithms (argument > 0). Composition f(g(x)) is substitution, not multiplication. To invert, swap x and y, then solve; the domain of f⁻¹ equals the range of f. For transformations, a·f(b(x − h)) + k stretches vertically by |a|, compresses horizontally by 1/|b|, shifts right h, and shifts up k, with reflections when a or b is negative.
Growth, Logarithms, Sequences
Linear functions change by a constant difference; exponential functions change by a constant ratio, f(x) = a·b^x with b = 1 + r. Arithmetic sequences use a_n = a_1 + (n − 1)d; geometric sequences use a_n = a_1·r^(n−1). Logarithms invert exponentials: log_b(xy) = log_b x + log_b y and log_b(x^p) = p·log_b x.
Worked Example
Rewrite f(x) = 2x² − 12x + 7 in vertex form. Factor the leading coefficient from the variable terms: f(x) = 2(x² − 6x) + 7. Complete the square: x² − 6x = (x − 3)² − 9. Substitute: f(x) = 2[(x − 3)² − 9] + 7 = 2(x − 3)² − 11. Vertex (3, −11), axis of symmetry x = 3, minimum value −11, range [−11, ∞).
Traps That Cost Marks
- Dividing both sides by a variable and losing a root: x² = 4x gives x = 0 or x = 4, not just 4.
- Squaring both sides of a radical equation creates extraneous solutions - always verify.
- Treating f(x + h) as f(x) + h.
- Reporting a domain in x when the model's variable is t.