Algebra and Functions
Algebra and Functions
This domain tests symbolic manipulation and the interpretation of functions: simplifying expressions, solving linear and quadratic equations, working with inequalities, systems, exponents, and graphs.
Core rules
- Distribute carefully: a(b+c) = ab + ac. Combine only like terms.
- Solve a linear equation ax + b = c by inverse operations in reverse order.
- For a system, substitution or elimination works; the solution must satisfy both equations.
- For ax² + bx + c = 0, the discriminant D = b² - 4ac. D > 0 gives two real roots, D = 0 one repeated root, D < 0 no real roots.
- Function notation: f(x) is the output for input x. Domain is the set of allowed inputs; range is the set of possible outputs.
- Graph transformations: f(x) + k shifts up k; f(x - h) shifts right h; af(x) stretches vertically by |a|; -f(x) reflects in the x-axis.
Worked example
Solve 2x + 3y = 12 and x - y = 1. From the second equation, x = y + 1. Substitute: 2(y + 1) + 3y = 12, so 2y + 2 + 3y = 12, 5y = 10, y = 2. Then x = 3. Check: 2(3) + 3(2) = 12 and 3 - 2 = 1. Both hold, so (3, 2) is the solution.
Traps
- Dropping a sign when subtracting an expression: -(x - 2) = -x + 2.
- Multiplying an inequality by a negative reverses the inequality sign.
- Dividing by a variable and losing the solution where it equals zero.
- Confusing f(x + 2) with f(x) + 2.
- Substituting into the quadratic formula with a sign error in b.