Mathematics: Algebra, Functions, and Trigonometry
What the Mathematics section tests
The Mathematics domain samples algebra, functions, trigonometry, and introductory calculus or precalculus. Expect manipulation of expressions, solving equations and inequalities, interpreting graphs, working with exponents and logarithms, and applying trigonometric identities. Some items are set in chemical or physical contexts, such as concentration or rate, but the mathematics is the same.
Core rules
- Apply exponent laws before factoring: a^m a^n = a^(m+n), (a^m)^n = a^(mn), and a^(-n) = 1/a^n. For logarithms, log_b(xy) = log_b x + log_b y and log_b(x^n) = n log_b x.
- For quadratics, check whether factoring is quick; otherwise use x = (-b ± sqrt(b^2 - 4ac))/(2a). The discriminant b^2 - 4ac tells you the number of real roots.
- In trigonometry, work in radians unless degrees are stated. Know the unit-circle signs by quadrant and the Pythagorean identity sin^2 θ + cos^2 θ = 1.
- For functions, composition f(g(x)) is not multiplication, and the domain of f(g(x)) excludes values that make g(x) undefined or outside f's domain.
- In calculus, the derivative gives instantaneous rate and the definite integral gives accumulated change; check units.
Worked example
Solve 2^(x+1) = 16^(x-1). Write 16 = 2^4, so the right side is 2^(4(x-1)) = 2^(4x-4). Equal bases give x + 1 = 4x - 4. Then 5 = 3x, so x = 5/3. Substituting back: left exponent 8/3; right exponent 4(5/3) - 4 = 8/3, so the solution checks.
Traps that cost marks
Do not cancel terms across a sum; cancel factors only. When squaring both sides, check for extraneous roots. Keep track of negative signs and of whether a calculator is in degree or radian mode. In word problems, convert units before computing.