Mathematics: Algebra, Functions, and Reasoning
Core concepts to master
Algebraic fluency is essential: manipulating expressions, solving linear and quadratic equations, factoring, and working with inequalities. Functions require understanding domain, range, composition, and inverses, plus the graphs of linear, quadratic, exponential, and logarithmic functions. Geometry covers similar triangles, the Pythagorean theorem, circle properties, area, and volume. Trigonometry involves the sine, cosine, and tangent ratios and the unit circle. Sequences and series, including arithmetic and geometric patterns, appear frequently. Probability and basic statistics, such as mean, median, and combinations, round out the syllabus.
Traps to watch
When solving inequalities, remember that multiplying or dividing by a negative number flips the sign. Do not cancel terms across an addition sign; only factors can be cancelled. In quadratics, check both roots and confirm they satisfy the original equation. Distinguish permutations (order matters) from combinations (order does not). Watch for extraneous solutions introduced by squaring both sides. Read whether a question asks for the value or for a simplified expression.
Worked example
Solve x squared minus 5x plus 6 equals zero. Factor to (x minus 2)(x minus 3) equals zero, so x equals 2 or x equals 3. Verify: 4 minus 10 plus 6 equals zero, and 9 minus 15 plus 6 equals zero. Rule of thumb: for quadratics, try factoring first, then the quadratic formula if factoring fails, and always substitute answers back.