Algebra and Functions
Core content
Candidates must solve linear equations and inequalities, systems of equations, quadratic equations, and simple polynomial, rational, exponential, and logarithmic equations. Understand function notation, domain and range, slope and intercepts, transformations, composition, and inverses. Recognise key features of graphs, including zeros, extrema, asymptotes, and end behaviour. Quadratic tools include factoring, completing the square, the quadratic formula, and the discriminant.
Trap watch
Multiplying or dividing an inequality by a negative number reverses the inequality sign. Squaring both sides can introduce extraneous solutions, so every proposed answer must be checked in the original equation. Rational expressions are undefined where a denominator is zero, and logarithms require a positive argument. In composition, f(g(x)) is not usually g(f(x)); in inverses, swap x and y and check domain restrictions. Do not assume a function is invertible without the horizontal line test.
Worked example
Solve sqrt(x+3) = x - 3. Squaring gives x+3 = x^2 - 6x + 9, so x^2 - 7x + 6 = 0 and x = 1 or x = 6. Checking: x = 1 gives 2 on the left and -2 on the right, so reject it; x = 6 gives 3 = 3, so the only solution is x = 6.
Rule of thumb: solve, then test each result in the original domain before accepting it.