Algebra and Functions
Algebra and Functions
Algebra and functions is the language layer of the Cameroon GCE A-Level Pure Mathematics with Mechanics paper: it supplies the manipulation skills that coordinate geometry, calculus and mechanics questions all assume. The exam tests domain and range, composite and inverse functions, the modulus function, polynomials (factor and remainder theorems), partial fractions, inequalities and graph transformations.
Functions: one-one, composite, inverse
A function is one-one only if every output value comes from exactly one input, and that is precisely the condition for an inverse to exist as a function. For f∘g the domain is the set of x in the domain of g for which g(x) lies in the domain of f. The range of f⁻¹ equals the domain of f, so state it explicitly. To find an inverse, write y = f(x) and solve for x in terms of y.
Polynomials and partial fractions
Remainder theorem: the remainder when f(x) is divided by (x − a) is f(a); dividing by (ax − b) means evaluating at x = b/a. Factor theorem: (x − a) is a factor exactly when f(a) = 0. Before splitting an improper rational function into partial fractions, divide out first so the numerator degree is lower than the denominator.
Modulus
|X| = k gives X = ±k. |X| < k is equivalent to −k < X < k, while |X| > k splits into X > k or X < −k. Sketch the graph before writing the solution set.
Worked example
f(x) = (3x − 1)/(x + 2) for x ≠ −2. Find f⁻¹ and its domain. Let y = (3x − 1)/(x + 2). Then y(x + 2) = 3x − 1, so yx + 2y = 3x − 1 and x(y − 3) = −1 − 2y, giving x = (2y + 1)/(3 − y). Hence f⁻¹(x) = (2x + 1)/(3 − x). The denominator shows x ≠ 3, and indeed f(x) = 3 − 7/(x + 2) can never equal 3, so the domain of f⁻¹ is x ≠ 3, matching the range of f.
Traps that cost marks
- Giving an inverse without its domain restriction.
- Squaring a modulus inequality and keeping solutions outside the valid interval.
- Starting partial fractions on an improper fraction instead of dividing out first.
- Stating the range as an inequality in x rather than in y.
- Cancelling a term across a sum instead of a factor.