Functions
Functions opens the SPM Additional Mathematics syllabus and tests whether you can use function notation as a working language rather than just recite definitions. Expect composite functions, inverse functions, and questions that link the domain, the range and the graph.
Core rules
- A function assigns exactly one image to each object in its domain. The set of images is the range, read from the y-axis of the graph.
- The composite fg means "apply g first, then f", so fg(x) = f(g(x)). In general fg ≠ gf.
- An inverse f⁻¹ exists only when f is one-to-one. The domain of f⁻¹ equals the range of f, and the graph of f⁻¹ is the reflection of y = f(x) in the line y = x.
- To find an inverse, let y = f(x), rearrange to make x the subject, then replace y with x.
Worked example
Given f(x) = (2x+3)/(x−1), x ≠ 1, find f⁻¹. Let y = (2x+3)/(x−1). Then y(x−1) = 2x+3, so yx − y = 2x+3. Collect the x terms: x(y−2) = y+3, giving x = (y+3)/(y−2). Replace y with x: f⁻¹(x) = (x+3)/(x−2), x ≠ 2. The restriction x ≠ 2 is the value that f never takes, confirming that the range of f becomes the domain of f⁻¹.
Traps
Writing fg when gf is required; treating f⁻¹(x) as 1/f(x); omitting the domain restriction on the inverse; assuming a many-to-one function has an inverse.