Algebra and Functions (A2 Unit 3: Pure Mathematics B) — WJEC A-Level Mathematics
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Algebra and Functions (A2 Unit 3: Pure Mathematics B) explained
A composite function fg(x) = f(g(x)) applies function g first and then function f to the output of g.
Read the full explanation
For fg(x) to exist, the range of g must be a subset of the domain of f. Composition is not generally commutative: fg(x) ≠ gf(x). An inverse function f⁻¹(x) exists if and only if f(x) is one-to-one (injective) over its domain. The domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f, satisfying f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. Geometrically, the graph of y = f⁻¹(x) is the reflection of y = f(x) in the line y = x; points of intersection between a function and its inverse lie along y = x. If a function is not one-to-one (like a quadratic), its domain must be restricted (e.g. x ≥ h) to define an inverse.
Your focus
- Form and simplify composite functions fg(x) and evaluate their domain and range.
- Determine conditions under which a function has an inverse and calculate f⁻¹(x) algebraically.
- Sketch inverse functions as reflections in the line y = x and locate intersection points.
Algebra and Functions (A2 Unit 3: Pure Mathematics B) exam tips
Marking Points
- forming composite functions by substituting g(x) into f(x) and simplifying
- determining the domain and range of composite and inverse functions
- finding the inverse function f⁻¹(x) algebraically by rearranging x = f(y)
- sketching y = f(x) and y = f⁻¹(x) showing reflection in the line y = x
Examiner Tips
- 💡To find f⁻¹(x): write y = f(x), swap x and y, and rearrange to make y the subject.
- 💡Remember that the range of f is the domain of f⁻¹, which must be stated explicitly if asked.
Common Mistakes
- evaluating composite functions in the wrong order, calculating gf(x) instead of fg(x)
- confusing the inverse function f⁻¹(x) with the reciprocal 1/f(x)
- forgetting to restrict the domain of a many-to-one function before finding its inverse