Relations and Functions
Relations and Functions is the language of mapping between sets. The exam tests Cartesian products, types of relations, domain and range, injective, surjective and bijective functions, composition and inverse functions.
Core rules
- A relation from A to B is any subset of A × B; a function assigns exactly one output to each input.
- An equivalence relation is reflexive, symmetric and transitive.
- f is one-one if f(a)=f(b) implies a=b; it is onto if every codomain element is an image.
- For f: A → B and g: B → C, (g ∘ f)(x)=g(f(x)); f has an inverse only when bijective.
Worked example
Let f(x)=(3x-2)/(x+1), x ≠ -1. Find f inverse. Put y=(3x-2)/(x+1). Then y(x+1)=3x-2, so yx+y=3x-2 and x(y-3)= -y-2. Hence x=(y+2)/(3-y). Therefore f inverse(y)=(y+2)/(3-y), with y ≠ 3. The domain of the inverse is the range of f, R minus {3}.
Traps
- Do not read f inverse as 1/f.
- Check the denominator and the restricted domain before simplifying.
- Composition order matters: f ∘ g is not usually g ∘ f.
- A relation can fail the vertical-line test; a function cannot have two outputs for one input.
- For equivalence, missing transitivity loses the mark even if reflexive and symmetric hold.