Relations and Functions
Relations and Functions
UPMSP Intermediate Mathematics tests whether you can move between a relation, its graph, and a function rule, and whether you can justify inverse and composition claims. A relation from A to B is any subset of A×B; a function is a relation in which every input in the domain has exactly one output. The set of outputs is the range, while the codomain is the declared target set, so range⊆codomain and equality is not automatic.
Core rules
- One-one (injective): f(x1)=f(x2) implies x1=x2.
- Onto (surjective): every element of the codomain is f(x) for some x.
- Bijective: both; only then does f^{-1} exist as a function from codomain to domain.
- Composition: (g∘f)(x)=g(f(x)); generally g∘f≠f∘g.
- A binary operation on a set must be closed and may be commutative, associative, or have an identity and inverses.
Worked example
Let f:R→R, f(x)=2x+3. Write y=2x+3, so x=(y-3)/2. Thus f^{-1}(x)=(x-3)/2. Check f(f^{-1}(x))=2((x-3)/2)+3=x and f^{-1}(f(x))=((2x+3)-3)/2=x. The line has nonzero slope, so it is one-one and onto R.
Traps
Mixing codomain with range; assuming every relation is a function; writing f^{-1} for 1/f; reversing composition; forgetting a restricted domain such as x≥0 for a square-root inverse.