Relations and Functions (introductory)
Relations and Functions (introductory) tests ordered pairs, Cartesian products, domain and range, and the definition of a function. The exam expects you to distinguish a relation from a function and to work with simple algebraic and arrow-diagram examples.
Core rules
- Cartesian product A×B={(a,b): a∈A, b∈B}; n(A×B)=n(A)n(B).
- A relation from A to B is any subset of A×B; its domain is the set of first coordinates, range the set of second coordinates.
- A function f:A→B assigns exactly one element of B to each element of A; no input has two outputs.
- Domain of f(x)=1/(x−a) excludes x=a; domain of √(x−a) requires x≥a.
- Composition (f∘g)(x)=f(g(x)) exists only when g(x) lies in the domain of f.
Worked example
Let A={1,2,3} and f(x)=x²−1 on A. Then f(1)=0, f(2)=3, f(3)=8. The domain is A and the range is {0,3,8}. The relation {(1,2),(1,3)} is not a function because input 1 has two outputs.
Traps
Do not call every relation a function. A repeated second coordinate is allowed; a repeated first coordinate with different second coordinates is not. When finding domain, solve the restriction, not the simplified expression only. Range values must actually be attained.