Revise: Functions
Product sets and their cardinality, relations and how to represent them, and what makes a relation a function — domain, range, the vertical line test, and function notation.
A × B = every ordered pair (a, b) with a ∈ A, b ∈ B.
{1,2} × {x,y} = {(1,x),(1,y),(2,x),(2,y)}.
A relation is any subset of A × B — domain = first coordinates, range = second.
{(1,2),(2,4)} has domain {1,2}, range {2,4}.
A function: every input has exactly one output.
{(1,4),(1,5)} is not a function — 1 has two outputs.
A graph is a function if no vertical line crosses it more than once.
A circle fails the test — a vertical line crosses it twice.
f(x) is the output of f at input x — substitute and simplify.
f(x)=2x²−3x+1 → f(4) = 21.
A × B is the set of ordered pairs (a,b); order matters.
{1,2} × {x,y} = {(1,x),(1,y),(2,x),(2,y)}.
A relation is any subset of A × B; domain/range come from the pairs used.
{(1,2),(2,4),(3,4)} → domain {1,2,3}, range {2,4}.
The same relation can be a set of pairs, a table, an arrow diagram, or a graph.
Table (1,5),(2,7),(3,9) → R={(1,5),(2,7),(3,9)}.
A function: no input repeats with a different output.
{(1,4),(2,4),(3,5)} is a function.
More than one crossing on any vertical line means it is not a function.
A non-vertical straight line always passes the test.
Domain = valid inputs; range = resulting outputs. Watch for zero denominators.
f(x)=1/(x−3) → domain excludes x=3.
f(x) names an output — substitute the input value for every x.
g(x)=x²+2x → g(-3) = 3.