Function notation and mappings

A function assigns exactly one output to each input in its domain. The domain is the permitted input set; the range is the set of outputs actually attained. Write f:x to x2, or f(x)=x2. A one–one function never sends distinct inputs to the same output. A many–one function can: f(-2)=f(2)=4. A one–many relation is not a function. On a graph, vertical lines test the function rule; horizontal lines test one–one behaviour. For f(x)=x2, -2 <= x 3, the range is 0 (x) 9. The vertex is inside the domain. Restricting to 0 <= x 3 makes the function one–one. f2(x)=f(f(x)), whereas [f(x)]2 squares the output. For f(x)=x+2, these are x+4 and (x+2)2.