function: let A and B be noempty sets. A function f from A to B is an assignment of exactly one element of B to each element of A. We write f(a)=b if b is the unique element of B assigned by the function f to the element a of A. If f is a function from A to B, we write f: A->B, (we also call it f maps A to B)
if f: A->B
domain of f: A
codomain of f: B
if f(a)=b, we say that b is the image of a, and a is a preimage of b
the range of f: the set of all images of elements of A
Let f1 and f2 be functions from A to R, then f1+f2 and f1f2 are also functions from A to R
f1+f2: (f1+f2)(x) = f1(x) + f2(x)
f1f2: (f1f2)(x)= f1(x)f2(x)
one-to-one function f (injective): if and only if f(a)=f(b) -> a=b for all a and b in the domain of f. It is the same as if and only if f(a)!=f(b) whenever a!=b.
increasing: A function f whose domain and codomain are subset of the set of real numbers is called increasing if f(x)<=f(y) whenever x<y and x,y in the domain of f
strictly increasing: A function f whose domain and codomain are subset of the set of real numbers is called strictly increasing if f(x)<f(y) whenever x<y and x,y in the domain of f
similarly, decreasing and strictly decreasing
From these definitions, a function that is either strictly increasing or strictly decreasing must be one-to-one. However, a function that is increasing, but not strictly increasing, or decreasing but not strictly decreasing, is not necessarily one-to-one
function f is onto, or surjective: if and only if for every element b of B, there is an element a of A with f(a)=b
function f is one-to-one correspondence, or bijection: if f is both one-to-one and onto
inverse function: let f be a one-to-one correspondence from set A to B, the inverse function of f is the function that assigns to an element b belonging to B the unique element a in A such that f(a)=b.
A one-to-one correspondence function is invertible because we can define an inverse of this function. A function is not invertible if it is not a one-to-one correspondence, because this inverse of such a function does not exise.
composition of function f and g: let g be a function from the set A to B, and let f be a function from the set B to C, the composition of function f and g is f(g(x))
To find f(g(a)) we first apply the function g to a to obtain g(a), and then apply f to the result g(a) to obtain f(g(a)). Note that the composition of f and g cannot be defined unless the range of g is a subset of the domain of f.
graph of function: Let f be a function from the set A to B, the graph of the f is the set of ordered pairs {(a,b)| a belongs A and f(a)=b}
The graph of function is often displayed pictorially to aid in understanding the behavior of the function
floor function: assigns to the real number x the largest interger that is less than or equal to x
ceiling function: assigns to the real number x the smallest interger that is greater than or equal to x
a useful approach for proving or disproving a conjection about floor function is to let x=n+e, where n is floor(x), and 0<=e<1. Similarly, when for ceiling functions, we can let x=n-e, where n=ceiling(x), and 0<=e<1