Let f :AB and g : BC be two functions. Then a function gof : AC defined by (gof) (x) = g(f(x)), for all xA
is called the composition of f and g.
Let A, B and C be three non-void sets and let f : AB, g: BC be two functions. Since f is a function from A to B, therefore for each xA there exists a unique element f(x) B. Again, since g if a function from B to C, therefore corresponding to f(x) B there exists a unique element g(f(x)) C. Thus, for each xA there exists a unique element g(f(x)) C.
It follows from the above discussion that f and g when considered together define a new function from A to C.This function is called the composition of f and g and is denoted by gof. We define it formally as follows.
a) gof is defined only if for each xA, f(x) is an element of g so that we can take its g-image. Hence for the composition gof to exist, the range of f must be a subset of the domain of g.
b) gof exists iff the range of f is a subset of domain of g. Similarly, fog exists if range of g is a subset of domain of f.
Properties of Composite Functions
P-1 The composition of functions is not commulative
i.e. fog ≠ gof
P-2 The composition of functions is associative
i.e. If f, g, h are any three functions such that (fog) oh and fo(goh) both exist, then
(fog)oh = fo(goh)
P-3 The composition of two bijective function is also a bijective function.
P-4 The composition of any function with the identity function is the function itself.
i.e. If f : A→B, then
foIA = IBof = f