
The basic idea of functions is that kind of a chain reaction in which the functions occur one after the other. Let f be a function whic associates with each person that person's mother and g be a function which associates with each person that person's father. Then f = {(x,y): x is a person and y is the mother of x} and g = {(x,y): x is a person and y is the father of x} We compose f with g by "applying" f and then g. We shall use the symbol "g(f)" to denote the composition of f with g. f(John) = John's mother Mary and g (Mary) = Mary's mother Thus g(Mary) = g(f(John)) and in the sense the "g" comes before "f". Therefore, we say that g(f)(John) = g(f(John)) = John's maternal grandfather. Similarly, f(g(Hari)) = f(Peter) = Peter's mother = John's paternal grandmother.
Definition: Given two functions, f and g, the function x > g(f(x)) is called a composite of f and g and is denoted by gf. The domain of gf is the set of all elements x in the domain of f for which f(x) is in the domain of g. The operation of formng a composite of two functions is called composition.
Example: Given that f: x > 3x2 and g:x >x^{5} for all x € R, find (gf)(x)
Directions: Find the composite functions of the following. 