SignificanceComposition is the operation of taking the output from one function and using that as the input to a second function. It, together with the algebraic operations on functions, is what allows us to take the basic building blocks of "pure" function types discussed in the Field Guide and build the large variety of functions which is the focus of calculus.
Notation and DefinitionIn terms of "function machines", the composition f g of the two functions f and g is the function which feeds an input to g and feeds the output of g to f.
Example: If f(x) = 3 sin (5 x) and g(x) = x2, then
Further RemarksThe domain of the composition f g is the domain of g, the codomain (or range) of this composition is the codomain (or range) of f, and the image of f g is contained in the image of f. The situation is calculus is very nice -- our functions all have domains and ranges which are subsets of the real numbers. Given a set of functions whose domains are all real numbers and whose ranges are subsets of the real numbers, we can compose them at will. This is how we build up complicated expressions like
Broken down into small components, this function may be thought of as the composition f(g(h(k(x)))) with f(x) = x4, g(x) = sin(x), h(x) = sqrt(x), and k(x) = x1/3 + ex.
Inverse FunctionsThere is one function which is distinguished from all others with respect to function composition, namely the identity function. The identity function on any set X is the function defined by
In the context of functions from the Field Guide, the identity function is the function y = x.
Along with an identity come inverses. In the case of functions,