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Concept

Composite Function

A composite function, or a composition of functions, combines two or more functions, which produces a new function. In a composition, the outputs produced by one function are the inputs of the other function. The composition of the functions and is denoted as or
Given four different functions, the composition of two/three of them is found.
Performing the composition of two functions is similar to evaluating one function into the other. For example, let and To find the variable in must be substituted with
Substitute for and evaluate
Note that makes sense only when the outputs of belong to the domain of Also, be aware that the composition of functions is not commutative — that is, in general, This can be checked with the same two functions.
Two machines simulating the composition of two functions. The functions used are f(x)=2x+1 and g(x)=x-5. The order of the composition can be set. Any input between -100 and 100 is accepted.
Just as changing the order of the machines in a factory could alter the final product, changing the order in which the functions are applied could produce different outputs. For example, here and are different values.

Extra

Parts of a Composition
Explore the parts of a composition by moving the magnifying glass.
Three sets labeled as A, B, and C respectively. Set A is the domain of g and has x inside; Set B is the range of g and also the domain of f. It has g(x) inside it; Set C is the range of f and has f(g(x)) inside; A bendy arrow points from A to B representing the function g; A bendy arrow points from B to C representing the function f; A bendy arrow points from A to C representing the composite function f(g(x)).