# Composite of two functions

From Calculus

## Definition

Suppose are two functions. The **composite function** is defined as the function:

Note that the function written at the right end of the composition is the function performed *first*, and the function written at the left end of the composition is the function performed *next*. We say that composition of functions is *done right-to-left*.

## Relation with various operations

Below, we discuss how a particular operation done for functions can be done for a composite of two functions:

Operation | Verbal description | How it's done |
---|---|---|

Graph | We are given the graphs of and (without necessarily having algebraic, numerical, or verbal descriptions of the functions) and we need a geometric method to sketch the graph of | graphing the composite of two functions |

Obtain explicit expression for | We are given explicit algebraic expressions for and and need an explicit algebraic expression for . | simple case: finding the composite of two functions by plugging in expressions case of piecewise functions: finding the composite of two piecewise functions |

Find limit of at a point | We have techniques for finding limits for both functions, we need a technique for finding the limit of the composite. | composition theorem for continuous functions |

Differentiate . | We have expressions for the derivatives and , we need an expression for . | chain rule for derivatives: . |

Integrate . | We want to integrate in terms of integration of simpler functions. | We can try integration by u-substitution or integration by parts. |