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 ![]() ![]() ![]() |
graphing the composite of two functions |
Obtain explicit expression for ![]() |
We are given explicit algebraic expressions 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 ![]() |
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 ![]() ![]() ![]() |
chain rule for derivatives: ![]() |
Integrate ![]() |
We want to integrate ![]() |
We can try integration by u-substitution or integration by parts. |