# Function composition

A combined function is a function that can be interpreted as two (or more) separate functions. The combined function is usually interpreted as an outer and inner function. An inner function is the function that is placed in the outer function.

Example 1

We have two functions

We construct a combined function such that the function u is an outer function and s is an inner function. That is, the function s is placed in the function u in place of the variable x.

With the previous functions we can also form a combined function in the opposite direction. That is, the function s as an outer function and the function u as an inner function.

Example 2

Let the function f be interpreted as a combined function. That is, it is defined by the functions s and u

An inner function is the function that is substituted in another function in place of a variable. Here it would be

An outer function is the function in which the inner function is substituted. Here it is

A combined function in which the function u is an outer function and the function s is an inner function is denoted by

The notation is read as "u circle s"

Example 3

Find the value of the combined function (u⚬ s) (1)

Lets form a combined function

The value when x = 1

Example 4

Interpret the function f as a combined function.

Clearly, the square root is an outer function, for a rational function is substituted inside it. Then the outer and inner functions are

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