Function composition

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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