# Composite Functions

## Understanding Composite Functions: Simplifying Operations with Multiple Functions

A composite function is the combination of two or more functions. This method is commonly used to transform numbers from one set to another. For example, if we have a function that maps numbers from set A to set B and another function that maps numbers from set B to set C, we can use a composite function to directly map numbers from set A to set C. The diagram below illustrates this concept:

- Composite functions allow for mapping between sets

It's important to remember these key properties of composite functions:

- In a composite function, the inside function is always performed first, and its output becomes the input for the outside function
- The domain of a composite function is determined by the domain of the first function applied
- When combining two functions, a single input is used, and the output of both functions is obtained
- To find the value of a composite function, we input a value into the original function and then use the resulting output as the input for all other functions

Let's go through an example to see how this process works:

Imagine we have the functions **f(x) = 2x + 1** and **g(x) = x^2**, and we want to find the value of **f(g(3))**. First, we calculate **g(3) = 9**. Next, we use this value as the input for **f(x)**, giving us **f(9) = 19**. Therefore, **f(g(3)) = 19**.

Composite functions can become more complex when dealing with quadratic, trigonometric, and reciprocal functions. However, the process remains the same as with simpler linear examples. Let's look at a few more worked examples:

- To find the value of
**f(g(x))**, where**f(x) = 1/x**and**g(x) = sin(x)**, we first calculate**g(x) = 1/sin(x)**. Then, we use this as the input for**f(x)**, resulting in**f(1/sin(x))**. - Consider the functions
**f(x) = 2x + 1**and**g(x) = x^2**. To find the value of**f(g(4))**, we start by calculating**g(4) = 16**. Next, we use this value as the input for**f(x)**, resulting in**f(16) = 33**. Therefore,**f(g(4)) = 33**.

**Key Takeaways:**

- Composite functions are created by combining two or more functions
- The inside function is always performed first in a composite function
- The domain of a composite function is determined by the first function applied
- To find the value, we input a value into the original function and then use the resulting output as the input for all other functions