# Composite Functions

## Definition and Understanding

• A composite function is a function that involves two or more functions.
• It is formed when the output of one function becomes the input of another function.
• A composite function is denoted as `(fog)(x)` or `f(g(x))`, meaning the function `f` is applied to the output of function `g`.

## Creating Composite Functions

• To create a composite function, you apply one function to the output of another.
• Example: if we have two functions, `f(x) = x + 3` and `g(x) = x^2`, their composite function `f(g(x))` would be `(x^2) + 3`.

## Evaluating Composite Functions

• To evaluate a composite function, you start from the innermost function and work outwards.
• Example: to evaluate `f(g(x))` at `x = 2`, firstly apply `g(x)` to get `g(2)` = `2^2`= `4`, then apply `f(x)` on the output, `f(4) = 4 + 3 = 7`. Hence `f(g(2))`= `7`.

## Inverse of Composite Functions

• The inverse of a composite function can be found by reversing the order of operations and replacing each function by its inverse, if it exists.
• Example: If `f(g(x))` is a composite function and both `f^-1(x)` and `g^-1(x)` are existing inverses, the inverse of the composite function would be `(g^-1of^-1)(x)`.