A composite function is a function that is created by applying one function to the results of another function. It's like a 'function within a function'.

If we have two functions, `f(x)` and `g(x)`, the composite function can be written as f(g(x)) or (f ∘ g)(x). This is read as 'f of g of x'.

The output of the inner function, `g(x)`, becomes the input for the outer function, `f(x)`.

To evaluate `f(g(x))` at a specific value, you always work from the inside out.

Step 1: First, calculate the value of the inner function, `g(x)`. Let's call this result 'k'.

Step 2: Then, use that result 'k' as the input for the outer function, `f(k)`. The final result is the value of the composite function.

You can also create a new single function that represents the entire composite operation.

To find the expression for `f(g(x))`, you substitute the entire expression for `g(x)` into every instance of 'x' in the function `f(x)`.

It's important to note that the order matters. In general, f(g(x)) is not the same as g(f(x)).

The domain of a composite function `f(g(x))` is determined by two conditions:

1. The input 'x' must be in the domain of the inner function `g(x)`.

2. The output of the inner function, `g(x)`, must be in the domain of the outer function `f(x)`.

Finding this domain can be the trickiest part of working with composite functions.

Composite functions are excellent for modeling situations that involve a sequence of operations or dependencies.

Economics: The cost of production might depend on the number of units, and the number of units might depend on the market price. A composite function can link cost directly to market price.

Physics: The kinetic energy of an object depends on its velocity, and its velocity might be a function of time. A composite function can describe the object's kinetic energy at any given time.

Computer Graphics: Applying multiple transformations (like rotation, scaling, and translation) to an object can be modeled as a composition of functions.