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Understanding the concept of inner and outer functions is critical to mastering function composition. Think of it as a two-step process where one function feeds into another. The 'inner function' is the first function that you evaluate, and its output becomes the input of the so-called 'outer function'.

For instance, in the exercise with functions f(x) and g(x), when we see (fg)(-6), we should interpret this as 'apply g to -6, then take that result and plug it into f'. It's a chain reaction where g(-6) is the inner function and f(x) (with x being the output from g(-6)) is the outer function. This is fundamental because without understanding which function to apply first, one cannot accurately evaluate compositions of functions.

Function evaluation might seem daunting, but it's essentially about substituting a given value into a function. To properly evaluate, one follows the order of operations, just as when dealing with numerical expressions. In our example, evaluating the inner function g(-6) means replacing every instance of x in g(x) = x - 4 with -6. The result of this calculation will then serve as the new input for the outer function f(x).

This step-by-step approach is not only systematic but also helps avoid errors. Always remember, we can evaluate functions at numbers, other functions, or even expressions - the key lies in correct substitution and following through with the proper arithmetic.

Substitution and simplification go hand-in-hand when dealing with functions. Substitution refers to replacing the variable in a function with a given number or expression, which is exactly what we do when we 'plug in' a value into our function. Following that, simplification is all about breaking down the complex expression into its simplest form.

The aim is to make the expression as straightforward as possible, often through arithmetic operations or by applying algebraic rules. Taking the result of g(-6), which we find through substitution, we 'plug' this result into f(x) to continue our work. Once substituted, we then simplify the expression by performing operations like addition, subtraction, or in the case of f(x), squaring the number. Simplification unveils the most reduced version of the function's output, giving you the final answer to the composition.