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Understanding how functions combine is essential, particularly in the context of function composition. In simpler terms, function composition is the process of applying one function to the results of another. For example, if you have two functions, say, f(x) and g(x), the composite function f \( \circ \) g means you first apply g(x), then take that result and apply f(x) to it.

The process can be visualized like a two-step journey: starting at x, moving to g(x), and finally landing at f(g(x)). It's important to take note of the order, as f \( \circ \) g may yield a different result than g \( \circ \) f. In the given exercise, we saw how to compute each composition and noted that the order of functions matters because |-x -1| is not the same as 3 - |x - 4|.

In practice, be sure to substitute the entire expression from the inner function where the variable x appears in the outer function. This method aids in maintaining accuracy when dealing with more complex expressions or functions.

The domain of a function represents all possible values that can be input into the function and is foundational when examining how functions behave. In technical terms, it's the set of all 'x-values' that will output real 'y-values'. Whenever you determine the domain, consider the limitations: fractions can't have zero denominators, square roots can't have negative radicands, and logarithms can't have non-positive arguments.

With respect to absolute value functions, like the one in our example f(x) = |x - 4|, the domain includes all real numbers because no matter what value 'x' is, the absolute value ensures a real number result. The same applies to linear functions, like g(x) = 3 - x, which inherently accept all real numbers. As a crucial step, we always confirm that the composite functions derived also have domains consisting of all real numbers, as seen in both f \( \circ \) g(x) and g \( \circ \) f(x), which handle real numbers without restriction.

The concept of absolute value is profoundly simple yet powerful. It's the distance from zero on the number line, and it's always non-negative. Mathematical notation uses two vertical bars, like this: |x|. The absolute value of a number or an expression tells us how far away it is from zero without bothering about the direction.

This attribute makes the absolute value incredibly useful in various areas of mathematics, including function composition. In our exercise, we dealt with the absolute value in f(x) = |x - 4| and its composition with g(x). The key takeaway is that no matter what the input is, whether it's a direct value of x or an expression resulting from another function, taking the absolute value ensures that the output will be non-negative. Understanding absolute value can significantly help in grasping how functions behave, especially when they interact through composition.