Function decomposition is the process of breaking down a composite function into its individual components. This is essentially the reverse of function composition, where two functions are combined to form a new function. Understanding how to decompose functions can be challenging, but with practice, it can become an enjoyable and creative exercise.

To illustrate function decomposition, consider the function \( h(x) = \sqrt{x - 2} \). The goal is to express this function as a composition of two functions, \( f \) and \( g \), such that \( h(x) = f(g(x)) \). A common strategy is to identify the inner function first. In this case, since \( h(x) \) involves a square root, we can set \( g(x) \) equal to the expression inside the square root, which is \( x - 2 \). Consequently, we define \( f(x) \) as the square root function applied to \( x \), specifically \( f(x) = \sqrt{x} \). Thus, we have:

\( g(x) = x - 2 \)

\( f(x) = \sqrt{x} \)

When we compose these functions, \( f(g(x)) = f(x - 2) = \sqrt{x - 2} \), which matches our original function \( h(x) \).

However, there are multiple valid ways to decompose a function. For instance, we could also define \( g(x) \) as the entire function \( \sqrt{x - 2} \) and set \( f(x) = x \). This gives us:

\( g(x) = \sqrt{x - 2} \)

\( f(x) = x \)

When composed, \( f(g(x)) = f(\sqrt{x - 2}) = \sqrt{x - 2} \), which again results in \( h(x) \).

Moreover, creativity in decomposition allows for even more unconventional approaches. For example, we could define \( g(x) = \sqrt{x - 2} - 1000 \) and \( f(x) = x + 1000 \). This results in:

\( g(x) = \sqrt{x - 2} - 1000 \)

\( f(x) = x + 1000 \)

When composed, \( f(g(x)) = f(\sqrt{x - 2} - 1000) = (\sqrt{x - 2} - 1000) + 1000 = \sqrt{x - 2} \), which is still valid.

In summary, function decomposition allows for various interpretations and methods to express a composite function. The key is to identify the inner function and creatively define the outer function, ensuring that the composition returns to the original function. With practice, this process can enhance your understanding of functions and their relationships.