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Function composition involves combining two functions to create a new function. This process can be thought of as plugging one function into another. It's an essential tool when checking if two functions are inverses.
For our exercise, we found the inverse function of a given function and check its accuracy using composition. The check involves plugging the inverse function back into the original function and verifying the result is equal to the input.
In mathematical terms, if we have two functions, say \( f(x) \) and \( g(x) \), then the composition of \( f \) and \( g \) is written as \( f(g(x)) \). To confirm if \( f(x) \) and \( f^{-1}(x) \) are inverses, we must ensure:

• \( f(f^{-1}(x)) = x \)
• \( f^{-1}(f(x)) = x \)

In our problem, this was demonstrated by showing \((\sqrt{x-4})^2 + 4 = x\), confirming the inverse calculation.

Understanding the domain and range of functions and their inverses is vital. The domain of a function is all the possible input values (typically \( x \)), while the range is all possible outputs (\( f(x) \)).
For our function \( f(x) = x^2 + 4 \), given \( x \geq 0 \), this constraint defines its domain. Consequently, the range is all real numbers greater than or equal to 4 because squaring non-negative numbers starts from 0, and adding 4 shifts this upward.
When finding an inverse function, the domain and range swap roles. Thus, the inverse \( f^{-1}(x) = \sqrt{x - 4} \) has a domain where \( x \geq 4 \) and a range consisting of all non-negative real numbers \( y \geq 0 \). This switch ensures the function and its inverse satisfy all values originally covered or produced.

Solving the equation to find the inverse function is a systematic process. After performing the switch between \( f(x) \) and \( x \), we solve algebraically to isolate the inverse term.
Starting with \( x = f^{-1}(x)^2 + 4 \), rearrange to isolate \( f^{-1}(x) \) on one side. We subtract 4 from both sides to find \( x - 4 = f^{-1}(x)^2 \).
Next, applying the square root to both sides, we solve for \( f^{-1}(x) \):

• Take the square root: \( f^{-1}(x) = \sqrt{x - 4} \).
• Ensure the values fulfill the original domain condition \( x \geq 0 \).

This process highlights key solving techniques, emphasizing careful consideration of domains and ensuring results are feasible given initial function constraints.