Understanding the domain of composed functions is essential for evaluating them correctly. When dealing with composed functions, such as \( f(g(x)) \), it is crucial to identify the restrictions imposed by both the inner function \( g(x) \) and the outer function \( f(x) \).

Consider the functions \( f(x) = \frac{1}{x - 2} \) and \( g(x) = \sqrt{x} \). To find the domain of the composition \( f(g(x)) \), we follow a systematic approach. First, we determine the restrictions for the inner function \( g(x) \). Since \( g(x) = \sqrt{x} \), it is clear that \( x \) must be greater than or equal to 0, as the square root of a negative number is undefined. Thus, the restriction for \( g(x) \) is:

\( x \geq 0 \)

Next, we substitute \( g(x) \) into \( f(x) \) to form the composed function:

\( f(g(x)) = f(\sqrt{x}) = \frac{1}{\sqrt{x} - 2} \)

Now, we need to identify any additional restrictions that arise from this composition. Specifically, the denominator of the fraction cannot equal zero. Therefore, we set up the equation:

\( \sqrt{x} - 2 \neq 0 \)

Solving this gives:

\( \sqrt{x} \neq 2 \)

Squaring both sides results in:

\( x \neq 4 \)

At this point, we have two restrictions: \( x \geq 0 \) from \( g(x) \) and \( x \neq 4 \) from the composition. To express the domain of the composed function \( f(g(x)) \), we combine these restrictions. The domain can be described as all values from 0 to 4, excluding 4, and all values greater than 4:

Domain: \( [0, 4) \cup (4, \infty) \)

This comprehensive approach ensures that we account for all restrictions when determining the domain of composed functions, allowing for accurate evaluations and further mathematical operations.