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The problem asks to determine the domain and range of the function \( f(x) = \frac{1}{\sqrt{x - 4}} \).

### Domain:
For the function \( f(x) = \frac{1}{\sqrt{x - 4}} \), we must ensure that:

1. The expression inside the square root, \( x - 4 \), must be non-negative, meaning \( x - 4 \geq 0 \). Therefore, \( x \geq 4 \).
2. The denominator cannot be zero, meaning \( \sqrt{x - 4} \neq 0 \), which implies \( x - 4 > 0 \) or \( x > 4 \).

Thus, the domain of the function is all \( x \) such that \( x > 4 \).

\[
\text{Domain: } (4, \infty)
\]

### Range:
For the range of the function:

1. The function \( f(x) = \frac{1}{\sqrt{x - 4}} \) is a reciprocal of the square root function, which means \( f(x) \) will always be positive (since both the square root and reciprocal are positive).
2. As \( x \to 4^+ \) (i.e., as \( x \) approaches 4 from the right), \( \sqrt{x - 4} \) approaches 0, making \( f(x) \to \infty \).
3. As \( x \to \infty \), \( \sqrt{x - 4} \to \infty \), making \( f(x) \to 0^+ \) (but never reaching zero).

Thus, the range of the function is:

\[
\text{Range: } (0, \infty)
\]

### Final Answer:
- Domain: \( (4, \infty) \)
- Range: \( (0, \infty) \)

Would you like further clarification on any step? Here are some related questions to extend your understanding:

1. What is the domain and range of \( f(x) = \frac{1}{\sqrt{4 - x}} \)?
2. How does the reciprocal nature of the function affect the range?
3. What happens if we remove the square root from the denominator?
4. Can the function \( f(x) = \frac{1}{\sqrt{x-4}} \) be defined for negative values of \( x \)?
5. How does the function behave as \( x \to \infty \)?

**Tip:** Always check both the conditions for square roots and denominators when determining the domain of a function.

Dapatkan domain dan range dari fungsi berikut: f(x) = 1 / sqrt(x - 4)

Discover the domain and range of the function f(x) = 1 / sqrt(x - 4). Learn how to handle domain restrictions for square roots and how the reciprocal nature affects the range. Suitable for students in Grades 10-12.

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