https://file.aiquickdraw.com/mathbot/file/1727740155722vpfnr97y.jpg

The given function is:

\[
f(x) = \frac{3}{\sqrt{x - 1}}
\]

To find the domain of this function, we need to identify all values of \(x\) for which the function is defined.

### Steps:

1. **Square Root Constraint**: The square root is only defined for non-negative numbers. Therefore, the expression under the square root, \(x - 1\), must be non-negative:
   \[
   x - 1 \geq 0
   \]
   Solving this inequality:
   \[
   x \geq 1
   \]
   Thus, \(x\) must be greater than or equal to 1.

2. **Denominator Constraint**: Since the function involves a denominator, we need to ensure that the denominator is not zero. The denominator is \(\sqrt{x - 1}\), and this equals zero when \(x = 1\). Therefore, \(x \neq 1\).

### Conclusion:
Combining both conditions, the function is defined for \(x > 1\). Therefore, the domain of the function is:
\[
x \in (1, \infty)
\]

Would you like more details on this, or do you have any questions?

Here are five related questions for further exploration:
1. How would the domain change if the function was \( f(x) = \frac{3}{\sqrt{2x - 5}} \)?
2. What is the range of the given function \( f(x) = \frac{3}{\sqrt{x - 1}} \)?
3. Can we find the inverse of this function? If so, what is it?
4. What happens if the square root is replaced with a cube root?
5. How does this function behave as \(x\) approaches infinity?

**Tip**: When dealing with square root functions, always consider the conditions under which the expression inside the root is non-negative.

Find the domain of the function f(x) = 3 / √(x - 1)

Learn how to find the domain of the function f(x) = 3 / √(x - 1). This problem involves understanding the restrictions imposed by the square root and the denominator. Follow a step-by-step solution suitable for high school students in Grades 9-12.

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