Find the domain of the following functions:
`(1)/(sqrt(x-6))`

Answer

To find the domain of the function \( f(x) = \frac{1}{\sqrt{x - 6}} \), we need to ensure that the expression under the square root is positive and that the denominator is not zero. Here are the steps to determine the domain:

### Step 1: Identify the condition for the square root
The expression under the square root must be greater than zero for the function to be defined. Thus, we set up the inequality:
\[ x - 6 > 0 \]

### Step 2: Solve the inequality
To solve the inequality, we add 6 to both sides:
\[ x > 6 \]

### Step 3: Consider the denominator
Since the function is in the form of \( \frac{1}{b} \), we also need to ensure that the denominator is not zero. The denominator is \( \sqrt{x - 6} \), which means:
\[ \sqrt{x - 6} \neq 0 \]
This condition is satisfied when \( x - 6 \neq 0 \), which leads us to:
\[ x \neq 6 \]

### Step 4: Combine the conditions
From the conditions derived, we have:
1. \( x > 6 \)
2. \( x \neq 6 \)

Since \( x > 6 \) already excludes \( x = 6 \), we can conclude that the domain of the function is:
\[ \text{Domain} = (6, \infty) \]

### Final Answer
Thus, the domain of the function \( f(x) = \frac{1}{\sqrt{x - 6}} \) is:
\[ (6, \infty) \]

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