Find the domain and range of the following functions:
`(1)/(sqrt(x-3))`

Answer

To find the domain and range of the function \( f(x) = \frac{1}{\sqrt{x - 3}} \), we can follow these steps:

### Step 1: Determine the Domain
The domain of a function consists of all the values of \( x \) for which the function is defined.

1. **Identify restrictions**: The function has a square root in the denominator, which means that the expression inside the square root must be positive (greater than zero) to avoid division by zero and to ensure the square root is defined.
\[ x - 3 > 0 \]
This simplifies to:
\[ x > 3 \]

2. **State the domain**: Therefore, the domain of the function is all \( x \) values greater than 3. In interval notation, this is expressed as:
\[ \text{Domain} = (3, \infty) \]

### Step 2: Determine the Range
Next, we find the range of the function, which consists of all possible output values \( f(x) \).

1. **Analyze the function**: Rewrite the function:
\[ f(x) = \frac{1}{\sqrt{x - 3}} \]
As \( x \) approaches 3 from the right (i.e., \( x \to 3^+ \)), \( \sqrt{x - 3} \) approaches 0, making \( f(x) \) approach infinity:
\[ f(x) \to \infty \quad \text{as} \quad x \to 3^+ \]

2. **Behavior as \( x \) increases**: As \( x \) increases beyond 3, \( \sqrt{x - 3} \) increases, causing \( f(x) \) to decrease. Thus, \( f(x) \) will take on values from a very large number down to 0, but never actually reaching 0.

3. **State the range**: Therefore, the range of the function is all positive values:
\[ \text{Range} = (0, \infty) \]

### Final Answer
- **Domain**: \( (3, \infty) \)
- **Range**: \( (0, \infty) \)