The function given is \( y = \frac{1}{\sqrt{x-2}} \).

### Understanding the Function:
This function is a reciprocal square root function. Let's break down its components:

- **Square Root**: The square root function is defined as \( \sqrt{x-2} \).
- **Reciprocal**: The reciprocal of a function is \( \frac{1}{\text{function}} \).

### Key Characteristics of \( y = \frac{1}{\sqrt{x-2}} \):

1. **Domain**: The function \( \sqrt{x-2} \) is defined only when \( x - 2 \geq 0 \), meaning \( x \geq 2 \). Therefore, the domain of \( y = \frac{1}{\sqrt{x-2}} \) is \( x > 2 \), because the square root cannot be zero (as the function is in the denominator).

2. **Range**: The range of the function is \( y > 0 \), because the square root and reciprocal both produce positive values.

3. **Behavior Near the Domain's Boundary**:
   - As \( x \) approaches \( 2 \) from the right (i.e., \( x \to 2^+ \)), \( \sqrt{x-2} \) approaches \( 0 \), making \( y = \frac{1}{\sqrt{x-2}} \) approach infinity.
   - This indicates a vertical asymptote at \( x = 2 \).

4. **Asymptotic Behavior**: 
   - There is a vertical asymptote at \( x = 2 \).
   - As \( x \) increases, \( y \) decreases and approaches \( 0 \) but never actually reaches \( 0 \).

### Graphical Representation:
- The graph of \( y = \frac{1}{\sqrt{x-2}} \) will be a curve starting from \( x = 2 \) (but not including \( x = 2 \)), moving to the right, with \( y \) starting from a very large positive value and decreasing toward zero as \( x \) increases.

Would you like more details or have any questions about this?

Here are five related questions to deepen your understanding:

1. How would the graph change if the function were \( y = \frac{1}{\sqrt{x-1}} \)?
2. What happens to the function \( y = \frac{1}{\sqrt{x-2}} \) if you add a constant outside the square root?
3. Can you identify another function with a vertical asymptote? How do you find it?
4. How would you determine the horizontal asymptote for a function like \( y = \frac{1}{\sqrt{x-2}} + c \)?
5. How would the domain change if the function were \( y = \frac{1}{\sqrt{2-x}} \) instead?

**Tip:** To graph functions with square roots and reciprocals, first understand the behavior of the inner function and then apply the transformations step by step.

Explore the function y = 1/sqrt(x-2) with detailed insights into its domain, range, and asymptotic behavior. Learn how to graph the function and understand its characteristics. Suitable for advanced high school students and anyone interested in calculus and function analysis.

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