Find the domain and range of the following:
(a). f(x) = 1/√(x − 3)

Let's analyze the function \( f(x) = \frac{1}{\sqrt{x - 3}} \) to determine its domain and range.

### 1. **Domain**:
The domain of a function is the set of all possible input values (x-values) for which the function is defined.

- The expression under the square root, \( x - 3 \), must be non-negative because the square root of a negative number is not real. Therefore, we need:
  \[
  x - 3 \geq 0
  \]
  \[
  x \geq 3
  \]
  
- Additionally, the function \( f(x) \) involves a fraction, and the denominator \( \sqrt{x - 3} \) cannot be zero because division by zero is undefined. Therefore, we require:
  \[
  \sqrt{x - 3} \neq 0
  \]
  \[
  x - 3 \neq 0
  \]
  \[
  x \neq 3
  \]

Thus, combining these two conditions, the domain is:
\[
\text{Domain of } f(x): \, (3, \infty)
\]
This means \( f(x) \) is defined for all \( x > 3 \).

### 2. **Range**:
The range of a function is the set of all possible output values (y-values).

- \( f(x) = \frac{1}{\sqrt{x - 3}} \) is positive because both the numerator (1) and the square root function are positive for \( x > 3 \).
  
- As \( x \) approaches \( 3 \) from the right (i.e., \( x \) is slightly greater than 3), \( \sqrt{x - 3} \) approaches 0, and therefore, \( f(x) \) becomes very large. Thus, the function can take very large positive values.
  
- As \( x \) increases, \( \sqrt{x - 3} \) increases, making \( f(x) \) decrease. However, since the square root function increases without bound as \( x \) increases, \( f(x) \) approaches 0 but never actually reaches 0.

Thus, the range of the function is:
\[
\text{Range of } f(x): \, (0, \infty)
\]

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

Would you like more details or have any questions?

Here are some related questions to consider:
1. How would the domain change if the function were \( f(x) = \frac{1}{\sqrt{5 - x}} \)?
2. What is the domain and range of \( f(x) = \sqrt{x - 2} \)?
3. How does adding a constant inside the square root affect the domain and range?
4. What is the domain and range of \( f(x) = \frac{1}{\sqrt{x^2 - 1}} \)?
5. How would the range change if the function were \( f(x) = \frac{2}{\sqrt{x - 3}} \)?

**Tip:** Always remember that square roots require the radicand to be non-negative, and denominators cannot be zero!

Discover how to find the domain and range of the function f(x) = 1/√(x - 3) through detailed analysis. Understand the conditions for the function's domain and explore the range of possible output values. Suitable for high school students studying functions and square roots.

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