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The given function is 

\[
f(x) = \frac{1}{x^2 - 5x - 14} + \sqrt{\frac{x + 1}{x - 1}}.
\]

To analyze this function, we can explore several aspects like its domain, singularities, and potential simplifications:

### 1. Domain
- **Rational Expression**: The expression \(\frac{1}{x^2 - 5x - 14}\) is undefined when the denominator is zero. So, we need to find the roots of the quadratic equation \(x^2 - 5x - 14 = 0\). Using the quadratic formula:
  
  \[
  x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{5 \pm \sqrt{25 + 56}}{2} = \frac{5 \pm \sqrt{81}}{2} = \frac{5 \pm 9}{2}.
  \]
  
  This gives the roots \(x = 7\) and \(x = -2\). Therefore, the expression \(\frac{1}{x^2 - 5x - 14}\) is undefined at \(x = 7\) and \(x = -2\).

- **Square Root Expression**: The expression \(\sqrt{\frac{x + 1}{x - 1}}\) is defined when both the numerator and denominator are positive, or both are negative (since the square root of a negative number is not real if we are considering real-valued functions):
  
  1. \(\frac{x + 1}{x - 1} \geq 0\).
  
     - **Numerator** \(x + 1\) is positive when \(x > -1\) and negative when \(x < -1\).
     - **Denominator** \(x - 1\) is positive when \(x > 1\) and negative when \(x < 1\).

     To satisfy \(\frac{x + 1}{x - 1} \geq 0\):
     - Both are positive: \(x > 1\).
     - Both are negative: \(x < -1\).

     Thus, this expression is defined for \(x \in (-\infty, -1] \cup [1, \infty)\).

- Combining the conditions from both parts of the function:
  
  \[
  x \neq 7, \quad x \neq -2, \quad x \in (-\infty, -1] \cup [1, \infty).
  \]

  Therefore, the domain of \(f(x)\) is:
  
  \[
  (-\infty, -2) \cup (-2, -1] \cup [1, 7) \cup (7, \infty).
  \]

### 2. Singularities
- There are vertical asymptotes at \(x = 7\) and \(x = -2\) where the denominator of the rational expression equals zero.

### 3. Further Analysis
- This function can be analyzed further for limits, derivatives, and possible asymptotic behavior if needed.

Would you like a more detailed explanation or exploration of this function?

### Relative Questions
1. How do we determine the horizontal asymptotes of the given function?
2. What is the derivative of \(f(x)\) for this function?
3. How can we graph this function to better understand its behavior?
4. What happens to the function as \(x \rightarrow \pm \infty\)?
5. Can we simplify the square root expression further?

### Tip
When analyzing the domain of a function, always look out for where the expression might be undefined, such as division by zero and square roots of negative numbers.

f(x) = 1 / (x^2 - 5x - 14) + sqrt((x + 1) / (x - 1))

Explore the detailed analysis of the function f(x) = 1 / (x^2 - 5x - 14) + sqrt((x + 1) / (x - 1)). Learn how to find its domain, singularities, and solve quadratic equations. This problem covers key algebraic concepts including rational expressions, quadratic equations, and square roots. Suitable for Grades 10-12.

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