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After reviewing the problem again, the domain of the rational function \( f(x) = \frac{x + 1}{2x + 9} \) is indeed all real numbers except where the denominator equals zero. 

Following the solution steps:

1. **Identify the restriction in the domain:**  
   Set the denominator \( 2x + 9 \) equal to zero:
   \[
   2x + 9 = 0
   \]
2. **Solve for \( x \):**
   \[
   2x = -9
   \]
   \[
   x = -\frac{9}{2}
   \]

This means that \( x \) cannot be \( -\frac{9}{2} \).

Therefore, the correct domain is:
\[
\{ x \mid x \neq -\frac{9}{2} \}
\]

### Answer
The correct option is (C): \(\{ x \mid x \neq -\frac{9}{2} \}\).

Would you like further explanation on this topic?

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Here are some related questions to deepen understanding:

1. How would you determine the domain if the function had multiple fractions?
2. What if the function was \( f(x) = \frac{x + 1}{2x^2 + 9x} \); how would that change the solution?
3. How does the domain of a function affect its graph?
4. What is the process for finding the domain of a function with square roots in the denominator?
5. How would you define the domain of a polynomial function?

**Tip:** For rational functions, always check for any values that make the denominator zero. These are the values excluded from the domain.

Find the domain of the rational function f(x) = (x + 1) / (2x + 9).

Learn how to determine the domain of the rational function f(x) = (x + 1) / (2x + 9). This problem covers key concepts in algebra, including the exclusion of values that make the denominator zero. Suitable for high school students in Grades 10-12.

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