To find the domain of the rational function:

\[
f(x) = \frac{5x+15}{2x+19}
\]

### Steps:
1. The domain of a rational function consists of all real numbers **except** where the denominator is zero. Therefore, we need to find when the denominator is equal to zero.
   
2. Set the denominator equal to zero and solve for \(x\):
   \[
   2x + 19 = 0
   \]
   \[
   2x = -19
   \]
   \[
   x = \frac{-19}{2}
   \]

3. Therefore, the function is undefined at \( x = -\frac{19}{2} \).

### Domain:
The domain is all real numbers **except** \( x = -\frac{19}{2} \).

Thus, the domain is:

\[
x \in (-\infty, -\frac{19}{2}) \cup (-\frac{19}{2}, \infty)
\]

Would you like further details or clarifications?

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Here are 5 related questions to explore:
1. How do you find the domain of a more complex rational function?
2. What happens to the graph of a function at points where the denominator is zero?
3. How do you find the range of a rational function?
4. What are vertical and horizontal asymptotes, and how do they relate to domain restrictions?
5. Can you apply the same method to find domains of square root functions?

**Tip:** Always check for both the numerator and denominator when determining the domain, especially if there are radicals or logarithms involved.

Learn how to find the domain of the rational function (5x+15)/(2x+19) with a clear, step-by-step solution. This problem involves key algebraic concepts like identifying where a function is undefined by solving for when the denominator is zero. Suitable for students in Grades 9-12.

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