To find the domain of the function:

$f(x) = \frac{3}{x - 11} + \frac{4}{x + 3}$

we need to determine where the function is undefined. A rational function like this is undefined when the denominator of any term is zero because division by zero is not allowed.

Step-by-step:

1. The first term is $\frac{3}{x - 11}$. The denominator $x - 11$ becomes zero when $x = 11$.
2. The second term is $\frac{4}{x + 3}$. The denominator $x + 3$ becomes zero when $x = -3$.

Conclusion:

The function is undefined at $x = 11$ and $x = -3$.

Domain:

The domain of the function is all real numbers except $x = 11$ and $x = -3$.

In interval notation, the domain is: $(-\infty, -3) \cup (-3, 11) \cup (11, \infty)$

Would you like more details or have any questions?

Here are five related questions:

1. How do you find the domain of more complex rational functions?
2. What happens to the graph of a function at points where it is undefined?
3. How do vertical asymptotes relate to the domain of a rational function?
4. Can a function have more than two points where it is undefined?
5. How would the domain change if a square root or logarithmic term was added?

Tip: Always check for zeroes in the denominator when determining the domain of rational functions.