Let's solve the problem step by step.

The function provided is:

$f(x) = \frac{x + 9}{x^2 + 8x + 7}$

To find the domain of the function, we need to ensure that the denominator is not equal to zero, because division by zero is undefined.

Step 1: Set the denominator equal to zero and solve for $x$.

$x^2 + 8x + 7 = 0$

This is a quadratic equation. To solve it, we can factor it:

$x^2 + 8x + 7 = (x + 7)(x + 1) = 0$

So, the solutions are:

$x = -7 \quad \text{or} \quad x = -1$

Step 2: Exclude these values from the domain.

The domain consists of all real numbers except $x = -7$ and $x = -1$, because at these points, the denominator becomes zero.

Set-Builder Notation:

The domain in set-builder notation is:

$\{ x \mid x \neq -7 \text{ and } x \neq -1 \}$

Interval Notation:

In interval notation, the domain is:

$(-\infty, -7) \cup (-7, -1) \cup (-1, \infty)$

If you have any further questions or need more details, feel free to ask! Here are some related questions to explore:

1. How do you determine the domain for more complex rational functions?
2. What happens to the graph of a function at points where the denominator is zero?
3. How can quadratic equations be solved if factoring doesn't work?
4. Can a rational function have an infinite domain?
5. What is the connection between the domain of a function and its range?

Tip: Always check if the denominator of a rational function is zero to avoid undefined values!