https://file.aiquickdraw.com/mathbot/file/17296367861171xj9py7y.jpg

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!

Find the domain of the function f(x) = (x + 9) / (x^2 + 8x + 7).

Learn how to find the domain of the rational function f(x) = (x + 9) / (x^2 + 8x + 7) by excluding values that make the denominator zero. This problem covers key concepts of factoring quadratics and understanding rational functions, suitable for students in Grades 9-12.

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