State the domain of 
f(x) = 
(x + 9)(x + 1)
(x + 3)(x + 8)
,
 using interval notation. All answers should be exact.
The domain is

To find the domain of the function \( f(x) = \frac{(x + 9)(x + 1)}{(x + 3)(x + 8)} \), we need to determine when the denominator is equal to zero, as division by zero is undefined.

The denominator is \( (x + 3)(x + 8) \). Set this equal to zero and solve for \( x \):

\[
(x + 3)(x + 8) = 0
\]

This gives two solutions:

\[
x + 3 = 0 \quad \Rightarrow \quad x = -3
\]
\[
x + 8 = 0 \quad \Rightarrow \quad x = -8
\]

Thus, the function is undefined at \( x = -3 \) and \( x = -8 \).

The domain of \( f(x) \) is all real numbers except \( x = -3 \) and \( x = -8 \). In interval notation, this is written as:

\[
(-\infty, -8) \cup (-8, -3) \cup (-3, \infty)
\]

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How do you find vertical asymptotes for a rational function?
2. What is the range of the function \( f(x) = \frac{(x + 9)(x + 1)}{(x + 3)(x + 8)} \)?
3. How do you determine the horizontal asymptote of a rational function?
4. What is the behavior of \( f(x) \) near the excluded values \( x = -3 \) and \( x = -8 \)?
5. How would you find the zeros of \( f(x) \)?

**Tip:** When finding the domain of a rational function, always check where the denominator equals zero.

State the domain of f(x) = (x + 9)(x + 1) / (x + 3)(x + 8), using interval notation. All answers should be exact.

Learn how to find the domain of the rational function f(x) = (x + 9)(x + 1) / (x + 3)(x + 8) using interval notation. This problem covers algebraic concepts like rational functions and domain restrictions. Suitable for students in Grades 10-12.

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