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The rational function provided is \( f(x) = \frac{x - 9}{x^2 + 9} \).

To find the domain of this function, we need to determine when the denominator is equal to zero, since division by zero is undefined.

The denominator is \( x^2 + 9 \). Setting this equal to zero and solving for \( x \):

\[
x^2 + 9 = 0
\]
\[
x^2 = -9
\]

Since there are no real numbers whose square is negative, \( x^2 + 9 \neq 0 \) for any real \( x \). This means the denominator never becomes zero, and the function is defined for all real values of \( x \).

### Conclusion:
The domain of \( f(x) \) is all real numbers:
\[
\text{Domain: } (-\infty, \infty)
\]

Let me know if you need further clarification or details!

Here are 5 related questions for further practice:
1. What is the range of the function \( f(x) = \frac{x - 9}{x^2 + 9} \)?
2. How would the domain change if the denominator was \( x^2 - 9 \) instead of \( x^2 + 9 \)?
3. Can you find any vertical asymptotes for the given function?
4. What would the horizontal asymptote of \( f(x) = \frac{x - 9}{x^2 + 9} \) be?
5. How does adding or subtracting a constant to the denominator affect the domain of a rational function?

### Tip:
To find the domain of any rational function, always check where the denominator equals zero. If it never does, the domain is all real numbers!

Find the domain of the rational function f(x) = (x - 9) / (x^2 + 9)

Learn how to determine the domain of the rational function f(x) = (x - 9) / (x^2 + 9). The solution explores key algebraic concepts, focusing on the behavior of the denominator and why it's always positive. Suitable for high school students in Grades 9-12.

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