https://file.aiquickdraw.com/mathbot/file/1725609208214vut7vx26.jpg

The function provided in the image is:

\[
f(x) = \frac{9}{x^2 + 16}
\]

To determine the **domain** of this function, we need to find all values of \(x\) for which the function is defined.

Since the function is a rational function, it is defined wherever the denominator is non-zero. The denominator of the function is \( x^2 + 16 \).

Now, let’s analyze the denominator:

\[
x^2 + 16 \geq 16 \quad \text{for all real values of} \, x.
\]

This expression will never be zero for any real value of \(x\), because \(x^2 \geq 0\) and adding 16 ensures that the denominator is always positive. Thus, the denominator never equals zero.

Therefore, the function is defined for **all real values** of \(x\).

### Conclusion:
The domain of the function \(f(x) = \frac{9}{x^2 + 16}\) is:

\[
\text{Domain}: \, (-\infty, +\infty)
\]

Do you need further clarification or details on this? 

### Related Questions:
1. How do you determine the domain for other types of rational functions?
2. Can this function have any asymptotes? If so, what kind?
3. How would you find the range of this function?
4. What is the behavior of this function as \( x \) approaches infinity?
5. How would you differentiate this function?

### Tip:
When finding the domain of a function involving a fraction, always check where the denominator might be zero—these are the points to exclude.

Explore how to find the domain of the rational function f(x) = 9 / (x^2 + 16). This solution explains why the denominator never equals zero and identifies that the function is defined for all real values of x. Suitable for high school students in Grades 9-12.

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