The problem provided asks for the domain of two functions, $f(x) = \frac{x}{x^2 + 4}$ and $g(x) = \frac{x + 9}{x^2 - 81}$, expressed as intervals.

1. Domain of $f(x)$:

The domain of a rational function is defined by the values of $x$ for which the denominator is not zero. For $f(x)$:

$f(x) = \frac{x}{x^2 + 4}$

Here, the denominator $x^2 + 4$ is always positive (since $x^2 + 4 \geq 4$ for all real $x$). Therefore, there are no restrictions on $x$, meaning the domain of $f(x)$ is all real numbers.

$\text{Domain of } f(x): (-\infty, \infty)$

2. Domain of $g(x)$:

For $g(x)$:

$g(x) = \frac{x + 9}{x^2 - 81}$

The denominator $x^2 - 81 = (x - 9)(x + 9)$ equals zero when $x = 9$ or $x = -9$. These values must be excluded from the domain since division by zero is undefined.

$\text{Domain of } g(x): (-\infty, -9) \cup (-9, 9) \cup (9, \infty)$

Final Answer:

• Domain of $f(x)$: $(-\infty, \infty)$
• Domain of $g(x)$: $(-\infty, -9) \cup (-9, 9) \cup (9, \infty)$

Would you like any further clarification?

Here are five follow-up questions to deepen your understanding:

1. Why is the domain of $f(x)$ unrestricted despite having a rational function?
2. How do you factor the denominator of $g(x)$ to identify critical points?
3. How would the domain change if the function included a square root?
4. What would happen to the domain of $g(x)$ if the numerator was also zero at the excluded points?
5. How do you write a union of intervals in interval notation?

Tip: Always check the denominator for zero when finding the domain of rational functions.