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The given function is:

\[
f(x) = \frac{5x}{x^2 - 16}
\]

To find the domain of the function, we need to determine where the function is defined. Specifically, the function will be undefined when the denominator is zero.

The denominator is \( x^2 - 16 \), so we solve for when it equals zero:

\[
x^2 - 16 = 0
\]

\[
x^2 = 16
\]

\[
x = \pm 4
\]

Thus, the function is undefined at \( x = 4 \) and \( x = -4 \). Therefore, the domain of the function is all real numbers except \( x = 4 \) and \( x = -4 \).

The domain of \( f(x) \) is:

\[
(-\infty, -4) \cup (-4, 4) \cup (4, \infty)
\]

Would you like further details or have any questions?

Here are five related questions to expand on this topic:
1. How would the domain change if the function was \( \frac{5x}{x^2 + 16} \)?
2. What is the range of the given function \( f(x) \)?
3. How would you find the vertical asymptotes of this function?
4. What would the derivative of \( f(x) \) tell you about the behavior of the function?
5. How do you find horizontal asymptotes for rational functions like this?

**Tip:** When finding the domain of a function, always look for values that make the denominator zero or create undefined operations like taking the square root of a negative number.

Learn how to find the domain of the rational function f(x) = 5x / (x^2 - 16) by determining when the denominator is zero. This algebra problem covers domain restrictions for rational functions, factoring, and solving quadratic equations. Suitable for students in Grades 9-12.

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