Find the domain of the function.
​g(x)equalsStartFraction 5 x Over x squared minus 36 EndFraction

To find the domain of the function:

\[
g(x) = \frac{5x}{x^2 - 36}
\]

we need to identify where the denominator is zero, as the function is undefined when the denominator equals zero.

1. Set the denominator equal to zero and solve for \(x\):
   \[
   x^2 - 36 = 0
   \]
2. Factor the equation:
   \[
   (x - 6)(x + 6) = 0
   \]
3. Solve for \(x\):
   \[
   x = 6 \quad \text{or} \quad x = -6
   \]

Thus, the function is undefined at \(x = 6\) and \(x = -6\).

### Domain
The domain of the function consists of all real numbers except \(x = 6\) and \(x = -6\). In interval notation, this is:

\[
\text{Domain: } (-\infty, -6) \cup (-6, 6) \cup (6, \infty)
\]

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. How do you determine if a function is undefined?
2. What are the vertical asymptotes of this function?
3. Can you find the range of this function?
4. How do you find the x- and y-intercepts of a rational function?
5. How would you graph this function?

**Tip:** Always check the denominator of a rational function to avoid division by zero when determining the domain.

Find the domain of the function: g(x) = 5x / (x^2 - 36).

Explore how to find the domain of the rational function g(x) = 5x / (x^2 - 36) using factoring techniques. This problem focuses on understanding the restrictions imposed by the denominator and identifying the real number intervals where the function is defined. Suitable for high school students studying algebra and rational functions.

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