https://file.aiquickdraw.com/mathbot/file/1726303687571hgqnuien.png

The given function is:

\[
g(x) = \frac{3}{\sqrt{x^2 - 16}}
\]

### Domain of the Function

To determine the domain of \( g(x) \), we must identify the values of \( x \) for which the function is defined.

1. The square root, \( \sqrt{x^2 - 16} \), must be defined. This implies:
   \[
   x^2 - 16 \geq 0
   \]
   Simplifying this inequality:
   \[
   x^2 \geq 16
   \]
   \[
   |x| \geq 4
   \]
   Therefore, \( x \geq 4 \) or \( x \leq -4 \).

2. Additionally, the denominator must not be zero, so:
   \[
   \sqrt{x^2 - 16} \neq 0
   \]
   This implies:
   \[
   x^2 - 16 \neq 0 \quad \Rightarrow \quad x^2 \neq 16 \quad \Rightarrow \quad x \neq \pm 4
   \]

Thus, the domain of the function is:
\[
(-\infty, -4) \cup (4, \infty)
\]

Would you like further details or have any questions?

Here are some related questions for further exploration:

1. What is the range of the function \( g(x) \)?
2. How would the graph of \( g(x) \) look?
3. Can we find the vertical asymptotes of this function?
4. How can we differentiate \( g(x) \)?
5. What happens to \( g(x) \) as \( x \to \infty \)?

**Tip**: Always check for restrictions in the domain when square roots or denominators are involved in a function.

What is the domain of the function g(x) = 3 / √(x² - 16)?

Learn how to find the domain of the function g(x) = 3 / √(x² - 16). Explore the restrictions imposed by the square root and the denominator, suitable for students in Grades 10-12.

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