what is the domain of q(x)=$$ 8\text{ over }16-x^2 $$

To find the domain of the function $ q(x) = \frac{8}{16 - x^2} $, we need to ensure that the denominator is not equal to zero, since division by zero is undefined.

The denominator of the function is $ 16 - x^2 $. Set the denominator equal to zero and solve for $ x $:

\[
16 - x^2 = 0
\]

\[
x^2 = 16
\]

\[
x = \pm 4
\]

Thus, the function is undefined at $ x = 4 $ and $ x = -4 $, because these values would make the denominator zero.

Therefore, the domain of the function is all real numbers except $ x = 4 $ and $ x = -4 $.

In interval notation, the domain is:

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

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. What is the range of the function $ q(x) $?
2. How would the domain change if the function was $ q(x) = \frac{8}{(16 - x^2)^2} $?
3. What is the vertical asymptote of the function $ q(x) $?
4. Can you determine the intercepts of the function $ q(x) $?
5. How would the graph of $ q(x) $ look like?

**Tip**: Always check for restrictions in the domain by ensuring the denominator is non-zero in rational functions!

what is the domain of q(x)= 8 over 16-x^2

Learn how to determine the domain of the rational function q(x) = 8/(16 - x^2). This problem demonstrates important concepts like domain restrictions, solving quadratic equations, and ensuring the denominator is non-zero. Suitable for high school students in Grades 9-12.

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