To determine the interval over which the function [tex]\( g(x) = \sqrt[3]{x+6} \)[/tex] is negative, we need to understand when the expression inside the cube root results in a negative value.
### Step-by-Step Solution:
1. Identify the expression inside the cube root:
The function is given as [tex]\( g(x) = \sqrt[3]{x+6} \)[/tex].
Here, the expression inside the cube root is [tex]\( x + 6 \)[/tex].
2. Determine when the expression inside the cube root is negative:
The function [tex]\( g(x) \)[/tex] will be negative when:
[tex]\[
x + 6 < 0
\][/tex]
3. Solve for [tex]\( x \)[/tex] to find the critical value:
To find the value of [tex]\( x \)[/tex] that makes the expression inside the cube root zero, solve the inequality:
[tex]\[
x + 6 < 0
\][/tex]
Subtracting 6 from both sides:
[tex]\[
x < -6
\][/tex]
4. Write the interval where the function is negative:
The inequality [tex]\( x < -6 \)[/tex] tells us that [tex]\( g(x) \)[/tex] is negative for all [tex]\( x \)[/tex] less than -6.
5. Express the interval in interval notation:
In interval notation, this is written as:
[tex]\[
(-\infty, -6)
\][/tex]
So, the interval over which the function [tex]\( g(x) = \sqrt[3]{x+6} \)[/tex] is negative is [tex]\((- \infty, -6)\)[/tex].
### Conclusion:
The correct interval from the given choices is:
[tex]\[
(-\infty, -6)
\][/tex]
