To determine the interval on which the function [tex]\( g(x) = \sqrt[3]{x + 6} \)[/tex] is negative, we need to consider the properties of the cube root function and the argument inside the cube root.
1. Understanding the Cube Root Function:
The cube root function, [tex]\( \sqrt[3]{y} \)[/tex], outputs a negative value when its argument [tex]\( y \)[/tex] is negative. Therefore, in order for [tex]\( g(x) \)[/tex] to be negative, the expression inside the cube root, [tex]\( x + 6 \)[/tex], must be negative.
2. Setting up the Inequality:
We need to solve the inequality:
[tex]\[
x + 6 < 0
\][/tex]
3. Solving the Inequality:
Subtract 6 from both sides to isolate [tex]\( x \)[/tex]:
[tex]\[
x < -6
\][/tex]
4. Defining the Interval:
The solution to this inequality tells us that [tex]\( g(x) \)[/tex] will be negative for all [tex]\( x \)[/tex] values that are less than [tex]\(-6\)[/tex].
Therefore, the interval on which the function [tex]\( g(x) = \sqrt[3]{x + 6} \)[/tex] is negative is:
[tex]\[
(-\infty, -6)
\][/tex]
Thus, the correct choice from the given options is:
[tex]\[
(-\infty, -6)
\][/tex]
