To determine on which interval the function [tex]\( g(x) = \sqrt[3]{x - 5} \)[/tex] is negative, let's follow these steps:
1. Understand the Function: The function [tex]\( g(x) = \sqrt[3]{x - 5} \)[/tex] involves a cube root. For [tex]\( g(x) \)[/tex] to be negative, the expression inside the cube root [tex]\( x - 5 \)[/tex] must be negative. This is because the cube root of a negative number is also negative.
2. Set up the Inequality: To find when [tex]\( g(x) \)[/tex] is negative, set up the inequality:
[tex]\[
\sqrt[3]{x - 5} < 0
\][/tex]
3. Solve the Inequality: Solve the inequality [tex]\( \sqrt[3]{x - 5} < 0 \)[/tex].
- For the cube root to be negative, the radicand (the expression inside the cube root) must be negative:
[tex]\[
x - 5 < 0
\][/tex]
- Solve this inequality by isolating [tex]\( x \)[/tex]:
[tex]\[
x < 5
\][/tex]
4. Determine the Interval: The solution [tex]\( x < 5 \)[/tex] describes all values of [tex]\( x \)[/tex] less than 5.
Thus, in interval notation, the solution is:
[tex]\[
(-\infty, 5)
\][/tex]
Therefore, [tex]\( g(x) \)[/tex] is negative on the interval [tex]\(\boxed{(-\infty, 5)}\)[/tex].
