To determine the interval where the function [tex]\( g(x) = \sqrt[3]{x - 5} \)[/tex] is negative, we need to analyze the behavior of the cube root function [tex]\(\sqrt[3]{x - 5}\)[/tex].
1. Understanding the Cube Root Function:
- The cube root function, [tex]\(\sqrt[3]{y}\)[/tex], is defined for all real numbers [tex]\(y\)[/tex], and it changes sign at [tex]\(y = 0\)[/tex]. Specifically, [tex]\(\sqrt[3]{y}\)[/tex] is negative when [tex]\(y < 0\)[/tex] and positive when [tex]\(y > 0\)[/tex].
2. Finding When [tex]\( \sqrt[3]{x - 5} \)[/tex] is Negative:
- The expression inside the cube root is [tex]\(x - 5\)[/tex].
- We need to determine where [tex]\(x - 5\)[/tex] is negative because this will make the cube root function negative.
3. Setting Up the Inequality:
- To find when [tex]\(x - 5 < 0\)[/tex]:
[tex]\[
x - 5 < 0
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
x < 5
\][/tex]
4. Conclusion:
- The cube root function [tex]\( \sqrt[3]{x - 5} \)[/tex] is negative for values of [tex]\(x\)[/tex] less than 5.
- Therefore, the interval where the function is negative is:
[tex]\[
(-\infty, 5)
\][/tex]
Hence, the correct interval where the function [tex]\( g(x) = \sqrt[3]{x - 5} \)[/tex] is negative is [tex]\((- \infty, 5)\)[/tex].
