To determine the interval of negative values for the cube root function [tex]\( h(x) = \sqrt[3]{x} \)[/tex], let's analyze the behavior of this function:
1. Understanding the Cube Root Function:
- The cube root function, [tex]\( h(x) = \sqrt[3]{x} \)[/tex], is defined for all real numbers. This means there are no restrictions on the domain of this function.
- Unlike the square root function, which is only defined for non-negative numbers, the cube root function is also defined for negative numbers. This is because the cube root of a negative number is also a real number (e.g., [tex]\( \sqrt[3]{-8} = -2 \)[/tex]).
2. Interval of Negative Values:
- To find the interval of negative values for [tex]\( h(x) = \sqrt[3]{x} \)[/tex], we need to identify the domain where [tex]\( x \)[/tex] is negative.
- Negative values of [tex]\( x \)[/tex] are represented by the interval extending from negative infinity up to, but not including, zero.
3. Formal Representation:
- In interval notation, the set of negative numbers can be represented as [tex]\( (-\infty, 0) \)[/tex].
So, the interval of negative values for the cube root function [tex]\( h(x) = \sqrt[3]{x} \)[/tex] is [tex]\( (-\infty, 0) \)[/tex].
