Short Answer 
The cube root function can yield both positive and negative results, with the cube root of a negative number being the negative of the cube root of its positive counterpart. Unlike square roots, which produce distinct results and do not overlap, cube roots can produce negative values from negative inputs and exhibit relational symmetry.
Step 1: Understanding Cube Roots
The cube root function allows for both positive and negative inputs. When we take the cube root of a negative number, we get a negative result, while the cube root of a positive number results in a positive value. In mathematical terms, the functions can be stated as:
- For negative input: (y = sqrt[3]{-x})
- For positive input: (y = -sqrt[3]{x})
Step 2: Distinguishing Between Roots
It’s crucial to understand that the positive square root and the negative square root of a number produce distinct results. This difference means that the graphs of these functions represent different mathematical entities. For instance, the square root of a negative number does not yield a real number, unlike the cube roots which do.
- No overlap occurs between positive and negative square roots.
- Cube roots can produce negative values from negative inputs.
Step 3: Identifying Function Equivalence
When comparing the cube root of negative numbers with their positive counterparts, we find that they relate closely. Specifically, the cube root of a negative number is equivalent to the negative of the cube root of the positive representation of that number. Thus, both the fourth and sixth functions represent the same mathematical relationship.
- (y = sqrt[3]{-x}) is the same as (y = -sqrt[3]{x}).
- However, this is not true for positive cube roots.