Final answer:
The cube root function does exist in quadrants I and III because positive numbers yield positive cube roots in Quadrant I, and negative numbers yield negative cube roots in Quadrant III. The statement is true.
Step-by-step explanation:
The cube root function indeed exists in quadrants I and III on a two-dimensional (x-y) graph. The statement is true. To understand why, consider the properties of the cube root function. First, any positive number, when subjected to a cube root, will return a positive result.
Which places its representation in Quadrant I of the Cartesian plane, where both x and y values are positive. Conversely, cube rooting a negative number will yield a negative result, which will be represented in Quadrant III where both the x and y values are negative.
This is due to the fact that the cube of a negative number is negative, and taking the cube root brings it back to a negative value.