Final answer:
In the given options, C and D represent functions. Option C associates each x with one y-value based on x² + sin x, and option D associates each x with one y-value as well. Options A and B do not represent functions as they can associate one x-value with multiple y-values.
Step-by-step explanation:
The question is which of the given options represent a function. A function, in mathematics, is a relation between sets that associates to every element of a first set exactly one element of the second set.
Option A, |x| + |y| = 2, does not represent a function because for a given x-value, there can be more than one corresponding y-value. Option B, |x + y| = 2, also does not represent a function for similar reasons to option A, as it does not satisfy the criterion of having exactly one y-value for each x-value.
Option C, |y| = x² + sin x, does represent a function because for each x-value, there is exactly one y-value determined by the expression on the right side. Option D, |x|² - x, can be reconsidered as y = |x|² - x, which is also a function, because it associates each x-value with a single y-value.
Therefore, the correct answers are options C and D, as both represent valid functions.