A function is considered "closed" if the set of all its possible outputs is contained within its own domain. In other words, all possible inputs must have a corresponding output in order for a function to be considered closed.
A closed function ensures that every possible input has a defined output, which is necessary for mathematical operations and calculations. It also allows for a more accurate and complete understanding of the behavior of the function.
If a function has a domain that is not continuous or has "gaps" in its range, it is likely not a closed function. Additionally, if there are any undefined or infinite values in the function, it is not considered closed.
Cosine (cos(x)) is a trigonometric function that is defined for all real numbers, but its range is limited to values between -1 and 1. This means that there are inputs for which there is no corresponding output, making it a non-closed function.
Yes, non-closed functions can still be useful in certain contexts. For example, they can be used to model real-world phenomena or to approximate more complex functions. However, in most cases, closed functions are preferred as they provide a more complete and accurate representation of the relationship between inputs and outputs.