Explain why domain restrictions are necessary to create inverse trigonometric functions rather than relations.(4 points)
1. Inverse trigonometric functions are defined as functions that undo the actions of their corresponding trigonometric functions. Therefore, domain restrictions are necessary in order to ensure that the inverse functions are well-defined and unique.
2. Without domain restrictions, inverse trigonometric functions would not pass the horizontal line test, meaning that they would not be functions but rather relations. By restricting the domain, we can guarantee that the inverse trigonometric functions have a one-to-one correspondence with their original trigonometric functions.
3. Domain restrictions are also necessary to avoid ambiguity and ensure that the inverse trigonometric functions have a clear and consistent output for every input value within their restricted domain. This helps in making the inverse trigonometric functions useful in various mathematical applications.
4. Additionally, domain restrictions help in defining the range of the inverse trigonometric functions, which is important in determining the behavior and properties of these functions. By specifying a restricted domain, we can determine a specific range for the inverse trigonometric functions that align with their characteristics.