The existence of an inverse function is important in mathematics often because the function and its inverse give "dictionaries" by which one can translate information about the domain to the range and back again.
Inverse trigonometric functions
The trigonometric functions , , , , and are not invertible functions on their usual domains. However, by restricting the functions to appropriate smaller domains, they become invertible. There are multiple possible restricted domains for each function on which they become invertible, which causes ambiguity in the definition of the inverse trigonometric functions. This ambiguity often creates confusion for the newcomer to trigonometry.