Can a one to one function not be a function?

"y = f(x) is a function if it passes the vertical line test. It is a 1-1 function if it passes both the vertical line test and the horizontal line test. " -

In order to be a one-to-one function, it first has to BE a function and pass the vertical line test. For example, a relation on a graph like a circle that does not pass the vertical line test is not function nor one-to-one.