Does squareroot of a complex number have two different outputs?

Question

A function is defined as a mapping for which it is true that for every input there is exactly one output. Squareroot is a function. It is defined as the inverse function to the quadratic function. So why does a squareroot of a complex number nine have two outputs? They should be +3 and -3. I don't understand that. It is a contradiction to the definition of a function, ie to the part "... only one output."

Answer 1

$f(x)=x^2$ isn't one-to-one, and hence isn't invertible. It doesn't pass the horizontal line test. Not every function does.

In complex analysis you learn about branches of the square root function. Branch cuts are a way of dealing with certain multi-valued functions.

The theory of Riemann surfaces has to do with this phenomenon.