Is it possible to modify any function $f(x)$ such that the new function yields only the real component $\Re[f(x)]$ of the original function?

Question

For example, suppose that you have the function, $f(x) = \sqrt{x-1} + \sqrt{x+1}$, which yields a mixture of purely real, purely imaginary and complex values. Is there an operation, or a series of operations, that could be applied to a function like this to describe only the real component of $f(x)$?