Rational and Irrational Numbers in Functions

Consider the real numbers defined by the function:

$$f(x) =\frac{x^2 - 4}{x - 2}$$

Two values of $x$, namely $x_1$ and $x_2$, are chosen such that both $f(x_1)$ and $f(x_2)$ are irrational numbers. If $x_1$ is a rational number and $x_2$ is derived from $x_1$ by taking the square root of its result, what type of numbers can $f(x_1)$ and $f(x_2)$ be? Identify the absolute value of $f(x_1) + f(x_2)$.

Note: The function has a removable discontinuity. Consider the simplification that affects the behavior of this function.