Integer values of a rational function

Question

How does one analytically determine the integer values of a rational function $f(x)$$=$$\frac{40-8x}{8x+2}mod1$ where $x$ is an element of the rationals? I just gave the function listed as an example, I would like to know the generally preferred methods (if they exist) of analytically determining integer values of rational functions.

Answer 1

Multiply the equation $\frac{40 - 8 x}{8 x + 2} = y$ by the denominator and you have a linear equation in $x$: solve to get

$$ x = \dfrac{20-y}{4+4y} $$ Note that if $y$ is an integer (other than $-1$), $x$ is a rational number.