Example: Let f(x) = 0 if x is rational and f(x) = 1 if x is irrational. This function is called Dirichlet’s Function. The domain and range for f is R.
The preimage of R is the set of all elements such that f(x) is contained in R. Since R includes the numbers 0 and 1, the preimage of R under f is everything in the domain, i.e. R.

The preimage of [-1/2, 1/2] consists of all elements such that f(x) is contained in [-1/2, 1/2]. This set does not contain 1, so the preimage of that set can not contain any irrational number. Hence, the preimage of [-1/2, 1/2] is the set of rational numbers Q.

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