Definition:Rational-Valued Function

Definition
Let $f: S \to T$ be a function.

Let $S_1 \subseteq S$ such that $f \left({S_1}\right) \subseteq \Q$.

Then $f$ is said to be rational-valued on $S_1$.

That is, $f$ is defined as rational-valued on $S_1$ iff the image of $S_1$ under $f$ lies entirely within the set of rational numbers $\Q$.

A rational-valued function is a function $f: S \to \Q$ whose codomain is the set of rational numbers $\Q$.

That is, $f$ is rational-valued it is rational-valued over its entire domain.