Definition:Rational-Valued Function
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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 if and only if it is rational-valued over its entire domain.