One-dimensional extension of a rational-linear map ($h(d+qt)=f(d)+qa$)

Let $D$ be a rational vector subspace of the real numbers and let $t\notin D$. Every element of $D+\mathbb{Q}t$ has a unique expression $d+qt$. For any prescribed real number $a$, the rule $h(d+qt)=f(d)+qa$ extends a rational-linear map $f$ on $D$ and remains rational-linear. The uniqueness follows because a nonzero rational multiple of $t$ cannot lie in $D$.