Reading some material on rational maps between projective varieties, I've seen that some people define them differently. For example, some people define a rational map $f : \mathbb P^n \to \mathbb P^m$ by $f(x) = [f_0(x), \dots, f_m(x)]$ for homogeneous functions $f_0,\dots,f_m$ of degree $d$. However, I have also seen it defined by $f(x) = [f_0(x), \dots, f_m(x)]$ for $f_0,\dots,f_m \in k(\mathbb P^n)$ instead.
Are these two definitions equivalent? Namely, if we have $f_0,\dots,f_m \in k(\mathbb P^n)$ can we produce homogeneous functions $f'_0, \dots, f'_m$ all of degree $d$ where $f(x) = [f'_0(x), \dots, f'_m(x)]$ as well (and vice versa)?