  Let $\mathcal{E}$, $\mathcal{E}_1$, and $\mathcal{E}_2$ be equations, $n$ and
$k$ be positive integers. The rainbow number
$\operatorname{rb}([n],\mathcal{E})$ is difined as the minimum number of colors
such that for every exact $(\operatorname{rb}([n],\mathcal{E}))$-coloring of
$[n]$, there exists a rainbow solution of $\mathcal{E}$. The Gallai-Rado number
$\operatorname{GR}_k(\mathcal{E}_1:\mathcal{E}_2)$ is defined as the minimum
positive integer $N$, if it exists, such that for all $n\ge N$, every
$k$-colored $[n]$ contains either a rainbow solution of $\mathcal{E}_1$ or a
monochromatic solution of $\mathcal{E}_2$. In this paper, we get some exact
values of rainbow and Gallai-Rado numbers involving binary function equations.
We also provide an algorithm to calculate the rainbow numbers of nonlinear
binary function equations.
