To determine the value of \( k \) in the function \( g(y) = y + k \) that describes the translation from the point \( (4, -1) \) to the point \( (4, 2) \), we need to focus on the \( y \)-coordinates of these points.
When we translate the point from \( (4, -1) \) to \( (4, 2) \), the \( y \)-coordinate changes from \(-1\) to \(2\). The transformation can be described as follows:
\[ g(-1) = 2 \]
Using the function \( g(y) = y + k \), we substitute \( y = -1 \):
\[ g(-1) = -1 + k \]
Setting this equal to \( 2 \):
\[ -1 + k = 2 \]
To solve for \( k \):
\[ k = 2 + 1 \] \[ k = 3 \]
Thus, the value of \( k \) is \( \boxed{3} \).