To determine the value of \( k \) in the function \( g(y) = y + k \) that describes the translation from the point \( (4, -1) \) to \( (4, 2) \), we can focus on the change in the y-coordinate.
The starting y-coordinate is \(-1\) and the ending y-coordinate is \(2\). The translation can be described by the equation:
\[ g(-1) = 2 \]
Substituting \(-1\) into the function:
\[ g(-1) = -1 + k \]
Setting this equal to the final y-coordinate:
\[ -1 + k = 2 \]
Now, we solve for \( k \):
\[ k = 2 + 1 = 3 \]
Thus, the value of \( k \) is \( \boxed{3} \).