I would like to find a mathematically correct way of denormalizing a linear combination of previously normalized variables, using the information of original variables. The normalized variables are between $0$ and $1$, so it is NOT a z-score normalization. Let me explain the problem with an example.
Assume that we have two variables $X$ and $Y$, with minimum and maximum values of $X_{min}$, $X_{max}$, $Y_{min}$, and $Y_{max}$, respectively. I also define $\Delta X = X_{max} - X_{min}$, and $\Delta Y = Y_{max} - Y_{min}$. The normalization results in:
$X_n = norm(X) = (X-X_{min})/\Delta X, \quad 0 \leq X_n \leq 1$
$Y_n = norm(Y) = (Y-Y_{min})/\Delta Y, \quad 0 \leq Y_n \leq 1$
and if I want to denormalize back to original variables, I simply use
$X = denorm(X_n) = (X_n \cdot \Delta X) + X_{min}$
$Y = denorm(Y_n) = (Y_n \cdot \Delta Y) + Y_{min}$
Now let's define $Z_n = X_n + Y_n$ as a linear combination of the normalized variables. I can expand $Z_n$ as
$Z_n = (X-X_{min})/\Delta X +(Y-Y_{min})/\Delta Y = \dfrac{X\cdot \Delta Y+Y\cdot \Delta X-\Delta X\cdot Y_{min}-\Delta Y\cdot X_{min}}{\Delta X\cdot \Delta Y}$
Here is the question: How do I find $Z$, i.e. the denormalized version of $Z_n$, as a function of $X_{min}$, $\Delta X$, $Y_{min}$, and $\Delta Y$? In other words, is there a closed form for function $f$ below:
$Z = denorm(Z_n) = (Z_n \cdot \Delta Z) + Z_{min}= f(X_{min}, \Delta X, Y_{min}, \Delta Y)$