given a linear transformation of the mean will the standard deviation stay the same?

I have to make a linear transformation of a mean to "denormalize" it after a prediction comes out of a model. If the denormalization is just a linear transformation of the mean, does the standard deviation need to undergo any transformation as well?

The function used to normalize, (and reversely denormalize) is written below...

$$normalized = \frac{x - min}{max - min} * (max_{new} - min_{new}) + min_{new}$$

The standard deviation does not change when you add a constant, but it is affected when you multiply by a factor (gets multiplied by the absolute value of the factor). In your case $$\sigma (normalized)=\frac{max_{new}-min_{new}}{max-min}\sigma(x).$$