Why is duplicating inputs bad?
The issue with building a regression model on all 3 of these is that you are potentially introducing multicollinearity into the model. Although log(input) and sqrt(input) are not linear functions of the input a quick test (using Matlab) shows they are still highly correlated (depending on the range)

input=rand(1,100);
input_log=log(input);
input_rt=sqrt(input);
corrcoef(input, input_log) %-&gt;0.8549
corrcoef(input, input_rt) %-&gt;0.9779
corrcoef(input_sq, input_rt) %0&gt;0.9407


Multicollinearity will not reduce the predictive power of the model, but it will make the regression coefficients of these variables difficult to interpret since small adjustments in the data may cause the regression model to switch which of the 3 input variations it finds significant. It also adds unnecessary complexity to the model. I would train the models with the 3 separately if you want to observe the effects.

Additionally, applying things like sqrt and log to your output can be useful if the data currently can't be described by a linear relationship.