We consider additive function regression for association studies between functional covariate and scalar or functional outcome. We introduce the functional generalized additive model - a flexible approach to describe association when the response is scalar and discuss its extension/s for functional responses. The key idea is to model the effect of the functional covariate at some time point through a bi-variate or tri-variate unknown function depending on the value of the functional covariate at the specified time and the time point itself. We develop computationally efficient estimation methodology and discuss prediction of a new response value or trajectory. The modeling framework allows for inference in the form of confidence bands, prediction intervals or hypothesis testing. We investigate our methodology numerically, through simulations and real data applications.