For Generalized Additive Models (GAMs) in regression problems, to depict a non-linear relationship between predictors and their response, we start with the multiple linear regression model $y_i = \beta_0 + \beta_1x_{i1} + \beta_2x_{i2} + \dots + \beta_px_{ip} + \epsilon_i$. We then replace each linear component ($\beta_jx_{ij}$) with a non-linear function $f_j(x_{ij})$ so that the model becomes $y_i = \beta_0 + f_1(x_{i1}) + f_2(x_{i2}) + \dots + f_p(x_{ip}) + \epsilon_i$. A separate non-linear function $f_j$ is applied to every predictor variable, and these are then added together. This is why it is considered an additive model.