In this work we present methods to study the causal effect of a scalar treatment on a functional outcome based on observational data. We develop a semi-parametric estimator for a Functional Average Treatment Effect (FATE), based on outcome regression. Using recent results from functional data analysis, we show how to obtain exact valid inferences on the FATE under certain conditions: we give simultaneous confidence bands, which cover the parameter of interest with a given probability over the entire domain. Using simulation experiments, we compare the performance of the simultaneous confidence bands to that of pointwise bands that do not take the multiple comparison problem into account, and find that the former achieve the desired coverage rates, whereas the latter do not. In addition, we use the methods presented to estimate the effect of early adult location on subsequent income development for one Swedish birth cohort. Overall, we find a positive effect of living in an urban, as opposed to rural, area at the age of 20 on cumulative lifetime incomes, but there are differences by gender. For women, the effect is stronger and positive over the entire study period, whereas for men there is a negative effect during the first years.