experimental physics - How to guess the correct fitting function to some data?

Imagine we are plotting some date points $\left(x_i,f(x_i)\right)$ that we obtained experimentally, and that we want to know what $f(x)$ is. The way to do this is to use some software and try to fit the data to some guessed function. For example, if the behavior of the data points looks like exponential decay we then choose an exponential decaying function ..etc.

My question is: sometimes the data points are perfectly fitted to the exponential decaying trial function only on a certain region, but then the rest of the points show deviation away from the trial function.

1. How to proceed in this case? is there a catalog (something like mathematical tables) for functions and their plots that one can use as a guide?


2. Is there a systematic way to get the best fit instead of that trial and error method?