Fiting a "glued together" function

i have another fitting question.
I m trying to fit a sum of many functions. The problem is that the original function has three subranges and it s for example exponential on one subrange and linear on another. The original function is not a sum of a linear and exp. but sort of glued together.I would like to describe this original function with a parameter( such as the position of the transformation between the exp and linear subranges)and then fit the sum of these original functions for these parameters. Is it possible to create my own personal function with one or two parameters and then use the Fit to fit a sum of these functions?
what would be my best strategy. Thanks in advance