As I understand it, you would be helped by two functions, call them g(x) and h(x), where g(x) starts at g(0) = 1 and tends toward 0 as x --> infinity; and h(x) starts at h(0) = 0 and tends toward 1 as x --> infinity. Then you would be able to build e^(f(x)*g(x)) + x*h(x). This would start off like y = e^(f(x)) and smoothly approximate y = x + 1. Not knowing any particular values you want for this graph, we can get a family of approximately desired functions by taking g(x) = e^(-x) and I'm still working on a good answer for h(x), but maybe you can come up with your own, or tell me whether I'm understanding what you want correctly.