Find an exponential function with given condition How can I have an example of an exponential function defined in the X range 1 - infinity, with values starting at 40 and converging to 1?

It can't be a pure exponential, since a decaying exponential function decays to $0$. But we can look for a function of the kind $1+ke^{-x}$. Then our condition of having value $40$ at $1$ becomes the equation
$$1+ke^{-1}=40.$$
Solve. We get $ke^{-1}=39$, so $k=39e$.
Slightly more naturally, we can look for a function of the type $1+ce^{-(x-1)}$. Then we find that $c=39$.
We have freedom in adjusting the rate of decay, by looking for functions of the shape
$$1+ce^{-\lambda(x-1)}.$$
Pick any positive $\lambda$ that you like, and let $c=39$.
There is no need to use $e$ as the base. Let $a$ be your favourite base.  We can look for a function of shape  $1+c a^{-(x-1)}$. Again we will get $c=39$.  
