Okay so I have what seems to be a fairly simple problem. Find the domain and range of f(x)=(e^x) - 1. Then find the inverse and its domain and range.
Normally I can do this, but because of e I'm stuck.
dom f = R (you should already know this). To find the range of any function you should first draw a graph. It will then be clear that Ran f = (-1, +oo).
The rule for the inverse function $\displaystyle y = f^{-1}(x)$ is found by solving $\displaystyle x = e^y - 1$. If you're familiar with logarithms (and I assume you are since you've been set this problem) this equation should be easy to solve. Where do you get stuck?
Finally, you should know that $\displaystyle \text{dom } f^{-1} = \text{ran } f$ and $\displaystyle \text{ran } f^{-1} = \text{dom } f$.
The domain and range of $\displaystyle e^x$ are all real numbers and all real numbers larger than 0, respectively. That is, in $\displaystyle y= e^x$ x can be any number and y can be any positive number. Now what about $\displaystyle y= e^x- 1$? If you can calculate $\displaystyle e^x$ for any value of x, can't you then subtract 1 from any number? If $\displaystyle y= e^x$ can be any positive number and then you subtract 1 from it what do you get?