Question

g(x) = 1/ (1-e^(x-1))
What are the natural domain and range of g?
anyone knows how to solve this? thank you

Answer 1

\[g(x) = \frac{1}{(1-e^{x-1})}\]

So, the problem is to determine the domain. Which is another way of saying, what are the values of x which provide me with a real answer.

In this case, the denominator can never equal zero. So, solve:
\[1-e^{x-1} = 0\]
to determine where your function is undefined.
\[e^{x-1} = 1\]
Which simplifies (by taking ln of each side)
\[\ln(e^{x-1}) = \ln(1)\]
\[x-1 = \ln(1)\]
\[x = \ln(1)+1\]
So, when x is equal to ln(1) + 1, my function is undefined. For other values of x, I'm good to go.

\[Domain = (-\infty, \ln(1)+1) \cup (\ln()+1),\infty)\]