Calculus of a Single Variable, Chapter 5, 5.4, Section 5.4, Problem 10

We are asked to use the derivative of y=5000/(1+e^(2x)) to determine if the function has an inverse. There is a theorem that states that if a function is monotonic (meaning the function is always non-increasing or non-decreasing) then it has an inverse. So, to show a function is monotonic, we need to check if its derivative is either positive or negative for all values in the domain.
By the quotient rule, the derivative is:
y'=(-5000(2e^(2x)))/(1+e^(2x))^2
The denominator is positive for all x. The 2e^(2x) term is positive for all x so the numerator is negative for all x. Therefore the derivative is always negative and the function decreases on its entire domain.
Thus the function has an inverse.
The graph of the function (black):