If f: R → (0, 2) defined by is invertible, find f-1.
We have f: R → (0, 2) and
Given that f-1 exists.
Let y ϵ (0, 2) such that f(x) = y
⇒ 2e2x = y (e2x + 1)
⇒ 2e2x = e2xy + y
⇒ 2e2x – e2xy = y
⇒ e2x (2 – y) = y
Taking ln on both sides, we get
We have f(x) = y ⇒ x = f-1(y)
But, we found f(x) = y ⇒
Hence,
Thus,