Question

Question #04706

Answer 1

I assume that you have Rolle's Theorem and the Mean Value Theorem to work with.

Explanation:

`f` is continuous and differentiable on its domain.

If there were `x_1 != x_2` with `f(x_1)=f(x_2)`, then there would be a `c` in `(x_1, x_2)` with `f'(c) = 0` (Rolle's or MVT)

But, `f'(x)` is (strictly) negative on the domain of `f`.

Therefore, there cannot be `x_1 != x_2` with `f(x_1)=f(x_2)`.

So `f` is one-to-one and has an inverse.