I am currently trying to get a deeper understanding of the properties of inverse functions and integrals, so I would very much appreciate it if you could give me a little hand. This is a collection of some tasks I was unable to entirely grasp on my own.
Some background information: F(x) is the inverse function of $$G(x) = \int_{1}^{x}\frac{1}{t}dt$$ for x > 0.
I have proven that both F(x) and G(x) are continuous and differentiable. $$V_F = [0, \infty]\wedge D_F = R$$
We know that G(e) = 1, and therefore F(1) = e.
The issue here is proving:
$$F(a + b) = F(a)*F(b)$$ $$F(-a) = \frac{1}{F(a)}$$ $$F(a)^n = F(na)$$ $$F(n) = e^n$$
Without assuming G(x) = ln(x) or F(x) = e^x.
We also know that $$G(ab) = G(a) + G(b)$$ and $$G(\frac{1}{x}) = -G(x)$$
I am sorry if these problems are somewhat self-explanatory or bothersome, but I'd love it if you could help me understand! :))