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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! :))

MCrypa
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  • See here: http://math.stackexchange.com/questions/2176741/prove-that-fxy-fx-fy-when-fx-is-not-lnx#comment4477418_2176741 and here: http://math.stackexchange.com/questions/2177866/prove-if-fxy-fxfy-then-fx-ax – Simply Beautiful Art Mar 08 '17 at 17:49
  • Wow, thank you :)) Although one of the links were another question of mine haha – MCrypa Mar 08 '17 at 17:52
  • :P No problem! (I'm sure there's other similar questions, but those were the ones I could immediately find.) – Simply Beautiful Art Mar 08 '17 at 17:53
  • If you have already proven the formulas for $G$, then these follow immediately from $F$ being the inverse function of $G$: $G(F(a + b)) = ?$, and $G(F(a)F(b)) = ?$ Both of these can be answered using the product formula for $G$ and the fact that it is the inverse of $F$. – Paul Sinclair Mar 09 '17 at 04:43

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