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I have to show that arcsinh is the inverse function to sinh. I checked that $\sinh(\operatorname{arcsinh}(x))=x$. Is that sufficient or do I also need to show that $\operatorname{arcsinh}(\sinh(x))=x$?

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You also need to show the second equality. For example, $\sqrt{x}$ is not the inverse of $x^2$ even though $\sqrt{x}^2=x$, because $\sqrt{x^2}\neq x$ for negative $x$. –  Alex Becker Jan 10 '12 at 9:38
    
Or, maybe another comment, you should state the domain and range of each of your functions. For example, sinh is not injective on the complex numbers... –  GEdgar Jan 10 '12 at 13:29
    
I see the problem. I'll go for the other equation tomorrow. Our domain is just the real numbers. I would have to argue that it is injective (sinh) in order to get by with just one equation? –  queueoverflow Jan 10 '12 at 15:13
    
I have now stated that sinh is fully defined in the real numbers, is continuous (smooth), since it is a combination of e-functions. And I have proven that it is strictly monotonic. Therefore it is bijective, and that means that I can turn the inverses as I like? –  queueoverflow Jan 10 '12 at 19:47

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