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Finding the domain of a composite function

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You need to take the square root in both equalities. So x lies in (-sqrt(2e),-sqrt(2/e)) U (sqrt(2/e),sqrt(2e))
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As people are pointing out, your answer is (almost) right. Since your teacher says log = ln, the domain is definitely in terms of e.

I don't know if anyone mentioned, but those should be closed intervals not open. You can certainly take the arcsin of +-1.

Also, it's slightly bothering me that you didn't check the domain of log(x\^2/2). That domain is R \\ {0}, all real numbers but 0. 0 is excluded from the domain anyway so that doesn't affect the final answer, but I feel you should have checked it.
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Shouldn't the antilog of the middle part of your inequality be x^2 /2?

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