Is the parity of function related to the parity of inverse function

Even functions have no inverse functions except those whose domain of definition is x = 0
Odd function, inverse function or odd function

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1. Parity of inverse function of even function? I know that the inverse of an odd function is an odd function. What about the inverse of an even function? Is it "even function" or "not necessarily even function" or "not necessarily even function"?
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