Example for a function which is both even and odd
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Answer:
A function f is even if and only if f(−x)=f(x) for all x in its domain. A function f is odd if and only if f(−x)=−f(x) for all x in its domain.
Thus, for a function to be both even and odd at the same time, f(−x)=f(x)=−f(x) for all x in its domain, which is only possible if f(x)=0 for all x in its domain.
For a function to have an inverse, it needs to be one-to-one. If a function is even, then y=f(x)=f(−x) for all x in its domain, so its inverse would presumably have x=±f−1(y), so its inverse wouldn't be a function. Remember: a function is a relation that maps elements in a domain to a unique element in the range; the relation x=±f−1(y) does not map elements uniquely.
A periodic function would have f(x)=f(x+p) for some periodic time p, so again, the same problem ensues: if y=f(x)=f(x+p) for all x in its domain, its inverse relation would relate y to both x and x+p (and, indeed, to any element of the form x+kp, k∈Z), so the relation wouldn't be a function.