Question

Prove that g(x)=1/f(x) is even if and only if f(x) is even.

Answer 1

i would use a contradiction, so lets assume f(x) is odd in that f(-x)=-f(x).so g(-x)=1/f(-x) = 1/-f(-x)=-1/f(-x). Hence if f(x) is odd g(x) is odd as if g(x) were even then g(-x)=g(x) yet -1/f(x) does not equal 1/f(x) CONTRADICTION . Now consider that f(x) is neither odd or even. In this case f(-x) does not equal-f(x) nor does it equal f(x), hence using the same method as before we show that for each of these cases g(-x) does not equal g(x) so g(x) is not even hence the CONTRADICTION. now let f(x) be even. f(-x)=f(x) so g(-x)=1/f(-x)=1/f(x)=g(x). So it when f(x) is even g(x) is also even. Thus we have proven that g(x) is even if and only if f(x) is even QED. Come at me bro :D.

Answer 2

that was a decent question.