Odd/even functions and fourier transforms

[anj158]

Hi, I'm really stuck on this question and have attempted to solve it, I think I'm getting close, but I just need someone to point me in the right direction?

I have to prove that if f $$\in$$ M(R) is odd/even, then $$\hat{f}$$ is odd/even

(where M(R) is the set of moderately decreasing functions)



I have used the rules that if f is even: f(x)=f(-x)
and if f is odd: -f(x)=f(-x)

and I have tried to link this with the definition of $$\hat{f}$$ but I am not getting the conclusion that I want, can anyone help? Thank you!

 

[vela]

Try starting with the definition of the Fourier transform and use the substitution t→-t.