What you have posted doesn't make sense to me. Please post the original question exactly as it's stated. Is f(x + y) meant to be the joint pdf of X and Y? Or are you taling about the pdf of the random variable X + Y? Note: The sum of two exponential random variables is NOT what you are claiming.

For the verification part of the question, use the change of variable theorem (see for example Walpole, Myers and Myers (1998): Probability and Statistics for Engineers and Scientists, p 185).

If you need more help, please show all your work and say where you get stuck.