question with 2 answers

Q. If f(x + f(y)) = f(x) + y for all x and y belonging to R, and f(0) = 1, then what is f(7)?

Ans.

Method 1:
f(x + f(o)) = f(x)+0
or f(x+1)=f(x).
so, f(0+1)=f(0)=1=f(1).
similarly, f(1+1)=f(1)=1=f(2)
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so, f(7)=1.

Mehtod 2:
f(0+f(0))=f(0)+0=1=f(1)
f(1+f(1))=f(1)+1=2=f(2)
f(1+f(2))=f(1)+2=3=f(3)
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similaly,f(7)=7.

Now, please tell me, sir, which of the above methods is more authentic and correct and why.