Find all solutions to the functional equation $f(x) +f(x+y)=y+2 $

Question

I've started studying functions and I am having trouble with the following question:

Find all solutions to the functional equation $f(x) +f(x+y)=y+2 $

Using the substitution technique when $y=0$ I have $f(x)=1$.

This implies that also $f(x+y)=1 $ and since $f(x)+f(x+y)=y+2$ , I am left with the conclusion that there are not solutions for the above functional equation.

Is this correct ?

Answer 1

You are correct. Setting $y = 0$ gives us $$ \forall x \in \mathbb{R} : f(x) = 1$$ In particular $$ \forall y \in \mathbb{R} : 1 + 1 = y + 2 \iff y = 0$$ which clearly is a contradiction!