Solving the functional equation $f(a + b) = a + f(b)$

Question

How would you solve: $$f(a+b)=a+f(b) ?$$ It seems similar to the Cauchy equation $$f(a+b)=f(a)+f(b),$$ but I'm not sure what to do with this. I have a feeling the only solution is $f(k)=k$ but idk. Thanks for the help, Guys!

Answer 1

Hint Setting $b = 0$ gives $$f(a) = a + f(0),$$ so any such function is determined by $f(0)$.

Answer 2

If $f(a + b) = a + f(b)$ for every $a$ and $b$, then, putting $c = f(0)$, $f(x) = f(x + 0) = x + f(0) = x + c$ for any $x$. Conversely for any given $c$ if you define $f(x) = x + c$, then $f$ satisfies $f(a+b) = a + b + c = a + f(b)$. So the solutions are all functions of the form $f(x) = x + c$.

Answer 3

Another approach would be to notice that in $$f(a+b)=a+f(b)$$ the left hand side is symmetric in $a$ and $b$, while the right hand side is not, and we can exploit this: swap $a$ and $b$. The left hand side remains unchanged, and hence so must the right hand side, and so we must have that $$a+f(b)=b+f(a)$$ for all $a$ and $b$. This gives us that for all $a$ and $b$, $$f(a)-a=f(b)-b$$ and so $$f(x)-x$$ is constant. We thus have that $f(x)=x+c$ for all $x$ for some constant $c$, and we can easily check that any such function satisfies the functional equation.