Finding all functions on $ \mathbb R $ satisfying $ f ( x - | x | ) + f ( x + | x | ) = x $

Question

I'm not sure if I'm correct in this example:

Find all functions such that $ f ( x - | x | ) + f ( x + | x | ) = x $, where $ x \in \mathbb R $.

So my answer is the are only one function satisfying this condition $ f ( x ) = \frac x 2 $. Am I correct?

Answer 1

It's immediate that $f(0) = 0$. For $x > 0$, we then have

$$x = f(x + |x|) + f(x - |x|) = f(2x) + f(0) = f(2x) \implies f(2x) = x$$

Thus, for $x > 0$, we have $f(2x) = x/2$. Similarly, if $x < 0$, we find that

$$x = f(x - |x|) = f(2x)$$ and proceed as above. So you're correct.

Answer 2

• If $x\ge0$ then the given equality gives $$f(2x)+f(0)=x\iff f(x)=\frac x 2 -f(0)$$
• if $x=0$ we have easily $f(0)=0$
• if $x\le0$ we have $$f(2x)+f(0)=x\iff f(x)=\frac x 2 -f(0)$$ hence $$\forall x\quad f(x)=\frac x 2 $$