Functions

Algebra Level 4

$f(x)+f(f(f(x))+y)=f(f(x))+x+f(f(y))$

If the above equation is true for all real $x$ and $y$ , what is the number of values $f(1)-f(0)$ can take?

Infinitely many 0 1 2

1 solution

Wen Z
Jun 16, 2016

Firstly we prove injectivity.

If $f(a)=f(b)$ ,

$f(f(f(a))+y)-f(f(a))=f(f(y))+a$ and $f(f(f(b))+y)-f(f(b))=f(f(y))+b$

But the LHSs are the same so

$f(f(y))+a=f(f(y))+b \Rightarrow a=b$

Now set $x=0$ .

This gives

$f(y+c)=f(f(y)) \Rightarrow f(y)=y+c \: \forall \: y \in R$

where $c=f(f(0))$

The conclusion follows from here.