Cool functions

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that

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

for all real numbers $x$ and $y$ .

#Algebra #Sharky

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Let $P(x,y)$ be the statement $f(f(x)+y) = x+f(f(y))$ .

$P(x,0)$ implies $f(f(x)) = x + f(f(0))$ . Applying this to $P$ gives $f(f(x)+y) = x+y+f(f(0))$ .

$P(x,f(0))$ implies $f(f(x)+f(0)) = x+f(0)+f(f(0))$ . $P(0,f(x))$ implies $f(f(0)+f(x)) = f(x)+f(f(0))$ . Equating the two gives $x+f(0)+f(f(0)) = f(x)+f(f(0))$ , or $f(x) = x + c$ for some fixed real number $c$ for all real $x$ . It can be easily verified to satisfy the equation.

Ivan Koswara - 5 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$