Functional Equation

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

$f(x+y) - 2f(x-y) + f(x) - 2f(y) = y-2.$

#FunctionalEquations #VictorLoh

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Comments

Let $P(x,y)$ be the statement $f(x+y)-2f(x-y)+f(x)-2f(y)=y-2$ .

$P(0,0)\implies f(0)=1$

$P(x,x)\implies f(2x)=f(x)+x$

$P(x,-x)\implies f(x)+2f(-x)=3-x\stackrel{x\to -x}\implies f(-x)+2f(x)=3+x$

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

The last one was got by adding the 2 previous equations.

$(2-f(x))+2f(x)=3+x\implies f(x)=x+1$

Thus $f(x)=x+1$ is the only possible solution. After checking it we see it works. $\square$

mathh mathh - 6 years, 10 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}`	$\boxed{123}$