Functional Equation

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

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

Note by Victor Loh
3 years, 10 months ago

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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.$$\square$$

- 3 years, 10 months ago