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Question Number 63485 by ANTARES VY last updated on 04/Jul/19

$$\mathbf{f}(\mathbf{x}-3)+\mathbf{f}\left(\mathbf{x}\right)=2\mathbf{x}-3$$$$\mathbf{F}\left(2\right)=0.$$$$\mathbf{F}(-2)=?$$

Answered by MJS last updated on 04/Jul/19

$$f\left(x\right)=ax+b$$$$f(x-3)+f\left(x\right)=2x-3$$$$2ax-3a+2b=2x-3$$$$2a=2$$$$-3a+2b=-3$$$$\Rightarrow a=1\wedge b=0$$$$f\left(x\right)=x$$$$F\left(x\right)=\int xdx=\frac{x^2}{2}+c$$$$F\left(2\right)=0\Rightarrow c=-2$$$$F\left(x\right)=\frac{x^2}{2}-2$$$$F(-2)=0$$

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