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Question Number 150311 by naka3546 last updated on 11/Aug/21

$$x,y\in \mathbb{R}$$$$Findallfunctionsthatsatisfythiscondition:$$$$f(x+y)=f\left(x\right)\cdot f\left(y\right)-f(x\cdot y)+1$$$$$$$$Findallfunctionsthatsatisfythiscondition:$$$$f\left(f\left(x\right)\right)=f\left(x\right)+x$$

Answered by aleks041103 last updated on 11/Aug/21

$$1\Rightarrow f(x+0)=f\left(x\right)=f\left(0\right).f\left(x\right)-f\left(0\right)+1$$$$(1-f\left(0\right))f\left(x\right)=-(1-f\left(0\right))$$$$Iff\left(0\right)\ne 1thenf\left(x\right)=-1=const.but$$$$this{doesn}^{\prime }tsatisfy2.$$$$Thisleavesf\left(0\right)=1,whichtriviallysatisfies$$$$1.wheny=0.$$$$Thenby2.$$$$f\left(f\left(0\right)\right)=f\left(1\right)=f\left(0\right)+0=1$$$$f\left(f\left(1\right)\right)=f\left(1\right)=f\left(1\right)+1\to contradiction$$$$$$$$\Rightarrow Nosuchfunctionsexist!$$

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