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Question Number 188407 by HeferH last updated on 01/Mar/23

$$xf\left(x\right)=f(x+2)$$$$f\left(2\right)=2$$$$f\left(8\right)=?$$

Answered by horsebrand11 last updated on 01/Mar/23

$$\left(i\right)xf\left(x\right)=f(x+2)$$$$\left(ii\right)(x-2)f(x-2)=f\left(x\right)$$$$x=2\Rightarrow f\left(2\right)=0?$$

Answered by manxsol last updated on 01/Mar/23

$$xf\left(x\right)=f(x+2)$$$$$$$$x=22f\left(2\right)=f\left(4\right)$$$$f\left(4\right)=4$$$$x=4$$$$4f\left(4\right)=f\left(6\right)$$$$f\left(6\right)=16$$$$x=66f\left(6\right)=f\left(8\right)$$$$6\left(16\right)=f\left(8\right)$$$$f\left(8\right)=96$$$$$$

Commented by mr W last updated on 01/Mar/23

$$youjusttookf\left(2\right)=2,butthisisnot$$$$proved,evenwrong,sincef\left(2\right)=0.$$

Answered by mr W last updated on 01/Mar/23

$$xf\left(x\right)=f(x+2)$$$$0\times f\left(0\right)=f\left(2\right)\Rightarrow f\left(2\right)=0$$$$2\times f\left(2\right)=f\left(4\right)\Rightarrow f\left(4\right)=0$$$$4\times f\left(4\right)=f\left(6\right)\Rightarrow f\left(6\right)=0$$$$6\times f\left(6\right)=f\left(8\right)\Rightarrow f\left(8\right)=0$$$$generallyf\left(2n\right)=0$$

Commented by manxsol last updated on 01/Mar/23

$$datof\left(2\right)=2$$$$0\times f\left(0\right)=f\left(2\right)\Rightarrow f\left(2\right)=0$$$$f\left(2\right)hastwo$$$$imagen\Rightarrow fnoisfuntion.$$$$iamworkedwithfuntion$$$$notwithrelation$$$$$$

Commented by manxsol last updated on 01/Mar/23

$$0cannotbeinthedomain$$$$ofthegivenfuntion.the$$$$functiongoesthrough(2,2).$$$$Idonthavetotryit.$$

Commented by mr W last updated on 01/Mar/23

$$f\left(2\right)=2cannotbeobtained,evenwhen$$$$x\ne 0.$$

Commented by manxsol last updated on 01/Mar/23

$$thatisverygoodandtrue$$$$answer.Grateful.$$

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