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Question Number 44350 by Necxx last updated on 27/Sep/18

$$Iff(x+y)=f\left(x\right).f\left(y\right)forrealx,y$$$$andf\left(0\right)\ne 0.LetF\left(x\right)=\frac{f\left(x\right)}{1+{\left(f\left(x\right)\right)}^2}$$$$thenF\left(x\right)is$$$$a)evenb)oddc)neitherevennorodd$$

Commented by maxmathsup by imad last updated on 27/Sep/18

$$\Rightarrow f\left(x\right).f(-x)=f\left(0\right)\Rightarrow f{\left(0\right)}^2=f\left(0\right)\Rightarrow f\left(0\right)=1\Rightarrow F\left(0\right)=\frac{f\left(0\right)}{1+{\left(f\left(0\right)\right)}^2}=\frac{1}{2}$$$$duetoF\left(0\right)\ne 0FcantbeoddletseeifFiseven$$$$F(-x)=\frac{f(-x)}{1+{\left(f(-x)\right)}^2}=\frac{\frac{1}{f\left(x\right)}}{1+{\left(\frac{1}{f\left(x\right)}\right)}^2}=\frac{1}{f\left(x\right)\left\{\frac{f^2\left(x\right)+1}{f^2\left(x\right)}\right\}}=\frac{f\left(x\right)}{1+f^2\left(x\right)}=F\left(x\right)$$$$soFiseven.$$

Commented by Necxx last updated on 27/Sep/18

$$thankyousir$$

Commented by Necxx last updated on 27/Sep/18

$${that}^{\prime }sit$$

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