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Question Number 115960 by Fikret last updated on 29/Sep/20

$$f^2\left(x\right)=f\left(2x\right)+2f\left(x\right)-2$$$$f\left(1\right)=3\Rightarrow f\left(6\right)=?$$

Commented by prakash jain last updated on 29/Sep/20

$$\mathrm{power}\mathrm{converted}\mathrm{into}\mathrm{linear}$$$$f\left(x\right)=e^{ax}+b$$$$e^{2ax}+2{be}^{ax}+b^2=e^{2ax}+b+2e^{ax}+2b-2$$$$b=1$$$$f\left(x\right)=e^{ax}+1$$$$f\left(1\right)=3\Rightarrow 3=e^a-1\Rightarrow a=\ln 2$$$$f\left(x\right)=2^x+1$$$$f\left(6\right)=2^6+1=65$$

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