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Question Number 55621 by Tawa1 last updated on 28/Feb/19

$$\mathrm{The}\mathrm{function}\mathrm{pogof}\left(\mathrm{x}\right)={\mathrm{x}}^4+{2\mathrm{x}}^3+{2\mathrm{x}}^2\mathrm{is}\mathrm{divisible}\mathrm{by}\mathrm{the}\mathrm{half}\mathrm{of}\mathrm{the}$$$$\mathrm{function}\mathrm{of}\mathrm{p}.\mathrm{Find}\mathrm{g}\left(\mathrm{x}\right).$$

Answered by kaivan.ahmadi last updated on 28/Feb/19

$$2p\left(gof\left(x\right)\right)=2x^2(x^2+2x+2)$$$$dividep\left(x\right)\Rightarrow p\left(x\right)=2x^2$$$$\Rightarrow g\left(f\left(x\right)\right)=x^2+2x+2={(x+1)}^2+1$$$$theng\left(x\right)=x^2+1andf\left(x\right)=x+1is$$$$oneoftheanswers.$$$$generallyg\left(x\right)={(x-k)}^2+1$$$$andf\left(x\right)=x+k+1isanswer$$$$$$$$thisquestionhasnouniqueanswer$$$$$$$$t=f\left(x\right)\Rightarrow f^{-1}\left(t\right)=x$$$$g\left(t\right)={(f^{-1}\left(t\right)+1)}^2+1$$

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