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Question Number 193542 by SAMIRA last updated on 16/Jun/23

$$\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^3+{3\mathrm{x}}^2-1$$$$1)\mathrm{calcul}\mathrm{h}\left(\mathrm{X}\right)=\mathrm{f}(\mathrm{a}+\mathrm{X})-\mathrm{b}$$2) determine a and b such that h is odd

Commented by a.lgnaoui last updated on 17/Jun/23

$$\mathrm{odd}?$$

Commented by SAMIRA last updated on 17/Jun/23

uneven

Answered by mr W last updated on 17/Jun/23

$$2)$$$$h\left(x\right)=-h(-x)$$$$f(x+a)-b=-f(-x+a)+b$$$$f(x+a)=-f(-x+a)+2b$$$${(x+a)}^3+3{(x+a)}^2-1=-{(-x+a)}^3-3{(-x+a)}^2+1+2b$$$${(x+a)}^3+{(-x+a)}^3+3{(x+a)}^2+3{(-x+a)}^2=2+2b$$$$3(a+1)x^2+a^3+3a^2=1+b$$$$\Rightarrow a+1=0\Rightarrow a=-1$$$$\Rightarrow 1+b=a^3+3a^2\Rightarrow b=1$$

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