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Question Number 187714 by 073 last updated on 20/Feb/23

Commented by 073 last updated on 20/Feb/23

$$\mathrm{fog}\left(\mathrm{x}\right)=5\mathrm{x}-1$$$${\mathrm{gof}}^{-1}\left(\mathrm{x}\right)=6\mathrm{x}+2$$$$\mathrm{g}\left(\mathrm{x}\right)=?$$$$\mathrm{please}\mathrm{solution}??$$

Commented by 073 last updated on 20/Feb/23

$$\mathrm{i}\mathrm{need}\mathrm{it}$$

Answered by aleks041103 last updated on 20/Feb/23

$$f\left(g\left(x\right)\right)=5x-1$$$$g\left(f^{-1}\left(x\right)\right)=6x+2$$$$\Rightarrow g\left(g\left(x\right)\right)=g\left(f^{-1}\left(f\left(g\left(x\right)\right)\right)\right)=g\left(f^{-1}(5x-1)\right)=$$$$=6(5x-1)+2=30x-4$$$$supposeg\left(x\right)=ax+b$$$$\Rightarrow a(ax+b)+b=30x-4$$$$\Rightarrow a^2=30,(a+1)b=-4$$$$\Rightarrow a=\pm \sqrt{30},b=-\frac{4}{1\pm \sqrt{30}}=-\frac{4(1\mp \sqrt{30})}{1-30}=$$$$=\frac{4}{29}\mp \frac{4\sqrt{30}}{29}$$$$\Rightarrow g\left(x\right)=\sqrt{30}x+(\frac{4}{29}-\frac{4\sqrt{30}}{29})org\left(x\right)=-\sqrt{30}x+(\frac{4}{29}+\frac{4\sqrt{30}}{29})$$

Commented by 073 last updated on 20/Feb/23

$$\mathrm{thanks}\mathrm{alot}$$

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