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gof-x-2x-1-fog-x-1-3x-2-fof-3-




Question Number 201952 by mathlove last updated on 16/Dec/23
(gof)_x =2x−1    (fog)_x ^(−1) =3x+2  (fof)_3 =?
$$\left({gof}\right)_{{x}} =\mathrm{2}{x}−\mathrm{1}\:\: \\ $$$$\left({fog}\right)_{{x}} ^{−\mathrm{1}} =\mathrm{3}{x}+\mathrm{2} \\ $$$$\left({fof}\right)_{\mathrm{3}} =? \\ $$
Answered by cortano12 last updated on 17/Dec/23
  { (((g○f)(x)= 2x−1)),(((f○g)^(−1) (x)=3x+2)) :}⇒(f○f)(3)=?     ((f○g)^(−1) (g○f))(x)=(f○f)(x)= 3(2x−1)+2      ⇔ (f○f)(x)=6x−1       ⇔ (f○f)(3)= 17
$$\:\begin{cases}{\left(\mathrm{g}\circ\mathrm{f}\right)\left(\mathrm{x}\right)=\:\mathrm{2x}−\mathrm{1}}\\{\left(\mathrm{f}\circ\mathrm{g}\right)^{−\mathrm{1}} \left(\mathrm{x}\right)=\mathrm{3x}+\mathrm{2}}\end{cases}\Rightarrow\left(\mathrm{f}\circ\mathrm{f}\right)\left(\mathrm{3}\right)=? \\ $$$$\:\:\:\left(\left(\mathrm{f}\circ\mathrm{g}\right)^{−\mathrm{1}} \left(\mathrm{g}\circ\mathrm{f}\right)\right)\left(\mathrm{x}\right)=\left(\mathrm{f}\circ\mathrm{f}\right)\left(\mathrm{x}\right)=\:\mathrm{3}\left(\mathrm{2x}−\mathrm{1}\right)+\mathrm{2} \\ $$$$\:\:\:\:\Leftrightarrow\:\left(\mathrm{f}\circ\mathrm{f}\right)\left(\mathrm{x}\right)=\mathrm{6x}−\mathrm{1} \\ $$$$\:\:\:\:\:\Leftrightarrow\:\left(\mathrm{f}\circ\mathrm{f}\right)\left(\mathrm{3}\right)=\:\mathrm{17}\: \\ $$

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