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Question Number 180554 by cherokeesay last updated on 13/Nov/22

Answered by Rasheed.Sindhi last updated on 13/Nov/22

$$f\left(x\right)=2x-5,g\left(x\right)=x+1$$$${\left(f\left(g^{-1}\right)\right)}^{-1}\left(x\right)=?$$$$\bullet g^{-1}\left(x\right)=?$$$$y=x+1\Rightarrow x=y-1\Rightarrow y=x-1$$$$\Rightarrow g^{-1}\left(x\right)=x-1$$$$f\left(g^{-1}\right)\left(x\right)=2(x-1)-5=2x-7$$$${\left(f\left(g^{-1}\right)\right)}^{-1}\left(x\right)$$$$y=2x-7\Rightarrow x=\frac{y+7}{2}\Rightarrow y=\frac{x+7}{2}$$$${\left(f\left(g^{-1}\right)\right)}^{-1}\left(x\right)=\frac{x+7}{2}$$

Commented by cherokeesay last updated on 13/Nov/22

$$Nice!Thankyousomuch.$$

Answered by cortano1 last updated on 14/Nov/22

$${\left(\mathrm{f}\left({\mathrm{g}}^{-1}\right)\right)}^{-1}=\mathrm{g}\left({\mathrm{f}}^{-1}\left(\mathrm{x}\right)\right)$$$${\mathrm{f}}^{-1}\left(\mathrm{x}\right)=\frac{\mathrm{x}+5}{2}$$$$\mathrm{g}\left({\mathrm{f}}^{-1}\left(\mathrm{x}\right)\right)=\mathrm{g}\left(\frac{\mathrm{x}+5}{2}\right)=\frac{\mathrm{x}+5}{2}+1=\frac{\mathrm{x}+7}{2}$$$$$$

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