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Question Number 142052 by Rankut last updated on 26/May/21

$$\mathbf{Given}\mathbf{that}\mathbf{fog}\left(\mathbf{x}\right)=\frac{2\mathbf{x}-1}{\mathbf{x}}\mathbf{and}\mathbf{g}\left(\mathbf{x}\right)=5\mathbf{x}+2,$$$$\mathbf{Find}\mathbf{f}\left(\mathbf{x}\right).$$$$$$

Answered by iloveisrael last updated on 26/May/21

$$\Rightarrow f(5x+2)=\frac{2x-1}{x}$$$$\Rightarrow f(5x+2)=2-\frac{1}{x}$$$$\Rightarrow f\left(x\right)=2-\frac{1}{\left(\frac{x-2}{5}\right)}=2-\frac{5}{x-2}$$$$\Rightarrow f\left(x\right)=\frac{2x-9}{x-2}$$

Commented by Rankut last updated on 26/May/21

$$howdidyouget\left(\frac{x-2}{5}\right)$$

Answered by EDWIN88 last updated on 26/May/21

$$\mathrm{f}\left(\mathrm{g}\left(\mathrm{x}\right)\right)=\mathrm{f}(5\mathrm{x}+2)=\frac{2\mathrm{x}-1}{\mathrm{x}}$$$$\mathrm{let}5\mathrm{x}+2=\mathrm{u}\Rightarrow \mathrm{x}=\frac{\mathrm{u}-2}{5}$$$$\mathrm{we}\mathrm{get}\mathrm{f}\left(\mathrm{u}\right)=\frac{2\left(\frac{\mathrm{u}-2}{5}\right)-1}{\left(\frac{\mathrm{u}-2}{5}\right)}=\frac{2\mathrm{u}-4-5}{\mathrm{u}-2}$$$$\mathrm{f}\left(\mathrm{u}\right)=\frac{2\mathrm{u}-9}{\mathrm{u}-2}\Rightarrow \mathrm{so}\mathrm{f}\left(\mathrm{x}\right)=\frac{2\mathrm{x}-9}{\mathrm{x}-2}$$

Answered by mathmax by abdo last updated on 26/May/21

$$\mathrm{fog}\left(\mathrm{x}\right)=\frac{2\mathrm{x}-1}{5}\Rightarrow \mathrm{f}\left(\mathrm{g}\left(\mathrm{x}\right)\right)=\frac{2\mathrm{x}-1}{5}\Rightarrow \mathrm{f}(5\mathrm{x}+2)=\frac{2\mathrm{x}-1}{\mathrm{x}}$$$$\mathrm{let}5\mathrm{x}+2=\mathrm{t}\Rightarrow \mathrm{x}=\frac{\mathrm{t}-2}{5}\Rightarrow \mathrm{f}\left(\mathrm{t}\right)=2-\frac{1}{\frac{\mathrm{t}-2}{5}}=2-\frac{5}{\mathrm{t}-2}=\frac{2\mathrm{t}-4-5}{\mathrm{t}-2}$$$$=\frac{2\mathrm{t}-9}{\mathrm{t}-2}\Rightarrow \mathrm{f}\left(\mathrm{x}\right)=\frac{2\mathrm{x}-9}{\mathrm{x}-2}$$

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