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Question Number 202154 by cortano12 last updated on 22/Dec/23

$$\downharpoonleft \underset{―}{}$$

Commented by mr W last updated on 22/Dec/23

$$\mathrm{f}(\mathrm{xy}+1)=\mathit{\nu }$$

Answered by Rasheed.Sindhi last updated on 22/Dec/23

$$\downharpoonleft \underset{―}{}$$

Commented by Rasheed.Sindhi last updated on 22/Dec/23

$$\mathrm{f}(\mathrm{xy}+1)=\mathcal{J}$$

Answered by cortano12 last updated on 22/Dec/23

$$\{\begin{array}{l}\mathrm{f}(\mathrm{xy}+1)=\mathrm{f}\left(\mathrm{x}\right)\mathrm{f}\left(\mathrm{y}\right)-2\mathrm{f}\left(\mathrm{y}\right)-\mathrm{x}+5\\ \mathrm{f}(\mathrm{yx}+1)=\mathrm{f}\left(\mathrm{y}\right)\mathrm{f}\left(\mathrm{x}\right)-2\mathrm{f}\left(\mathrm{x}\right)-\mathrm{y}+5\end{array}$$$$\Rightarrow \left(1\right)-\left(2\right):0=-2\mathrm{f}\left(\mathrm{y}\right)+2\mathrm{f}\left(\mathrm{x}\right)+\mathrm{y}-\mathrm{x}$$$$2\mathrm{f}\left(\mathrm{y}\right)-2\mathrm{f}\left(\mathrm{x}\right)=\mathrm{y}-\mathrm{x}$$$$\mathrm{y}=0\Rightarrow 16-2\mathrm{f}\left(\mathrm{x}\right)=-\mathrm{x}$$$$\Rightarrow \mathrm{f}\left(\mathrm{x}\right)=\frac{16+\mathrm{x}}{2}$$$$\mathrm{x}=0\Rightarrow 2\mathrm{f}\left(\mathrm{y}\right)=\mathrm{y}+16$$$$\Rightarrow \mathrm{f}\left(\mathrm{y}\right)=\frac{\mathrm{y}+16}{2}$$$$\mathrm{f}\left(2024\right)=\frac{2040}{2}=1020$$

Commented by Rasheed.Sindhi last updated on 22/Dec/23

$$\mathrm{Cortano}\mathrm{sir}!$$$$\mathrm{f}\left(\mathrm{x}\right)=\frac{16+\mathrm{x}}{2}\Rightarrow \mathrm{f}\left(1\right)=\frac{17}{2}$$$$\mathrm{whereas}$$$$\mathrm{for}\mathrm{x}=\mathrm{y}=0:$$$$\mathrm{f}(\mathrm{xy}+1)=\mathrm{f}\left(\mathrm{x}\right)\mathrm{f}\left(\mathrm{y}\right)-2\mathrm{f}\left(\mathrm{y}\right)-\mathrm{x}+5\wedge \mathrm{f}\left(0\right)=8$$$$\Rightarrow \mathrm{f}(0\times 0+1)=\mathrm{f}\left(0\right)\mathrm{f}\left(0\right)-2\mathrm{f}\left(0\right)-0+5$$$$\Rightarrow \mathrm{f}\left(1\right)=8^2-2\left(8\right)+5=53$$$$\mathrm{How}\mathrm{f}\left(1\right)\mathrm{has}\mathrm{two}\mathrm{different}\mathrm{values}?$$

Commented by cortano12 last updated on 22/Dec/23

$$\mathrm{yes}\mathrm{sir},\mathrm{the}\mathrm{question}\mathrm{inconsistent}$$

Answered by MathematicalUser2357 last updated on 06/Jan/24

$$1020$$

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