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Question Number 193267 by Mingma last updated on 09/Jun/23
Answered by cortano12 last updated on 09/Jun/23
$$\:\:\left(\mathrm{i}\right)\:\mathrm{4f}\left(\mathrm{x}\right)+\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{x}}\right)=\mathrm{24x}+\mathrm{5}+\frac{\mathrm{6}}{\mathrm{x}} \\ $$$$\:\:\left(\mathrm{ii}\right)\:\mathrm{4f}\left(\frac{\mathrm{1}}{\mathrm{x}}\right)+\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{24}}{\mathrm{x}}+\mathrm{5}+\mathrm{6x} \\ $$$$\:\:\left(\mathrm{i}\right)×\mathrm{4}=\:\mathrm{4f}\left(\frac{\mathrm{1}}{\mathrm{x}}\right)+\mathrm{16f}\left(\mathrm{x}\right)=\mathrm{96x}+\mathrm{20}+\frac{\mathrm{24}}{\mathrm{x}} \\ $$$$\:\:\left(\mathrm{i}\right)−\left(\mathrm{ii}\right) \\ $$$$\:\:\:\Rightarrow\mathrm{15f}\left(\mathrm{x}\right)=\mathrm{90x}+\mathrm{15} \\ $$$$\:\:\Rightarrow\mathrm{f}\left(\mathrm{x}\right)=\:\mathrm{6x}+\mathrm{1} \\ $$$$\:\Rightarrow\mathrm{f}\left(\mathrm{337}\right)=\mathrm{6}×\mathrm{337}+\mathrm{1}=\mathrm{2022}+\mathrm{1}=\mathrm{2023} \\ $$
Commented by Frix last updated on 09/Jun/23
$${f}\left({x}\right)=\mathrm{18}{x}+\mathrm{1}\:\Rightarrow \\ $$$$\mathrm{4}{f}\left({x}\right)+{f}\left(\frac{\mathrm{1}}{{x}}\right)=\mathrm{72}{x}+\mathrm{5}+\frac{\mathrm{18}}{{x}}\neq\mathrm{24}{x}+\mathrm{5}+\frac{\mathrm{6}}{{x}} \\ $$
Commented by Mingma last updated on 09/Jun/23
What's your final answer?
Commented by Tinku Tara last updated on 09/Jun/23
$$\mathrm{15}{f}\left({x}\right)=\mathrm{90}{x}+\mathrm{15} \\ $$$$\Rightarrow{f}\left({x}\right)=\mathrm{6}{x}+\mathrm{1} \\ $$
Commented by Mingma last updated on 09/Jun/23
2023 is correct!
Commented by Mingma last updated on 09/Jun/23
2023 is correct!