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Question Number 139711 by bramlexs22 last updated on 30/Apr/21

$${\left[\mathrm{f}\left(\mathrm{x}\right)\right]}^2-{\left[\mathrm{f}(-\mathrm{x})\right]}^2=4\mathrm{x}$$

Commented by mr W last updated on 30/Apr/21

$$f\left(x\right)=\frac{x}{a}+a$$

Commented by hknkrc46 last updated on 30/Apr/21

$$\mathit{f}\left(\mathit{x}\right)=\mathit{a}\mathit{x}+\mathit{b}\Rightarrow \mathit{f}(-\mathit{x})=-\mathit{a}\mathit{x}+\mathit{b}$$$$\Rightarrow {\mathit{f}}^2\left(\mathit{x}\right)-{\mathit{f}}^2(-\mathit{x})=4\mathit{a}\mathit{b}\mathit{x}=4\mathit{x}$$$$\Rightarrow \mathit{a}\mathit{b}=1$$$$\mathit{f}\left(\mathit{x}\right)=\mathit{a}\mathit{x}+\frac{1}{\mathit{a}}=\frac{\mathit{x}}{\mathit{b}}+\mathit{b}$$

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