f-x-2-f-x-2-4x-

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

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

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

$$\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)\:=\:\boldsymbol{{ax}}\:+\:\boldsymbol{{b}}\:\Rightarrow\:\boldsymbol{{f}}\left(−\boldsymbol{{x}}\right)\:=\:−\boldsymbol{{ax}}\:+\:\boldsymbol{{b}} \\ $$$$\Rightarrow\:\boldsymbol{{f}}^{\mathrm{2}} \left(\boldsymbol{{x}}\right)−\boldsymbol{{f}}^{\mathrm{2}} \left(−\boldsymbol{{x}}\right)\:=\:\mathrm{4}\boldsymbol{{abx}}\:=\:\mathrm{4}\boldsymbol{{x}} \\ $$$$\Rightarrow\:\boldsymbol{{ab}}\:=\:\mathrm{1} \\ $$$$\boldsymbol{{f}}\:\left(\boldsymbol{{x}}\right)\:=\:\boldsymbol{{ax}}\:+\:\frac{\mathrm{1}}{\boldsymbol{{a}}}\:=\:\frac{\boldsymbol{{x}}}{\boldsymbol{{b}}}\:+\:\boldsymbol{{b}} \\ $$

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