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Question Number 124444 by benjo_mathlover last updated on 03/Dec/20

 If f((2/x)) = (3/(2+4x)) and f^(−1) (p)=2   then p =?

$$\:{If}\:{f}\left(\frac{\mathrm{2}}{{x}}\right)\:=\:\frac{\mathrm{3}}{\mathrm{2}+\mathrm{4}{x}}\:{and}\:{f}^{−\mathrm{1}} \left({p}\right)=\mathrm{2}\: \\ $$$${then}\:{p}\:=? \\ $$

Answered by bemath last updated on 03/Dec/20

 ⇔ f^(−1) (p)=2 then f(2)=p  we get  { (((2/x) = 2 ; x=1)),((p=(3/(2+4x)) = (3/(2+4(1)))=(1/2))) :}

$$\:\Leftrightarrow\:{f}^{−\mathrm{1}} \left({p}\right)=\mathrm{2}\:{then}\:{f}\left(\mathrm{2}\right)={p} \\ $$$${we}\:{get}\:\begin{cases}{\frac{\mathrm{2}}{{x}}\:=\:\mathrm{2}\:;\:{x}=\mathrm{1}}\\{{p}=\frac{\mathrm{3}}{\mathrm{2}+\mathrm{4}{x}}\:=\:\frac{\mathrm{3}}{\mathrm{2}+\mathrm{4}\left(\mathrm{1}\right)}=\frac{\mathrm{1}}{\mathrm{2}}}\end{cases} \\ $$

Answered by mathmax by abdo last updated on 03/Dec/20

f^(−1) (p)=2 ⇒p=f(2) but f((2/x))=(3/(2+4x))  let take x=1 ⇒  f(2)=(3/6)=(1/2) ⇒p=(1/2)

$$\mathrm{f}^{−\mathrm{1}} \left(\mathrm{p}\right)=\mathrm{2}\:\Rightarrow\mathrm{p}=\mathrm{f}\left(\mathrm{2}\right)\:\mathrm{but}\:\mathrm{f}\left(\frac{\mathrm{2}}{\mathrm{x}}\right)=\frac{\mathrm{3}}{\mathrm{2}+\mathrm{4x}}\:\:\mathrm{let}\:\mathrm{take}\:\mathrm{x}=\mathrm{1}\:\Rightarrow \\ $$$$\mathrm{f}\left(\mathrm{2}\right)=\frac{\mathrm{3}}{\mathrm{6}}=\frac{\mathrm{1}}{\mathrm{2}}\:\Rightarrow\mathrm{p}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$

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