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Question Number 165153 by mathlove last updated on 26/Jan/22

f(x)=3e^(2x) f^(−1) (x)=?

$${f}\left({x}\right)=\mathrm{3}{e}^{\mathrm{2}{x}} \\ $$$${f}^{−\mathrm{1}} \left({x}\right)=? \\ $$

Answered by mahdipoor last updated on 26/Jan/22

⇒x=3e^(2y) ⇒(1/2)ln(x/3)=y=f^(−1) (x)

$$\Rightarrow{x}=\mathrm{3}{e}^{\mathrm{2}{y}} \Rightarrow\frac{\mathrm{1}}{\mathrm{2}}{ln}\frac{{x}}{\mathrm{3}}={y}={f}^{−\mathrm{1}} \left({x}\right) \\ $$

Answered by MathsFan last updated on 27/Jan/22

let y=3e^(2x) x=3e^(2y) → logx=2ylog3e → 2y=log_(3e) x → y=((log_(3e) x)/2) f^(−1) (x)=((log_(3e) x)/2)

$${let}\:\:{y}=\mathrm{3}{e}^{\mathrm{2}{x}} \\ $$$$\:{x}=\mathrm{3}{e}^{\mathrm{2}{y}} \:\rightarrow\:\:{logx}=\mathrm{2}{ylog}\mathrm{3}{e} \\ $$$$\rightarrow\:\mathrm{2}{y}={log}_{\mathrm{3}{e}} {x}\:\rightarrow\:{y}=\frac{{log}_{\mathrm{3}{e}} {x}}{\mathrm{2}}\: \\ $$$${f}^{−\mathrm{1}} \left({x}\right)=\frac{{log}_{\mathrm{3}{e}} {x}}{\mathrm{2}} \\ $$$$ \\ $$

Answered by alephzero last updated on 28/Jan/22

f(x) = 3e^(2x) f^(−1) (x) = ? x = 3e^(2y) ⇒ (x/3) = e^(2y) ⇒ ln (x/3) = 2y ⇒ ((ln (x/3))/2) = ((ln x − ln 3)/2) = y ⇒ f^(−1) (x) = ((ln x − ln 3)/2)

$${f}\left({x}\right)\:=\:\mathrm{3}{e}^{\mathrm{2}{x}} \\ $$$${f}^{−\mathrm{1}} \left({x}\right)\:=\:? \\ $$$${x}\:=\:\mathrm{3}{e}^{\mathrm{2}{y}} \\ $$$$\Rightarrow\:\frac{{x}}{\mathrm{3}}\:=\:{e}^{\mathrm{2}{y}} \\ $$$$\Rightarrow\:\mathrm{ln}\:\frac{{x}}{\mathrm{3}}\:=\:\mathrm{2}{y} \\ $$$$\Rightarrow\:\frac{\mathrm{ln}\:\frac{{x}}{\mathrm{3}}}{\mathrm{2}}\:=\:\frac{\mathrm{ln}\:{x}\:−\:\mathrm{ln}\:\mathrm{3}}{\mathrm{2}}\:=\:{y} \\ $$$$\Rightarrow\:{f}^{−\mathrm{1}} \left({x}\right)\:=\:\frac{\mathrm{ln}\:{x}\:−\:\mathrm{ln}\:\mathrm{3}}{\mathrm{2}} \\ $$

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