f-x-3e-2x-f-1-x-

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

f (x) = 3 e^{2 x}

f^{- 1} (x) = ?

Answered by mahdipoor last updated on 26/Jan/22

\Rightarrow x = 3 e^{2 y} \Rightarrow \frac{1}{2} l n \frac{x}{3} = y = f^{- 1} (x)

Answered by MathsFan last updated on 27/Jan/22

l e t y = 3 e^{2 x}

x = 3 e^{2 y} \to l o g x = 2 y l o g 3 e

\to 2 y = {l o g}_{3 e} x \to y = \frac{{l o g}_{3 e} x}{2}

f^{- 1} (x) = \frac{{l o g}_{3 e} x}{2}

Answered by alephzero last updated on 28/Jan/22

f (x) = 3 e^{2 x}

f^{- 1} (x) = ?

x = 3 e^{2 y}

\Rightarrow \frac{x}{3} = e^{2 y}

\Rightarrow \ln \frac{x}{3} = 2 y

\Rightarrow \frac{\ln \frac{x}{3}}{2} = \frac{\ln x - \ln 3}{2} = y

\Rightarrow f^{- 1} (x) = \frac{\ln x - \ln 3}{2}