Question

suppose f(x) = 2e^(3x-4). Find a formula for f^-1(x), the inverse function

Answer 1

let \[y = 2e^{3x - 4}\]

swap the x and y

\[x = 2e^{3y -4}\]

now may y the subject

\[\frac{x}{2} = e^{3y - 4}\]

take the base e log of both sides

\[\ln \frac{x}{2} = 3y - 4\]

add 4 to both sides

\[\ln (\frac{x}{2}) + 4 = 3y\]

divide by 3

\[y = \frac{1}{3} [\ln(\frac{x}{2}) +4]\]

Answer 2

then \[f^{-1} = \frac{1}{3} (\ln(\frac{x}{2}) + 4)\]