find the inverse, f

^{-1}(x). For the equation f(x)=2e^{3x}+1find the inverse, f^{-1}(x). For the equation f(x)=2e^{3x}+1

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f(x) = 2e^{3x}+1

y = 2e^{3x}+1

To find the inverse function, f^{-1}(x), swap the x's and y's and solve for y:

x = 2e^{3y}+1

x-1 = 2e^{3y}

(1/2)(x-1) = e^{3y}

ln[(1/2)(x-1)] = 3y [take the ln of both sides]

(1/3)(-ln(2)+ln(x-1)) = y [ln(1/2) = ln(2^{-1}) = -ln(2), ln(ab) = ln(a) + ln(b)]

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