Webwork: Functions – Inverse Functions: Problem 10

Computing the inverse function.

if F(x)=8e^2x-8/6e^2x+3,  find the inverse function.

step 1: Switch the x and y variables.

x=8e^2y-8/6e^2y+3

step 2: Multiply the denominator to both sides.

(6e^2y+3)(x)=(8e^2y-8)

step 2: Distribute the y to both coefficients.

6xe^2y+3x=8e^2y-8

step 3: Then isolate the natural logs to one side and move the coefficients to the other.

6ye^2y-8e^2y=-8-3x

step 4: Then you factor out the e, because they have the same exponents

e^2y(6x-8)=(-8-3x)

step 5: Then you isolate e by dividing the factors to the other side.

e^2y=(-8-3x)/(6x-8)

step 6: After you have to remove e by returning it to natural log to both sides.

ln(e^2y)=ln(-8-3x)/(6x-8)

step 7: Once e is removed we divide it by 2 to isolate the y value.

y=(ln(-8-3x)/(6x-8))/2

step 8: Return the y and x- variables to the proper places.

f^-1(x)=(ln(-8-3x)/(6x-8))/2

p.s: I apologize for any errors