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Question

Find a formula for the inverse of the function.

y = (e^x)/(1+8e^x)

Answer 1

switch x and y:
x= (e^y) / (1+8e^y)

then isolate the y:

x(1+8e^y)=e^y
x+8xe^y=e^y
x= e^y -8xe^y

Factor out the e^y:
x= e^y(1-8x)
e^y = (x)/(1-8x)

Then isolate y by taking ln's:
y= ln [ (x)/(1-8x)]
y= ln(x) - ln (1-8x)= f^(-1) (x)

Answer 2

1/y = (e^-x)+8

Answer 3

call e^x= z
y= z/(1+8z)
8yz+y= z so z= y/(1-8y)
Taking ln as ln z= x
x= ln (y/(1-8y)
now intrchange x and y

y=ln ( x/(1-8x)) your inverse function

Answer 4

x = (e^y)/(1+8e^y)

x(1+8e^y) = e^y

x(1+8e^y) - e^y = 0

e^y * (8x - 1) + x = 0

e^y = -x/(8x-1)

y = ln ( -x/(8x-1) )