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How do you find the inverse of #y=e^x/(1+6e^x)#?

Answer 1

To find the inverse of ( y = \frac{e^x}{1 + 6e^x} ), follow these steps:

Swap the roles of ( x ) and ( y ): ( x = \frac{e^y}{1 + 6e^y} ).
Solve for ( y ).

[ x = \frac{e^y}{1 + 6e^y} ] [ x(1 + 6e^y) = e^y ] [ x + 6xe^y = e^y ] [ 6xe^y - e^y = -x ] [ e^y(6x - 1) = -x ] [ e^y = \frac{-x}{6x - 1} ] [ y = \ln\left(\frac{-x}{6x - 1}\right) ]

This is the expression for the inverse function ( f^{-1}(x) ) of the given function ( f(x) = \frac{e^x}{1 + 6e^x} ).

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Answer 2

Cora Alexander

#y=ln(x/(1-6x))#

Switch #x# and #y# then solve for #y#.

#x=e^y/(1+6e^y)#

#x(1+6e^y)=e^y#

#x+6xe^y=e^y#

#6xe^y-e^y=-x#

#e^y(6x-1)=-x#

#e^y=-x/(6x-1)#

#e^y=x/(1-6x)#

#ln(e^y)=ln(x/(1-6x))#

#yln(e)=ln(x/(1-6x))#

#y=ln(x/(1-6x))#

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.