How do you find the inverse of #f(x) =ln(4x-1)#?

Answer 1

Caleb Alexander

To find the inverse of ( f(x) = \ln(4x - 1) ), follow these steps:

Replace ( f(x) ) with ( y ).
Swap the roles of ( x ) and ( y ), making ( x ) the dependent variable and ( y ) the independent variable.
Solve the resulting equation for ( y ).
Replace ( y ) with ( f^{-1}(x) ).

So, for ( f(x) = \ln(4x - 1) ):

( y = \ln(4x - 1) )
Swap roles: ( x = \ln(4y - 1) )
Solve for ( y ): ( e^x = 4y - 1 ) ( e^x + 1 = 4y ) ( y = \frac{e^x + 1}{4} )
Replace ( y ) with ( f^{-1}(x) ): ( f^{-1}(x) = \frac{e^x + 1}{4} )

Therefore, the inverse of ( f(x) = \ln(4x - 1) ) is ( f^{-1}(x) = \frac{e^x + 1}{4} ).

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Answer 2

Chloe Alexander

#f^(-1)(x)=(e^x+1)/4#

As #f(x)=y=ln(4x-1)#

#4x-1=e^y#

or #x=(e^y+1)/4#

Hence #f^(-1)(x)=(e^x+1)/4#

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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.