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How do you find the inverse of # f(x)=e^(2x-1)#?

Answer 1

Sadie Ackerman

Refer to explanation

We have that

#f(x)=e^(2x-1)=>lnf(x)=lne^(2x-1)=>lny=(2x-1)=> x=1/2(lny+1)#

hence #f^(-1)(x)=1/2*(lnx+1)#

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Answer 2

Liam Brennan

To find the inverse of the function ( f(x) = e^{2x-1} ), follow these steps:

Replace ( f(x) ) with ( y ).
Swap the variables ( x ) and ( y ).
Solve the equation for ( y ).
Replace ( y ) with ( f^{-1}(x) ).

Starting with ( y = e^{2x-1} ):

Step 1: Swap ( x ) and ( y ) to get ( x = e^{2y-1} ).

Step 2: Solve for ( y ):

[ x = e^{2y-1} ] [ \ln(x) = 2y - 1 ] [ 2y = \ln(x) + 1 ] [ y = \frac{1}{2}(\ln(x) + 1) ]

Step 3: Replace ( y ) with ( f^{-1}(x) ) to get the inverse function:

[ f^{-1}(x) = \frac{1}{2}(\ln(x) + 1) ]

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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.