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How do you find the inverse of #f(x)=ln(9x+7)#?

Answer 1

Theodore Amidon

#x=(e^y-7)/9#

You need to isolate the #x# terms, so:

#y=ln(9x+7)#

Take the exponential on both sides:

#e^y=e^{ln(9x+7)}#

But #e^ln(z)=z#, so it becomes

#e^y=9x+7#

Subtract #7# from both sides:

#e^y-7 = 9x#

Divide both sides by #9#:

#(e^y-7)/9 = x#.

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Answer 2

Christian Antunez

To find the inverse of ( f(x) = \ln(9x + 7) ):

Replace ( f(x) ) with ( y ). [ y = \ln(9x + 7) ]
Swap ( x ) and ( y ). [ x = \ln(9y + 7) ]
Solve for ( y ). [ e^x = 9y + 7 ] [ e^x - 7 = 9y ] [ y = \frac{e^x - 7}{9} ]
Replace ( y ) with ( f^{-1}(x) ) to express the inverse function. [ f^{-1}(x) = \frac{e^x - 7}{9} ]

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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.