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For the function #f(x)=(x-3)^3+1#, how do you find #f^-1(x)#?

Answer 1

In order to find for any function #f(x)#, we must apply the 'transformation' #y=x#. In order to do that, we must define #x# in terms of #y#, ie., find #f(y)#, then set #y=x#.

Let #y=f(x)#

#y=(x-3)^3+1#

#y-1=(x-3)^3#

#x-3=root(3)(y-1)#

#x=3+root(3)(y-1)#

We've now found #f(y)#, so we must set #y=x# by replacing #x# with #y# and #y# with #x#.

#f^-1(x)=y=3+root(3)(x-1)#

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Answer 2

Benjamin Achtenberg

To find the inverse function ( f^{-1}(x) ) for the given function ( f(x) = (x - 3)^3 + 1 ), follow these steps:

Replace ( f(x) ) with ( y ): ( y = (x - 3)^3 + 1 ).
Swap the variables ( x ) and ( y ): ( x = (y - 3)^3 + 1 ).
Solve this equation for ( y ) to express ( y ) in terms of ( x ).
Find ( f^{-1}(x) ) by replacing ( y ) with ( f^{-1}(x) ).

Following these steps will give you the inverse function ( f^{-1}(x) ) for the given function ( f(x) ).

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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.