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How do you find the inverse of #f(x) = 1-x^3#?

Answer 1

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

Write as:

#y=1-x^3#

Make #x# the dependant variable and then swap the letters round. In that, where ever there is an #x# (there should only be one!) write #y# and wherever there is a #y# write #x#.

#x^3=1-y#

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

Now change the letters round!

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

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Answer 2

Xavier Alonso

To find the inverse of a function ( f(x) ), we need to swap the roles of ( x ) and ( y ) and then solve for ( y ).

For ( f(x) = 1 - x^3 ), let's denote the inverse function as ( f^{-1}(x) ).

So, we swap ( x ) and ( y ), giving us the equation:

[ x = 1 - y^3 ]

Now, we solve for ( y ):

[ x = 1 - y^3 ]

[ y^3 = 1 - x ]

[ y = \sqrt[3]{1 - x} ]

Therefore, the inverse of ( f(x) = 1 - x^3 ) is ( f^{-1}(x) = \sqrt[3]{1 - 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.