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How do you find the inverse of #f(x) = x^3 + 4#?

Answer 1

Sadie Ackerman

The inverse of the function is #f^-1(x)=(x+4)^(1/3)#

Let #y=x^3+4#

Then

#x^3=y+4#

#=>#, #x=(y+4)^(1/3)#

Inverse the #y# and the #x#

#=>#, #y=(x+4)^(1/3)#

The inverse of the function is #f^-1(x)=(x+4)^(1/3)#

The inverse function is symmetric about the line #y=x#

And

#f(f^-1(x))=x#

graph{(y-x^3+4)(y-(x+4)^(1/3))(y-x)=0 [-10, 10, -5, 5]}

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Answer 2

Caleb Alexander

To find the inverse of ( f(x) = x^3 + 4 ), follow these steps:

Replace ( f(x) ) with ( y ).
Swap ( x ) and ( y ).
Solve the new equation for ( y ).
Replace ( y ) with ( f^{-1}(x) ), the inverse function.

So, we start by swapping ( x ) and ( y ):

[ x = y^3 + 4 ]

Now, solve this equation for ( y ):

[ x - 4 = y^3 ]

[ y = (x - 4)^{\frac{1}{3}} ]

Replace ( y ) with ( f^{-1}(x) ):

[ f^{-1}(x) = (x - 4)^{\frac{1}{3}} ]

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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.