How do you find the inverse of #f(x)=x^3-2# and is it a function?

Answer 1

#f^-1(x)=color(white)(0)^3sqrtx+2#

We begin with #f(x)=x^3-2#. To find the inverse of any equation, just switch #x# and #y#. No, before we do thta, I'm going to change the equation. I'm going to rename #f(x)# to #y#, just so that I don't have to deal with the parentheses or anything. Now, all I'bve doe is change what #f(x)# is called, not it's actual value.
Anyways, we have #y=x^3-2#. Now we switch #x# and #y# and then solve for #y#.
Now we've got #x=y^3-2#. If we add #2# on both sides we have #x+2=y^3#. Now we just need to get #y# as simple as possible, which we'll do by cube rooting both sides of the equation. That leaves us with #y=color(white)(0)^3sqrtx+2#. Now we just change #y# back to #f(x)# and add a #color(white)(0)^-1# to write it in inverse notation, and we have #f^-1(x)=color(white)(0)^3sqrtx+2#.
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Answer 2

To find the inverse of ( f(x) = x^3 - 2 ) and determine if it is a function, follow these steps:

  1. Replace ( f(x) ) with ( y ).
  2. Swap the roles of ( x ) and ( y ).
  3. Solve the resulting equation for ( y ).
  4. Replace ( y ) with ( f^{-1}(x) ) to express the inverse function.
  5. Check if the inverse function passes the vertical line test to determine if it is a function.

Here are the steps applied to ( f(x) = x^3 - 2 ):

  1. Replace ( f(x) ) with ( y ): ( y = x^3 - 2 )
  2. Swap the roles of ( x ) and ( y ): ( x = y^3 - 2 )
  3. Solve for ( y ): [ x + 2 = y^3 ] [ y = \sqrt[3]{x + 2} ]
  4. Replace ( y ) with ( f^{-1}(x) ): [ f^{-1}(x) = \sqrt[3]{x + 2} ]
  5. To determine if it is a function, check if the inverse passes the vertical line test. Since the cube root function is a one-to-one function (each input has a unique output), the inverse ( f^{-1}(x) = \sqrt[3]{x + 2} ) is also a function.

Therefore, the inverse of ( f(x) = x^3 - 2 ) is ( f^{-1}(x) = \sqrt[3]{x + 2} ), and it is a function.

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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.

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.

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.

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.

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