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Determining the inverse function for a given function

  1. Mar 16, 2010 #1
    1. The problem statement, all variables and given/known data
    Determine the inverse function for the function,

    f(x)=((2x+1)^3)-2

    I think i know the steps but i want to know if my answer is correct
    1.Change f(x) to f(y)
    2.Re write so x=f(y)
    3.Swap x and y variables
    4.replace y with f^-1(x)

    3. The attempt at a solution
    1.y=((2x+1)^3)-2
    2.cube sqrt y=(2x+1)-2
    =cube sqrt y+2=2x+1
    =cube sqrt y+1=2x
    =(cube sqrt y+1)/2=x
    3.y=(cube sqrt x+1)/2
    4.f^-1(x)=(cube sqrt x+1)/2

    Is this right???? Can someone please tell me how to do it if its wrong?
     
  2. jcsd
  3. Mar 16, 2010 #2

    Mentallic

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    Homework Helper

    It would be easiest if you just switch the x and y variables at the start (don't deal with f(x) and f(y) etc. while solving the equation), and then once you found the answer, change your y into f-1(x).

    y=((2x+1)^3)-2
    cube sqrt y=(2x+1)-2

    No! [tex]\sqrt[3]{((2x+1)^3-2}\neq (2x+1)-2[/tex]

    To illustrate this, I'll give you a more basic example that's a common mistake. Take pythagoras' theorem: [tex]c=\sqrt{a^2+b^2}[/tex]. Now, many students that come across this formula outside of solving lengths for right-triangles often make the mistake that [tex]\sqrt{a^2+b^2}=a+b[/tex] since they can just take the square root of each term inside... WRONG!
    Obviously if this were true we would have simplified pythagoras' theorem to c=a+b :tongue:

    The rule is [tex]\sqrt{x^2}=x[/tex] and x can be just about anything you choose, such as a+b. But then [tex](a+b)^2=a^2+2ab+b^2\neq a^2+b^2[/tex]. Only until you can convert [tex](2x+1)^3-2[/tex] into [tex]a^3[/tex] where a is some variable involving x, then can you take the cube root.

    But this is impossible, so all you need to do is move the -2 over to the other side, then you can take the cube root.
     
  4. Mar 16, 2010 #3
    So how do you change the (2x+1)^3-2 so x=y (eg) how do you find the answer?? im confused :p
     
  5. Mar 16, 2010 #4
    I remember also thinking that inverse functions were tricky when I started.

    Your function is y = (2x+1)3-2.

    To find the inverse, switch x and y so that you get x = (2y+1)3-2.

    Now the tricky bit is rearraging this so you have it in terms of y. So we need y = ....

    First add 2 to both sides, then you'll be on your way.
     
  6. Mar 16, 2010 #5

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    You have the order wrong. If you were evaluating this for some specific x, you would subtract 2 last so finding the inverse, you add 2 first:
    [itex]y+ 2= (2x+ 1)^3[/itex]. Now take the cube root of both sides.

    And please don't say "cube sqrt"- the "sqrt" or "square root" is specifically the inverse of squaring. What you want here, the inverse of cubing is "cube root".

    You should have, at this point, cube root(y+ 2)= 2x+1. Solve that for x.

     
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