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Math Help - Inverse Functions

  1. #1
    Newbie Tom G's Avatar
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    Inverse Functions

    I've got two functions [f(x) and g(x)] which I need to find the inverse of [f-(x) and g-(x)], can someone please explain the steps taken when finding the inverse of funtions.

    1. f(x)= the square root of 'x' (i don't have a square root button on my keyboard)

    2. g(x)= 2x+1

    I would appreciate if someone could show each step they have taken to get the answer.
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Tom G View Post
    I've got two functions [f(x) and g(x)] which I need to find the inverse of [f-(x) and g-(x)], can someone please explain the steps taken when finding the inverse of funtions.

    1. f(x)= the square root of 'x' (i don't have a square root button on my keyboard)

    2. g(x)= 2x+1

    I would appreciate if someone could show each step they have taken to get the answer.
    The general method is to take a function y = f(x), reverse the roles of x and y ( x = f(y), then solve this equation for y ( y = g(x) = f^{-1}(x).)

    When doing this you need to be careful about domains and ranges. (The range of the function is the domain of the inverse function and visa versa.)

    For example:
    1) y = f(x) = \sqrt{x}
    Switch the roles of x and y: x = \sqrt{y}
    Solve for y: y = x^2

    Thus f^{-1}(x) = x^2.

    Note, though, that since f(x) is only defined on  [0, \infty ) that the inverse function is only defined on  [0, \infty ) rather than  (-\infty, \infty ).


    2) g(x) = 2x + 1.
    You do this one. I get g^{-1}(x) = \frac{x - 1}{2}.

    -Dan
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  3. #3
    MHF Contributor red_dog's Avatar
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    If f:A\to B and f(x)=y then f^{-1}:B\to A and f^{-1}(y)=x.
    So, to find f^{-1} you have to find y as a function of x.

    1) f:[0,\infty)\to[0,\infty),f(x)=\sqrt{x}.
    \sqrt{x}=y
    Square both sides:
    x=y^2 So, f^{-1}(y)=y^2
    Now, change the letter for the variable:
    f^{-1}:[0,\infty)\to[0,\infty), \ f^{-1}(x)=x^2

    Now, can you solve 2)?
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  4. #4
    Newbie Tom G's Avatar
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    Thanks, I've now managed to do g(x) and got the same answer as topsquark.

    I've got:

    g(x)=2x+1

    x=2y+1
    x-1=2y
    (x-1)/2=y

    (x-1)/2=g-(x)
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