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Thread: component of composite function: given fcns (f o g)(x) and g(x), how to find f(x) ?

  1. #11
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    Quote Originally Posted by hellawowser View Post
    (f o g)(x) is 2x^2 -4x + 1 and g(x) = 2x - 3. That means f(2x - 3) = 2x^2 -4x + 1 right? How do i get to know the f equation?
    Here the idea is to express the given function in terms of (2x-3). So

    2x^2 -4x + 1 = 1/2 * (4x^2 -8x + 2) = 1/2 * [(4x^2 -12x + 9) + 4x - 7] = 1/2 * [(2x-3)^2 + 2(2x -3) - 1]

    continue.....
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

  2. #12
    Elite Member mmm4444bot's Avatar
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    Quote Originally Posted by hellawowser View Post
    f(x) = 1/2x^2 + x - 1/2 ??
    You can check!

    Using your candidate for f(x), determine the composite f(g(x)).

    Does the simplified result match the given?
    "English is the most ambiguous language in the world." ~ Yours Truly, 1969

  3. #13
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    Quote Originally Posted by hellawowser View Post
    I've meant that when you only know (f o g)(x) you have several pairs of equation for f(x) and g(x) , but when you know one of them, you still can have a lot of possibilities for the other one?

    Like i said i've only tried to fill in the equation : f(2x - 3) = 2x^2 -4x + 1 but i couldnt operate with it.
    Recently i've tried to analysis the domain and range for each but also couldnt get anything useful from it.
    I observe that no one has suggested what to me is the best algebraic approach to this (no guessing).

    You know that f(2x - 3) = 2x^2 - 4x + 1, and you want to find an expression for f(u). To do that, you can just substitute
    u = 2x - 3

    and solve to express x in terms of u. Put this in place of x on the right side, and simplify to obtain f(u).

    Another way to describe this process, if you are comfortable with the notation of function composition, is that you want [tex]f \circ g = h[/tex], where h(x) = 2x^2 - 4x + 1 and g(x) = 2x - 3, but you don't know f. You can just compose each side with [tex]g^{-1}[/tex]:

    [tex](f \circ g) \circ g^{-1} = h \circ g^{-1}[/tex]

    The left side simplifies to f, so the right side is the answer.


  4. #14
    Quote Originally Posted by Dr.Peterson View Post
    I observe that no one has suggested what to me is the best algebraic approach to this (no guessing).

    You know that f(2x - 3) = 2x^2 - 4x + 1, and you want to find an expression for f(u). To do that, you can just substitute
    u = 2x - 3

    and solve to express x in terms of u. Put this in place of x on the right side, and simplify to obtain f(u).

    Another way to describe this process, if you are comfortable with the notation of function composition, is that you want [tex]f \circ g = h[/tex], where h(x) = 2x^2 - 4x + 1 and g(x) = 2x - 3, but you don't know f. You can just compose each side with [tex]g^{-1}[/tex]:

    [tex](f \circ g) \circ g^{-1} = h \circ g^{-1}[/tex]

    The left side simplifies to f, so the right side is the answer.

    Awesome! Found the exact same result as the "guessing" and it is much easier. Thank you!

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