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Math Help - Composite functions

  1. #1
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    Exclamation Composite functions

    Need help with this one (composite functions):

    Let f(x) = x + 4 and g(x)= (x - 2)^2 . Find a function u so that f(g(u(x))) = 4x^2 - 8x + 8
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  2. #2
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    Quote Originally Posted by madman1611 View Post
    Need help with this one (composite functions):

    Let f(x) = x + 4 and g(x)= (x - 2)^2 . Find a function u so that f(g(u(x))) = 4x^2 - 8x + 8
    Hello,

    1. Calculate f(g(x)) = (x-2)^2+4

    2. f(g(u(x))) = 4x^2-8x+8 = 4x^2-8x+{\color{red}+4+4} = (2x-2)^2+4

    3. Now compare the equation from 1. with the result from 2. Obviously the term 2x was plugged into the equation #1 instead of x. Thus u(x) = 2x
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  3. #3
    Senior Member tukeywilliams's Avatar
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    Whoops, didnt see that  g(x) was given.
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  4. #4
    MHF Contributor red_dog's Avatar
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    There is another solution.
    f(g(x))=x^2-4x+8\Rightarrow f(g(u(x)))=u^2(x)-4u(x)+8
    So we have to solve the functional equation u^2(x)-4u(x)+8=4x^2-8x+8
    which is equivalent to [u(x)-2x][u(x)+2x-4]=0.
    Then u(x)=2x or u(x)=-2x+4
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