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Math Help - Composite functions

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    Composite functions

    Let g(x)=x-3. Find a function f so that f(g(x))= x^2

    Let f(x)= x^2. Find a function g so that f(g(x))= x^2+8x+16

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    Quote Originally Posted by johett View Post
    Let g(x)=x-3. Find a function f so that f(g(x))= x^2
    We know that
    f(x - 3) = x^2

    Assume we have a function f(x). Then f(x - 3) is a translation of this function 3 units to the right. And we know this function is x^2. So if we translate the function f(x - 3) by 3 units to the left, we get f(x). So
    f(x) = (x + 3)^2

    -Dan
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by johett View Post
    Let f(x)= x^2. Find a function g so that f(g(x))= x^2+8x+16
    f(g(x)) = x^2 + 8x + 16 = (x + 4)^2

    Since f(x) = x^2 then we know the argument of f(g(x)) = (x + 4)^2 must be g(x) = x + 4.

    -Dan
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  4. #4
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    Hello, johett!

    Let g(x)=x-3
    Find a function f(x) so that: . f(g(x)) \:=\: x^2

    We want: . f(x-3) \:=\:x^2

    . . f(x) must transform x-3 into x^2.

    This can be done by: .adding 3 and squaring.

    Therefore: . f(x) \:=\:(x+3)^2




    Let f(x)\:=\:x^2.
    Find a function g(x) so that: . f(g(x)) \:=\: x^2+8x+16

    We want: . f(g(x)) \:=\:(x+4)^2

    Therefore: . g(x)\:=\:x+4

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