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Composite Function

  1. Sep 15, 2009 #1
    The problem statement, all variables and given/known data
    141sgif.jpg

    The attempt at a solution

    [tex]g(f(x)) = h(x)[/tex]
    [tex]4f(x) + y = 4x - 1[/tex]
    [tex]4x + 16 + y = 4x - 1[/tex]
    [tex]y = -1 - 16[/tex]
    [tex]y = -17[/tex]

    so, [tex]g(x)= 4x + y = 4x - 17[/tex]

    Is this the correct way of going about this question? I used a guessing approach to this question. Is enough work shown to get full marks? Thanks.
     
  2. jcsd
  3. Sep 15, 2009 #2

    Mark44

    Staff: Mentor

    ==> g(x + 4) = 4x - 1
    ==> g(x) = 4(x - 4) - 1 = 4x -16 -1 = 4x - 17
    Hence g(x) = 4x - 17
    The reasoning behind my second equation above is that g(x + 4) represents a translation of g(x) to the left by 4 units, so to get the graph of g, I need to translate it and the function on the right side by 4 units to the right.
    Maybe you can justify the step above, but I don't see it. If the answer was in the back of the book, a guessing approach isn't worth much credit.
     
  4. Sep 16, 2009 #3
    Thanks for the help. You cleared it up for me.
     
  5. Sep 16, 2009 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Another way to do this. Since g(f(x))= g(x+ 4)= 4x- 1, let y= x+ 4. Then x= y- 4 so 4x-1= 4(y- 4)- 1= 4y- 17. g(x+4)= g(y)= 4y- 17 and, since the "y" is just a "placeholder", g(x)= 4x- 17.

     
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