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Given f find g such that g(x)=f(x-1)

  1. Mar 26, 2012 #1
    1. The problem statement, all variables and given/known data
    Find two complex polynomials f and g such that f(0)=f(-1)=1, f(1)=3 and g(x)=f(x-1)

    2. The attempt at a solution
    Using Lagrange polynomial I got f such that f(0)=f(-1)=1, f(1)=3
    Such f is defined by

    f(x)=x2+x+1

    Now that I've found f I need to find g such that g(x)=f(x-1), but I don't have any idea of how to do that. Any hint would be appreciated.
     
    Last edited: Mar 26, 2012
  2. jcsd
  3. Mar 26, 2012 #2

    SammyS

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    Given that f(x) = x2+x+1

    Find f(x-1) .
     
  4. Mar 26, 2012 #3

    HallsofIvy

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    You clearly know how to do some fairly complicated Calculus so surely you know how to evaluate a function! Just replace each "x" in [itex]f(x)= x^2+ x+ 12[/itex] with x- 1.
     
  5. Mar 26, 2012 #4
    Notice that if you define:
    [tex]
    h(x) \equiv f(x) - 1
    [/tex]
    then x = 0, and x = -1 are zeros of h. The simplest polynomial that can be written is:
    [tex]
    h(x) = A x (x + 1)
    [/tex]
    Then, use the fact that [itex]h(1) = f(1) - 1 = 3 - 1 = 2[/itex] to determine A. You can go back to f trivially then.

    Once you have chosen a particular choice for f, finding g, as explained in the above posts, is fairly trivial (just substitute [itex]x \rightarrow x - 1[/itex], expand and simplify).
     
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