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Prove problem help

  1. May 25, 2008 #1
    This not homework(self-study book). I do not know where to begin to prove this. This is from "Essentials Calculus" page 45, problem # 11

    [tex]f(x) = \left\{ \begin{array}{rcl}{-1} & \mbox{if}& -\infty < x < -1, \\ x & \mbox{if} & -1\leq x\leq1, \\1 & \mbox{if} & 1 < x <\infty ,\end{array}\right[/tex]
    [tex]g(x) = \frac {1} {2} |x+1 | - \frac {1}{2}|x-1|[/tex]
    Prove that [tex]f(x)\equiv g(x)[/tex]

    Latex is awesome :biggrin: First time using it here!
     
  2. jcsd
  3. May 25, 2008 #2

    rock.freak667

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    where's your attempt at the proof?

    But consider what g(x) is for the conditions for f(x)
     
  4. May 26, 2008 #3

    Dick

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    Glad you like latex. But, yes, as rock.freak667 points out, split into the three cases that define f.
     
  5. May 26, 2008 #4
    I don't understand ... "split into the three cases that define f." the cases are listed.

    I have no idea where to begin to prove this; a profound explanation would be helpful.
     
  6. May 26, 2008 #5

    nicksauce

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    He means split g(x) into the three cases that define f(x). You must show that when x < -1, g(x) = -1, when -1 < x < 1, g(x) = x, and when x > 1, g(x) = 1.
     
  7. May 26, 2008 #6

    D H

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    Consider this simpler problem:

    [tex]f(x) = \begin{cases}-x & \text{if}\ x < 0 \\ \phantom{-}x & \text{if}\ x>=0 \end{cases}[/tex]

    [tex]g(x) = |x|[/tex]

    For [itex]x>=0[/itex], [itex]g(x)=|x|=x[/tex]. For [itex]x<0[/itex], [itex]g(x)=|x|=-x[/tex]. In both cases, [itex]g(x)=f(x)[/itex]. The functions are identical for all x.

    You can do the same thing with your [itex]f(x)[/itex] and [itex]g(x)[/itex]. In particular, what does [itex]g(x)[/itex] evaluate to in each of the three regions?
     
  8. May 31, 2008 #7
    If you want to see how to write it formally out you would do something like:
    Case 1: Suppose x < -1 then g(x) = ... = f(x)
    Case 2: Suppose x > 1 then g(x) = ... = f(x)
    Case 3: Suppose -1<=x<=1 then g(x) = ... = f(x)

    Two functions f,g are equal in an interval I if f(x)=g(x) for every x in I. (In your case your interval is all real #'s.)
     
  9. May 31, 2008 #8
    Ok. Figured it out, thanks!
     
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