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How to find a function of f with the following characteristics

  1. Aug 28, 2011 #1
    1. The problem statement, all variables and given/known data

    Give an expression for a function f which has all the following characteristics:

    f (0) = 4 , and
    f is increasing for x < 0 , and decreasing for x > 0 , and
    y = 1 is a horizontal asymptote of the graph of f as x→-∞ , and
    y = 0 is a horizontal asymptote of the graph of f as x→∞ .


    2. Relevant equations
    Any I guess

    3. The attempt at a solution

    Ok, since part a was to do with trigonometry I figured this being part b might follow suit but after having tried 4cos(x), 4/cos(x) and several other differing equations revolving around some manipulation of cos(0)=1 I've come up totally empty.

    So then I tried all manner of sorts of x+4 and 1/x +4 but still nothing.

    I'm not even sure of where to exactly start or how to approach this question aside from trial and error. I can find several equations to fit each characteristic but combining them is stumping me. The only other thing I can think of is a piece wise equation but again I'm not sure how to approach it or where to start.

    Tips and hints on where to begin are all I'm after, as long as I know I'm on the right path.
     
  2. jcsd
  3. Aug 28, 2011 #2

    tiny-tim

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    hi dazza555! :smile:
    piecewise is fine :smile:

    (nothing in the question requires the answer to have a continuous derivative)

    and can you think of a function with f(0) = 0 and f(+∞) = 1 ? :wink:
     
  4. Aug 28, 2011 #3
    Thank you tiny-tim, that helps quite a bit. I shall work on it some more and post back early tomoz of how I go.
     
  5. Aug 30, 2011 #4
    Ok, so I've come up with the following piecewise equation:

    f(x)= 4 / e^(x^2), x>=0

    1 - 1/e^(x^2), x<0

    I'm fairly confident it satisfies all the criteria but I know I'll feel better once somebody confirms it (especially the "f(x) is increasing for x < 0 , and decreasing for x > 0" part).
     
  6. Aug 30, 2011 #5

    tiny-tim

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    hi dazza555! :smile:

    (try using the X2 icon just above the Reply box :wink:)
    (wouldn't it be better to write them as 4e-x2 and 1 - e-x2 ? :wink:

    and why not use x or -x instead of x2 ?)

    the first one is ok, but the second is decreasing, from 1 at -∞ to 0 at 0

    (and they aren't continuous at 0 … i know the question doesn't actually specify that, but it would be neater if it was continuous)
     
  7. Aug 30, 2011 #6
    I'm an idiot, I can't believe I missed that bit on the second piece, I've just assumed graph go up, graph increasing. Well it's a simple fix, if I change my second part to 1+3e-x2 it's all fixed and it'll be continuous at (0,0).

    The reason I chose x2 is because it gives a nice bell shape curve rather than a sharp spike up. Since I'm graphing it also on a rather small axis I figured this way I wont get marked down because it looks linear. Also the reason I wrote it as 4/e^(x^2) was because I failed to notice the superscript button and it kinda looked confusing with little hats going everywhere. But the constructive criticism is welcome, without it I never would've looked for it:wink:
     
  8. Aug 30, 2011 #7
    Also thanks for all the help tiny-tim.
     
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