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Math Help - Piecewise function problem.

  1. #1
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    Piecewise function problem.

    Hi

    I have an exponetial function y=exp(f(x))

    As the function comes out of the sharp curve (sorry dont know the technical name for this) I would like the function to become linear, ie use the equation of a straight line to stop the response being exponetial.

    How can I do this and make sure the function remains smooth?

    Thanks

    Nic
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  2. #2
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    As I understand it, you would be helped by two functions, call them g(x) and h(x), where g(x) starts at g(0) = 1 and tends toward 0 as x --> infinity; and h(x) starts at h(0) = 0 and tends toward 1 as x --> infinity. Then you would be able to build e^(f(x)*g(x)) + x*h(x). This would start off like y = e^(f(x)) and smoothly approximate y = x + 1. Not knowing any particular values you want for this graph, we can get a family of approximately desired functions by taking g(x) = e^(-x) and I'm still working on a good answer for h(x), but maybe you can come up with your own, or tell me whether I'm understanding what you want correctly.
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  3. #3
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    Maybe h(x) = 1/(x+1).

    If you want to delay/hasten the on-set of the effects of g(x) or h(x) you can either shift them by g(x + a), h(x + b) or scale them by c*g(x), d*h(x), or some combination thereof.
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  4. #4
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    Wow, I guess I totally forgot what I wanted out of h(x). Still thinking about this...
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  5. #5
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    Oh, maybe something cheap like h(x) = -1/(x+1) + 1
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  6. #6
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    Hi ragnar (lodbrok?)

    I tried to upload an image to clarify what I'm asking but the uploader wont have any of it. I'll try and clarify. I have data set that is modelled as exponential upto a transition point, lets say x_trans, after which I would like to model it as a linear function. My problem is that I dont want to introduce a dicontinuity at x_trans.

    So my piecewise funcion (in my head at least) looks like this,

    If x < x_trans then y = e^(x)

    If x > x_trans then y = mx + c

    Does this clarify the situation?
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