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General Function question

  1. Dec 10, 2013 #1
    If a function s(t) exists, does a function t(s) always exist?

    Are there functions with no inverse relationships?

    Suppose

    [itex]s = \int^t_a e^{u^2} du[/itex]

    Can there be a t(s)?
     
  2. jcsd
  3. Dec 10, 2013 #2

    jedishrfu

    Staff: Mentor

    Look at the trig functions for sin and cos over 0 to 2pi. They are clearly functions if inverted will map to two angles for a given sin or cos value. So the inverse is not a function.

    For the sin if you restrict it 0 to pi/2 then its invertible and similarly for cos if you restrict it to 0 to pi.

    http://en.wikipedia.org/wiki/Inverse_function

    Try drawing a graph of e^u^2 and estimate the area under the curve and see if its invertible.
     
    Last edited: Dec 10, 2013
  4. Dec 10, 2013 #3

    Mark44

    Staff: Mentor

    Let's revise your notation a bit to make things more understandable.
    Suppose y = s(t) is a function. "s" is just the name of the function that maps values of t to values of y. Many functions do not have inverses that are themselves functions. A very simple example is y = f(x) = x2. Because f is not one-to-one, f does not have an inverse.
     
  5. Dec 10, 2013 #4
    Hmm, what about a solution involving step functions?

    x = u0 y1/2-(1-u0)y1/2

    Wolfram visual.
     
    Last edited: Dec 10, 2013
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