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Math Help - How many right inverses does the function have?

  1. #1
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    How many right inverses does the function have?

    Let f : \mathbb{R} \rightarrow [-1,1] be defined by f(x)= sin x

    I know there is one as sin^{-1} x exists
    I also know that sin x is a surjective function, however i am lost where to go next
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  2. #2
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    Rephrasing the question:

    What are the functions g : [-1, 1] to R such that
    f(g(x)) = x (or sin(g(x)) = x)?

    g = sin^-1(x) is one solution
    What about g(x) = sin^-1(x) + 2*pi?
    What about g(x) = sin^-1(x) + 4*pi?
    What about g(x) = -sin^-1(x) + pi?
    What about g(x) = -sin^-1(x) + 3*pi?

    Also, the function does not even need to be continuous. What about the function
    g(x) = sin^-1(x) + 2*pi for 0 <= x <= 1
    and g(x) = sin^-1(x) for -1 <= x < 0

    Can you think of the most general form for g(x)?
    Last edited by snowtea; December 13th 2010 at 08:41 AM.
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