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Image of a function

  1. Oct 1, 2008 #1
    1. The problem statement, all variables and given/known data

    I have the function g(s,t) = [(st+1)/(st-1),(s-t)/(st-1),(s+t)/(st-1)], and I need to find its image.

    2. Relevant equations

    I know that every point on the image of g lies on the hyperboloid x^2 + y^2 - z^2 = 1.

    3. The attempt at a solution

    I am very inexperienced with linear algebra, and I need to solve this problem for tomorrow. The problem is, I don't even really understand exactly what an image is. I read that it is like the function's range, but I don't even really know how to define the range of a function in 3D. Could someone please walk me through the steps for how to solve this problem?
     
  2. jcsd
  3. Oct 1, 2008 #2

    HallsofIvy

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    I don't know that this really has anything to do with linear algebra. It's really a problem in vector valued functions. You should have learned, back in basic algebra, that the "image" of the function y= f(x) is the set f all possible y values. Here, "y" is a point in 3 dimensions. A good way to start is to write
    x= (st+1)/(st-1),
    y= (s-t)/(st-1),
    z= (s+t)/(st-1).

    Now, what are the possible values of x, y, and z? For example, since s and t can be any numbers, st can be any number. If x= (a+1)/(a-1), what are the possible values of x? You might try graphing y= (x+1)/(x-1) to answer that question.
     
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