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Is this function injective, surjective, or both?

  1. Oct 3, 2009 #1
    1. The problem statement, all variables and given/known data

    The following function f is a function from R to R. Determine whether f is injective (one-to-one), surjective (onto), or both. Please give reasons.

    2. Relevant equations

    f(x) = (x+1)/(x+2) if x != -2
    f(x) = 1 when x = 2

    3. The attempt at a solution

    f'(x) = 1/(x+2)2 > 0 for all x

    and the limits at both infinities are 1 using l'hopital.

    So the way I see it is the function grows from 1+ when x is a large negative, and then theres a horizontal assymptote at x = -2 so just before x = -2 f(x) tends to infinity and just after x = -2 the f(x) goes from negative infinity and gradually increases to 1. and then ofcourse at the point x = - 2 f(x) = 1 because that value has been forced in the definition.

    So now by visualising the graph I have a strong suspicion that this function is bijective, but I have no idea how to prove it 'analytically'
     
  2. jcsd
  3. Oct 3, 2009 #2
    OK nevermind i figured it out. Is there a way to remove my original post? Or should I post my reasoning incase it could be helpful to others?
     
    Last edited: Oct 3, 2009
  4. Oct 3, 2009 #3

    Office_Shredder

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    Usually the latter is preferred
     
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