Results 1 to 2 of 2

Math Help - question on a proof

  1. #1
    Member
    Joined
    Sep 2009
    Posts
    104

    question on a proof

    I need a little confirmation on the following proof:
    The problem is this:
    For an interval I, with f:I -> R where f is strictly increasing with f(I) an interval, prove that f^(-1):R -> I is strictly increasing.

    My proof is this:
    Because f is strictly increasing, for y,z exists in I such that y<z,
    f(y)<f(z).
    Let y,z exist in I. Then f(y),f(z) exist in R. and f(y) < f(z).
    but f^(-1)[f(y)]<f^(-1)[f(z)] which implies y<z and thus f^(-1) is strictly increasing.
    This proof seems trivial to me, which makes me think I did something wrong. I possibly assumed too much in my proof, not sure. Any ideas?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Oct 2009
    From
    London
    Posts
    42
    Hello

    I suffer from similar habbits. Always assuming things are complicated. Your proof is correct in its simplicity. You are absolutely right to assume the existence of f^{-1}, its given in the problem statement. Hence you may use the identity function f^{-1}(f(x)) = x, by the definition of the inverse.

    In fact just to reassure you that your proof makes sense, remember that f is the inverse of the function f^{-1}. And a function only has an inverse if it is strictly increasing or decreasing (a standard theorem from set theory). You have of course proved f^{-1} is strictly increasing. And thus the inverse of f^{-1} exists. namely, f.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Here's the proof what's the question!
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: April 26th 2010, 12:28 PM
  2. Another proof question
    Posted in the Pre-Calculus Forum
    Replies: 4
    Last Post: October 31st 2009, 09:22 PM
  3. Question on proof
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 1st 2009, 11:12 AM
  4. a proof question..
    Posted in the Algebra Forum
    Replies: 4
    Last Post: November 25th 2008, 08:35 AM
  5. [SOLVED] Proof Question
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: August 29th 2006, 09:56 AM

Search Tags


/mathhelpforum @mathhelpforum