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Math Help - Problem with an exercise related to functions

  1. #1
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    Problem with an exercise related to functions

    I've been wondering about the following for a while:

    Function f: [1, \infty[ \rightarrow R is non-negative and implements the equation x + f(x) = e^{f(x)}. Find f(e-1) and f(1).

    I just cannot get anything out of it.
    Do you have any ideas?
    Last edited by Delaop; January 29th 2010 at 02:02 AM.
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  2. #2
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    I for one cannot read your post. The symbols are ‘scrambled’.
    Why do you use special fonts? Why can’t you use standard notation?
    If you expect reasonable help learn LaTeX.
    http://www.mathhelpforum.com/math-he...-tutorial.html
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    Thanks for the tip and sorry! I didn't realize that others couldn't see it as for me it shows it perfectly well. I'm fixing it soon.
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  4. #4
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    Quote Originally Posted by Delaop View Post
    I've been wondering about the following for a while:

    Function f: [1, \infty[ \rightarrow R is non-negative and implements the equation x + f(x) = e^{f(x)}. Find f(e-1) and f(1).

    I just cannot get anything out of it.
    Do you have any ideas?
    Let y= f(1). Then 1+ f(1)= e^{f(1)} becomes 1+ y= e^y. There is a "trivial" solution to that- do NOT try to "solve" the equation in any algebraic way.

    Let y= f(e-1). Then 1+ f(e-1)= e^{f(e-1)} becomes e- 1+ y= e^y. Again, there is "one" very simple solution. Solve it "by inspection".
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  5. #5
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    Thanks. I found out that obviously f(1) = 0 and f(e-1) = 1. But as I have now got the values by guessing, I should be able to state that these are the only solutions. If f was strictly monotonic, that would be the case, but can I show that f is strictly increasing or strictly decreasing? Am I completely wrong?

    Edit: If f(1) = 0 was also something else than 0, f wouldn't be a function. Same with the value 1. Right?
    Last edited by Delaop; January 29th 2010 at 03:37 AM.
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