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Math Help - Prove a function is onto

  1. #1
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    [Solved]Prove a function is onto

    I need to prove f(x) = ((x+1)/x) is an onto function. I'm kind of hazy here, am I supposed to subsitute and solve for x? Such as, y=((x+1)/x). IF that is the case, I can't get it to add up to prove that function is onto. I know answer to problem, its just I think I'm jackng up my algebra and want to be able to do all the steps involved.

    Thanks!
    Last edited by SlapnutsGT; April 12th 2009 at 03:01 PM. Reason: solved!
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  2. #2
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    Onto what?
    What is the domain?
    What is the image set?
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  3. #3
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    Doh! Sorry all real numbers, domain is x not equal to 0 and co-domain not equal to 1.
    Last edited by SlapnutsGT; April 12th 2009 at 02:07 PM. Reason: Nevermind, what I came up with was wrong after all.
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  4. #4
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    Here's what you do:
    Suppose f is onto. Then there exists an x such that f(x) = (x+1)/(x) = y. Solve for x to get x = 1/(y-1). Now plug this value back into the equation, like this:
    f(1/(y-1)) = (x+1)/(x) = [(1/(y-1))+1]/[1/(y-1)]. Now solve this equation to end up with f(1/(y-1)) = y. There, now you have proved that f(x) is onto.
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  5. #5
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    Quote Originally Posted by spearfish View Post
    Here's what you do:
    Suppose f is onto. Then there exists an x such that f(x) = (x+1)/(x) = y. Solve for x to get x = 1/(y-1). Now plug this value back into the equation, like this:
    f(1/(y-1)) = (x+1)/(x) = [(1/(y-1))+1]/[1/(y-1)]. Now solve this equation to end up with f(1/(y-1)) = y. There, now you have proved that f(x) is onto.
    That is what I was orignally doing. I know the problem is I'm messing up my algebra. I can't solve for x, I always end up cancelling my x-variables. Sorry, my algebra is rusty.


    y=((x+1)/x) ...multiply both sides by x
    xy=x+1 ...subtract 1
    xy-1=x ...divide by (y-1)
    x=x/(y-1)

    Stuck here, not even sure if I did those steps properly.
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  6. #6
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    Yeah, I hear you, it happens to me too!

    y = (x+1)/x
    y = x/x + 1/x
    y = 1 + 1/x
    -1/x = 1-y
    1/x = -1+y
    1 = x(-1+y)
    1/(y-1) = x

    Hope that helps.
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  7. #7
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    Quote Originally Posted by spearfish View Post
    Yeah, I hear you, it happens to me too!

    y = (x+1)/x
    y = x/x + 1/x
    y = 1 + 1/x
    -1/x = 1-y
    1/x = -1+y
    1 = x(-1+y)
    1/(y-1) = x

    Hope that helps.
    Thats is what I was forgetting right there, thanks alot!
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