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Thread: One-to-one Or onto?

  1. #1
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    Question One-to-one Or onto?

    Hi all,

    can anyone help to solve this?


    Let f: R -> R defined as f(x) = 9X2. Determine whether or not the function is one-to-one and onto.

    Thanks in Advance
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by cu4mail View Post
    Hi all,

    can anyone help to solve this?


    Let f: R -> R defined as f(x) = 9X2. Determine whether or not the function is one-to-one and onto.

    Thanks in Advance
    it is not 1-1 nor onto..

    not 1-1:
    by def:
    $\displaystyle if f(x) = f(y) \implies x = y $
    take f(x) = 9, then x = 1 or x = -1.
    therefore not 1-1.

    not onto:
    $\displaystyle f: R \rightarrow R$
    for every element in the range, there must be an element in domain..
    take -1 in range, does there an element in the domain that will give you -1 when mapped using your function?
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  3. #3
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    what r the basic rules to identify one-to-one or onto? Just want to understand the methodoligy.

    the actual statement was as :
    $\displaystyle f: R \rightarrow R$ defined as $\displaystyle f (x) = 9 x^2$

    Thanks
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  4. #4
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by cu4mail View Post
    what r the basic rules to identify one-to-one or onto? Just want to understand the methodoligy.

    the actual statement was as :
    $\displaystyle f: R \rightarrow R$ defined as $\displaystyle f (x) = 9 x^2$

    Thanks
    the definitions themselves:
    Def:
    A function $\displaystyle f:R \rightarrow R$ is said to be one-to-one if $\displaystyle f(x)=f(y) $, then $\displaystyle x=y $.

    okay, let's take your function $\displaystyle f(x)=9x^2$ as an example.
    so, suppose $\displaystyle f(x)=f(y)$
    then $\displaystyle 9x^2 = 9y^2 \implies x^2=y^2 \implies x=|y|$
    here, x is either y or -y which contradicts the definition. hence, $\displaystyle f(x)=9x^2$ is not 1-1.

    another explanation is that, any value in the range corresponds to a unique value in the domain..

    Def:
    A function $\displaystyle f:X \rightarrow Y$ is said to be onto Y if the range of f is Y.

    in your function, $\displaystyle f:R \rightarrow R$ defined by $\displaystyle f(x)=9x^2$, the range is $\displaystyle [0, +\infty)$ and not R, therefore, f is not onto R.

    here, it says that for all the values in the range must corresponds to a value in the domain.
    Last edited by kalagota; Nov 2nd 2007 at 04:47 AM.
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