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Thread: Show f(x) is injective

  1. #1
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    Show f(x) is injective

    I want to show f(x) =x/(x+2) is injective.
    I know I need to assume a is not equal to a' but f(a)=f(a') and find contradiction.

    Here is my start:
    Assume a not equal to a'.
    f(a)=a/(a+2)
    f(a')=a'/(a'+2)

    Now I'm not sure where to go with this?
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  2. #2
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    Quote Originally Posted by kathrynmath View Post
    I want to show f(x) =x/(x+2) is injective.
    I know I need to assume a is not equal to a' but f(a)=f(a') and find contradiction.

    Here is my start:
    Assume a not equal to a'.
    f(a)=a/(a+2)
    f(a')=a'/(a'+2)

    Now I'm not sure where to go with this?
    Now assume that:

    $\displaystyle f(a)=f(a')$

    and that $\displaystyle a\ne -2$ and $\displaystyle a' \ne -2$ so:

    $\displaystyle \frac{a}{a+2}=\frac{a'}{a'+2}$

    then:

    $\displaystyle a(a'+2)=a'(a+2)$

    $\displaystyle aa'+a=a'a+a'$

    but multiplication is comutative so aa'=a'a, and so:

    $\displaystyle a=a'$

    a contradiction.

    The case where either equals $\displaystyle -2$ can easily be shown to imply the other also equals $\displaystyle -2$ since otherwise one side is undefined while the other is defined.

    CB
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