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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- Oct 29th 2008, 05:31 PMkathrynmathShow 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? - Oct 29th 2008, 09:34 PMCaptainBlack
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