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?
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