f(x) = (x+1)/(x+2)
a) Determine if the function is injective
b) Prove the answer to part a
c) Determine if the function is surjective
d) Prove the answer to part c
$\displaystyle f(x_1) = f(x_2)\implies \frac{x_1+1}{x_1+2} = \frac{x_2+1}{x_2+2}$. Thus, $\displaystyle (x_1+1)(x_2+2) = (x_1 + 2)(x_2+1)$ this means $\displaystyle x_1x_2+x_2+2x_1+2 = x_1x_2+2x_2+x_1+2 \implies x_1 = x_2$.
In general if $\displaystyle ab-cd\not = 0$ then $\displaystyle (ax+b)/(cx+d)$ is injective, try proving that.
Yes. You need to show $\displaystyle \frac{x+1}{x+2} = y$ has a solution for any $\displaystyle y$, can you do that? And in general $\displaystyle (ax+b)/(cx+d)$ is surjective as well (try proving that too).c) Determine if the function is surjective