Let $a$, $b$ and $c$ be real numbers and consider that $f$ maps $\mathbb{R}$ to $\mathbb{R}$.
For what values of $a$, $b$ and $c$ is $f(x) = ax^2 + bx + c$ (i) one-to-one? (ii) onto?
Let $a$, $b$ and $c$ be real numbers and consider that $f$ maps $\mathbb{R}$ to $\mathbb{R}$.
For what values of $a$, $b$ and $c$ is $f(x) = ax^2 + bx + c$ (i) one-to-one? (ii) onto?
Informal, to give you a start:
If $a\ne 0$, what does the curve $y=ax^2+bx+c$ look like? Bad, no? It is an upward or downward opening parabola, and one can see that both one to one and onto fail.
So for one to one, or onto, we need $a=0$. Suppose from now on that $a=0$.
If $b=0$, big trouble.
Show that if $a=0$ and $b\ne 0$, the function is one to one and onto. Geometrically, $y=bx+c$ is a line neither up and down nor parallel to the $x$-axis. The value of $c$ doesn't matter.
After you have figured out geometrically what's going on, doing the algebraic details (if required) will not be difficult.