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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?

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    I was thinking if a = 0 and b is not equal to zero, that is one case when f is both one-to-one and onto, but i couldn't figure out how to formulate the argument when a is not equal to zero. Also i tried to do an algebraic solution but ran out of ideas?2012-11-19

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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.

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    And @Gerry Myerson: You are right and I apologize for this, its just that I am new at using this forum, won't happen again. Thanks for pointing it out.2012-11-19