Can anyone explain one-to-one functions? Counterexamples?

Question

1) Are all constant functions one-to-one? If no, why? Doesn't f(a)=f(b) always? What would be an example?

2) Is p(x)=1+x^2 a one to one?

Answer 1

1) No, in fact constant functions are never one-to-one. The requirement for a one-to-one or "injective" fuction is that f(a) = f(b) if and only if a = b. With a constant function, f(a) = f(b) for all a and b, even when a != b. (That's != for unequal.)

2) No, because f(a) = f(-a) for all real numbers, and a != -a unless a = 0. Note that p(-2) = p(2) = 5, but we couldn't have p(-2) = p(2) for an injective function because -2 != 2.

Answer 2

here you can read what one on one means ( injective ) with images
http://en.wikipedia.org/wiki/Injective_function