Properties of the Function f:Z→Z with the Rule f(n) = 4n^3 - 1

[nicnicman]

Homework Statement

Is the function one-to-one, onto, both, or neither?
f: Z→Z has the rule of f(n) = 4n^3 - 1

Homework Equations

The Attempt at a Solution

My answer: one-to-one

Is this correct?

 

[STEMucator]

Well, how would you show it was one to one? How would you show it was onto?

 

[halo31]

I believe you are right that it is injective. It be wise that you show a proof to confirm its 1:1 and a counterexample to show its onto.

 

[nicnicman]

Well, the function is not onto because there is no integer n where 4n^3 - 1 = 1.

 

[nicnicman]

One-to-one proof:

4u^3 - 1 = 4v^3 - 1
4u^3 = 4v^3
u^3 = v ^3
u = v

 

[STEMucator]

Good. Now what about onto?

 

[nicnicman]

Thanks.

The function is not onto because there is no integer n where 4n^3 - 1 = 1.

 

[Dick]

One-to-one proof:

4u^3 - 1 = 4v^3 - 1
4u^3 = 4v^3
u^3 = v ^3
u = v

That seems fine to me. Neither of those is really a full scale proof. But I doubt you are expected to provide one. They are both good arguments that what you say is correct.

 

[nicnicman]

Yeah, the book doesn't even ask for a proof, but it's nice to know that the answer is right.

 

[HallsofIvy]

I disagree. What he gave are "full scale proofs". He showed that if f(u)= f(v) then u= v which is a perfectly good proof that f is "one to one". And he gave a counter example showing that it is NOT "onto".