Is $f : Z_9 \rightarrow Z_9$ injective, surjective, bijective och/eller inverterbar $d^{\circ}$ for

  1. $f(x) = x^3 + 3*x + 7$
  2. $f(x) = x^3 + 4*x + 5$
  3. $f(x) = x^3 + 5*x + 4$

How should I prove it? I think like a table but shall these values point to the same or?


As you say, the easiest way to do it is to draw up a table of the values that the function $f$ takes in each case.

If all of the values 0 to 8 appear in your table, then $f$ is surjective.

If no value is repeated, then $f$ is injective.

If both, then $f$ is bijective.

Of course, the "table" method only works for (small!) finite sets, and for these sets $f$ will either be bijective, or not injective and not surjective.

  • $\begingroup$ Which x-values should be same? $\endgroup$ – user3704516 Nov 5 '16 at 21:50
  • $\begingroup$ Ah, no $x$-values will be the same, it's the $f(x)$-values that might be the same! $\endgroup$ – Josh Hunt Nov 5 '16 at 21:51
  • $\begingroup$ So it can be different x-values but same f(x)? $\endgroup$ – user3704516 Nov 5 '16 at 21:52
  • $\begingroup$ I'm sorry, I'm not sure I understand your question. A function is said to be injective, if for every $x$ you have a different value of $f(x)$. That is, $f(x) = f(y)$ implies $x = y$ $\endgroup$ – Josh Hunt Nov 5 '16 at 21:57
  • $\begingroup$ I am test for all this x-values for b example $\endgroup$ – user3704516 Nov 5 '16 at 23:13

Calculate the tables for $x=0,1,\cdots,8$.

The function is injective if and only if all values $0,1,\cdots ,8$ appear.

Furthermore, the function is bijective if and only if it is injective if and only if it is surjective.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.