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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?

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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.

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  • $\begingroup$ 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
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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.

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