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I am wondering if it is possible to get domain and codomain of a function in Mathematica (by using only built-in functions if possible). For example, I would like to give as input the function $tan(x)$ and to obtain the output "domain" $x \neq \frac{\pi}{2} + k \, \pi $ with $k$ integer, and "codomain" $R$.

Thank you so much for your willingness.

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You can use FunctionDomain and FunctionRange:

FunctionDomain[Tan[x], x]

1/2 + x/π ∉ Integers

FunctionRange[Tan[x],x,y]

True

FunctionRange[Sin[x], x, y]

-1 <= y <= 1

Update on questions in comments:

  • how the answer $1/2 + x/π ∉ \mathbb{Z}$ is related to the right result $ x ≠ π / 2 + k π $?

The two expressions are equivalent: Move $\pi/2$ to the lhs and divide both sides of the second expression by $\pi$ to get $ x/π - 1/2 ≠ k $ ($k$ integer).

  • Why does Mathematica return the first expression (not the second) as the answer?

The first one is simpler by LeafCount:

1/2 + x/π ∉ Integers // LeafCount

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ForAll[k, Element[k, Integers], x != k + π/2] // LeafCount

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  • $\begingroup$ Hello @kglr thank you for your extremely clear reply. True means R or C? $\endgroup$ – Gennaro Arguzzi Nov 8 '18 at 8:48
  • $\begingroup$ @Gennaro, my pleasure. Thank you for the accept. $\endgroup$ – kglr Nov 8 '18 at 8:49
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    $\begingroup$ True means R or C? $\endgroup$ – Gennaro Arguzzi Nov 8 '18 at 8:50
  • $\begingroup$ @GennaroArguzzi, i think it means Reals. $\endgroup$ – kglr Nov 8 '18 at 9:08
  • $\begingroup$ Hi @kglr, I have another doubt: how the answer 1/2 + x/π ∉ Integers is related to the right result $x \neq \frac{\pi}{2} + k \, \pi$? $\endgroup$ – Gennaro Arguzzi Nov 10 '18 at 6:21

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