How to prove that a function is irrational? I need to know how to prove that a given function is irrational. Examples:
$$
f(x)=\sqrt{1+x^2}
$$
$$
f(x)=\tan(x) 
$$
Information about the definition of rational and irrational functions would be also helpful.
 A: Rational functions are functions that are equal to a quotient of polynomials.
As a consequence they have only finitely many zeros (finitely many poles, their derivatives are rational, and many other properties). This implies, that the example $\tan(x)$ can't be rational because it has infinitely many zeros.
You can prove that $\sqrt{1+x^2}$ is not rational by imitating the proof that $\sqrt{2}$ is not a rational number.

 Assume there are polynomials $p(x),q(x)$ with greatest common divisor $1$, such that >! $$\sqrt{1+x^2}=\frac{p(x)}{q(x)}$$
 Then $$(1+x^2)q(x)^2=p(x)^2$$
 Therefore $1+x^2$ divides $p(x)^2$. Since $1+x^2=(i-x)(-i-x)$ is square free, then $1+x^2$ divides $p(x)$, i.e. $p(x)=(1+x^2)r(r)$. Therefore 
$$(1+x^2)q(x)=(1+x^2)^2r(x)^2$$
It follows that $$q(x)=(1+x^2)r(x)^2$$
This means that $1+x^2$ should divide also $q(x)$. But this contradicts that $p(x)$ and $q(x)$ have greatest common divisor $1$.

If we were working with numbers such that $1+1=0$ then $$1+x^2=(1+x)^2$$ Therefore $\sqrt{1+x^2}=\sqrt{(1+x^2)^2}=1+x$ would be rational.
