Example of a bijection from the set of real numbers to a subset of irrationals I need an example of a bijection from the set of real numbers to a subset of the irrationals.
I tried something like
$f(x)=x+\sqrt{2}$,
but where should I map $-\sqrt{2}$?

Let $f(x) = \dfrac{\arctan x}{\pi}$, so $f^{-1}(x) = \tan \pi x$.  $f$ maps $\mathbb{R}$ to $(-\dfrac12,\dfrac12)$.
$$g(x) = \begin{cases}
x \in \mathbb{Q} & x + \sqrt{5}\\
x \notin \mathbb{Q} & x\\
\end{cases}$$
$$g^{-1}(x) = \begin{cases}
x > 1 & x - \sqrt{5}\\
x \le 1 & x\\
\end{cases}$$
So $g \circ f$ maps from $\mathbb{R}$ to a certain subset of the irrationals between $-\dfrac12$ and $\dfrac12 + \sqrt{5}$, and $f^{-1} \circ g^{-1}$ maps the inverse.
