For [0, 1) to [0, infinity) I have that this piecewise function that I believe is injective. For [0, infinity) to [0, 1), I consulted this post, but the set bounds are different.
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2$\frac{2}{\pi}\arctan(x)$ is a nice function from $[0,\infty)$ to $[0,1)$. And, $\tan(\frac{\pi}{2}x)$ is the inverse. – Michael Burr Dec 01 '16 at 22:16
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There are infinitely many of them! If you want us to list them all, we could go on for ever. – Crostul Dec 01 '16 at 22:17
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Not all! Unless you're so inclined :) – Nick Dec 01 '16 at 22:18
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By example $f(x)=x$ is injective in $[0,1)$. The same for $f(x)=x^2$, etc... and their images are contained in $[0,\infty)$. – Masacroso Dec 01 '16 at 22:19