How to create a bijection between $(0,1)$ and $(0, \infty)$?

Question

I don't understand how to do this. The tip I have for the question is to first find a bijection between $(0,1)$ and $(1,\infty)$.

Answer 1

Here is a continuous one: $x \mapsto \frac{1}{x}-1$

Answer 2

Consider first the bijection $f\colon x\mapsto \frac 1 {1-x}\colon (0,1) \to (1,+\infty)$, which is continuous and order-preserving.

Now $f-1$ is the bijection you want: $$ x\mapsto \frac x {1-x}\colon (0,1) \to (0,+\infty) $$ It too is continuous and order-preserving.

Answer 3

$$f : (0,1) \to (0,\infty) , \, x \mapsto -\ln(1 - x)$$

Answer 4

$\displaystyle f(x) = \tan (x{\pi/2})$