How Can We Map the Open Interval (0,1) to the Real Line R Using a Homeomorphism?

[GridironCPJ]

Homework Statement

Find an explicit homeomorphism from (0,1) to R.

Homework Equations

A homeomorphism from (-1,1) to R is f(x)=tan(pi*x/2).

The Attempt at a Solution

I'm horrible a modifying trig functions. Obviously, to shift by b you add b to (x) and you can change the frequency by including a scalar to pi. I tried f(x)=tax(2pi*(x+1)/2), but this doesn't do the trick.

 

[6.28318531]

You are close, you just need to change a few things. Also Do you have to use a trig function? What about x/1-x², that maps (-1,1) -> ℝ, could you modify that?

 

[Bachelier]

You are close, you just need to change a few things. Also Do you have to use a trig function? What about x/1-x², that maps (-1,1) -> ℝ, could you modify that?

Consider: $f: (0,1) → \mathbb{R} \\ \ \ x → \frac{2x-1}{1-(2x-1)^2}$

This mapping is a Homeomorphism. meaning a Bijection.
could someone specify a metric on (0,1) that defines (the same topology) as the abs. value (i.e. the usual) metric and makes this open interval into a complete set?