# Math Help - 1-1 from (0, 1) onto R

1. ## 1-1 from (0, 1) onto R

One can achieve a 1-1 correspondence from (0, 1) onto R easily by taking tan(pi * x - pi/2).

When I first looked at this problem, I came up with this. I'm curious if it works too.

Let f(x) be given by f(x) = 1/(e ^ x - 1) on {0 < x <= ln((e + 1)/2)}
f(x) = 1/(e - e ^ x) on {ln((e + 1)/2) < x < 1}

2. ## Re: 1-1 from (0, 1) onto R

Hey grandunification.

Hint: Check to see if your function is monotonic increasing (or decreasing) in the region of your function to see whether you have a 1-1 correspondence (also assume continuity and differentiation for the mapped transform).

The idea behind the above is that if something is monotonic and can be differentiated (with non-zero derivative at all points), then by the inverse function theorem an inverse function exists across the entire domain and co-domain and thus you have 1-1 properties.