Bijection from (0,1) to R

Question

I need help proving that f(x) = tan(xpi-pi/2) is a bijection where the map is from (0,1) to the Reals. I know I need to prove onto and one to one, but have little experience proving these facts with trig functions.

Answer 1

HINT:

1. $\tan x$ is continuous and monotonic increasing in $(-{\pi \over 2},{\pi \over 2})$ and hence injective.

2. $\lim\limits_{x \to -{\pi \over 2}}=-\infty$ and $\lim\limits_{x \to {\pi \over 2}}=\infty$ so it must be surjective.

Try to translate these facts in terms of your function.