a bijection is an injective (one-to-one) , surjective (onto) map between sets. if S = (0, 1) and T =R, find a map from S to T which is a bijection is an injective (one-to-one) , surjective (onto) map between sets. if S = (0, 1) and T =R, find a map from S to T which is 

my effort 
1)
(a) f(x) = x is a one to one function but it is not an onto function as each point in R is not mapped to the domain
(b) f(x) = 1/|x| is onto function but not one to one
(c) f(x) = -cot(πx)
is that correct 
any help will be appreciated 

For part c), possible answers are $\displaystyle f(x)=\frac{2x-1}{x(x-1)},\;f(x)=\frac{\ln 2x}{x-1},\;$ or $f(x)=\tan((x-\frac{1}{2})\pi)$, 
which is equivalent to your answer.
For part b), possible answers are $\displaystyle f(x)=\frac{(4x-1)(2x-1)(4x-3)}{x(1-x)}\;$ or 
$\hspace{2.1 in}f(x)=\tan((\pi(2x-\frac{1}{2}))$ for $x\ne\frac{1}{2}$, and $f(\frac{1}{2})=0$
