Let $$A = \left\{ {x| - 1 < x < 1} \right\} = B.$$      If $$f:A \to B$$   be bijective then a possible definition of $$f\left( x \right)$$  is :

Solution :
$$f\left( x \right) = \left| x \right|$$   is many-one and into.
$$f\left( x \right) = x\left| x \right|$$   is one-one but into as $$f\left( x \right)$$  will have only rational values.
$$f\left( x \right) = \sin \,\pi x$$   is onto but many-one $$\left[ {\because f\left( {\frac{3}{4}} \right) = f\left( {\frac{1}{4}} \right)} \right]$$