Determine which of the following functions $\displaystyle f_i: \mathbb{R} \to \mathbb{R} $ are injective, surjective, and bijective. If they are bijections write down their inverses.

(i) $\displaystyle f_{1}(x) = x-1 $ is a bijection (inverse function is $\displaystyle y = x+1 $)

(ii) $\displaystyle f_{2}(x) = x^3 $ is a bijection (inverse is $\displaystyle y = \sqrt [3]{x} $)

(iii) $\displaystyle f_{3}(x) = x^{3}-x $ is not injective, but is surjective.

(iv) $\displaystyle f_{4}(x) = x^{3}-3x^{2}+3x-1 $ is not bijective.

(v) $\displaystyle f_{5}(x) = e^{x} $ is bijective (inverse is $\displaystyle y = \ln x $)

(vi) $\displaystyle f_{6}(x) = \begin{cases} x^{2}\ \text{if}\ x \geq 0 \\ -x^{2}\ \text{if}\ x \leq 0 \end{cases} $ is bijective (inverse is same conditions with function as $\displaystyle \sqrt{x} $ and $\displaystyle -\sqrt{x} $)

Thanks