Let $f:(2, 3) \to (0, 1)$ be defined by $f(x) = x - [x]$. Then ${f^{ - 1}}(x)$ equals:

Suppose $f(x)=(x+1)^{2}$ for $x \geq -1$. If $g(x)$ is a function whose graph is the reflection of the graph of $f(x)$ in the line $y=x$,then $g(x) = $

If $f(x) = \frac{2x - 3}{3x - 4}$,$x \neq \frac{4}{3}$,then the value of $f^{-1}(x)$ is

If $g$ is the inverse of $f$ and $f^{\prime}(x)=\frac{1}{1+x^3}$,then $g^{\prime}(x)$ is

If $\alpha$ is the minimum value for which the inverse of $f(x)=x^2+3x-3$ exists in $[\alpha, \infty)$ and $g$ is the inverse of $f$, then find the value of $\frac{dg}{dx}$ at $x=\alpha+\frac{5}{2}$.

State with reason whether the following function has an inverse: $g : \{5, 6, 7, 8\} \rightarrow \{1, 2, 3, 4\}$ with $g = \{(5, 4), (6, 3), (7, 4), (8, 2)\}$.