If $f: R \rightarrow R$ is a mapping defined by $f(x)=x^{3}+5$,then $f^{-1}(x)$ is equal to

If $f: R \rightarrow R$ and $g: R \rightarrow R$ are defined by $f(x) = 5x - 3$ and $g(x) = x^2 + 3$,then $g \circ f^{-1}(3)$ is equal to

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 the function $f(x)=x^3+e^{\frac{x}{2}}$ and $g(x)=f^{-1}(x)$,then the value of $g^{\prime}(1)$ is

Let $f : A \to B$ be a function defined as $f(x) = \frac{x - 1}{x - 2}$,where $A = R - \{2\}$ and $B = R - \{1\}$. Then $f$ 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 $x=\alpha+\frac{5}{2}$.