Let $f(x) = (x + 2)^2 - 2, x \geq - 2$. Then $f^{-1}(x) =$

If $f(x) = \int\limits_0^x {\frac{1}{{\sqrt {1 + {t^3}} }}\,} dt$ and $h(x)$ is the inverse of $f(x)$,then the value of $\frac{{h''(x)}}{{{h^2}(x)}}$ is

Let $f: \{1, 2, 3\} \rightarrow \{a, b, c\}$ be a one-one and onto function given by $f(1) = a$,$f(2) = b$,and $f(3) = c$. Show that there exists a function $g: \{a, b, c\} \rightarrow \{1, 2, 3\}$ such that $g \circ f = I_X$ and $f \circ g = I_Y$,where $X = \{1, 2, 3\}$ and $Y = \{a, b, c\}$.

Let $f(x) = x^5 + 2e^{x/4}$ for all $x \in \mathbb{R}$. Consider a function $g(x)$ such that $(g \circ f)(x) = x$ for all $x \in \mathbb{R}$. Then the value of $8g'(2)$ is:

Let $g(x)$ be the inverse of an invertible function $f(x)$ which is differentiable at $x = c$. Then $g'(f(c))$ equals:

Let $f(x)=(x+1)^2-1, x \geqslant-1$,then the set $\{x : f(x)=f^{-1}(x)\}$ is