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:

If $f(x) = \frac{x^2-x}{x^2+2x}$,then the value of $\frac{d}{dx}(f^{-1}(x))$ at $x = 2$ is:

If $g(x)$ is the inverse function of $f(x)$ and $f^{\prime}(x) = \frac{1}{1+x^4}$,then $g^{\prime}(x)$ is

Consider $f: R_{+} \rightarrow [-5, \infty)$ given by $f(x) = 9x^{2} + 6x - 5$. Show that $f$ is invertible with $f^{-1}(y) = \frac{\sqrt{y+6}-1}{3}$.

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 $f: R - \{-\frac{4}{3}\} \rightarrow R$ be a function defined as $f(x) = \frac{4x}{3x+4}$. The inverse of $f$ is the map $g: \text{Range } f \rightarrow R - \{-\frac{4}{3}\}$ given by