If $f: R \rightarrow R$ is defined as $f(x) = (2020 - x^{2019})^{1 / 2019}$,$\forall x \in R$,find $(f \circ f \circ f \circ f) \left( \frac{2019}{2020} \right)$.

Let $f, g :(1, \infty) \rightarrow \mathbb{R}$ be defined as $f(x) = \frac{2x+3}{5x+2}$ and $g(x) = \frac{2-3x}{1-x}$. If the range of the function $f \circ g : [2, 4] \rightarrow \mathbb{R}$ is $[\alpha, \beta]$,then $\frac{1}{\beta-\alpha}$ is equal to

If $f$ is the greatest integer function defined on $R$ as $f(x) = [x]$ and $g$ is the modulus function defined on $R$ as $g(x) = |x|$,then the value of $(g \circ f)\left(\frac{-5}{3}\right)$ is

Let $f: R \rightarrow R$ and $g: R \rightarrow R$ be defined by $f(x)=2x+1$ and $g(x)=x^2-2$. Determine $(g \circ f)(x)$.

Let $f: \{1,3,4\} \rightarrow \{1,2,5\}$ and $g: \{1,2,5\} \rightarrow \{1,3\}$ be given by $f = \{(1,2), (3,5), (4,1)\}$ and $g = \{(1,3), (2,3), (5,1)\}$. Write down $g \circ f$.

If $f(x) = \frac{\alpha x}{x + 1}$,$x \ne -1$,for what value of $\alpha$ is $f(f(x)) = x$?