If $g(f(x)) = |\sin x|$ and $f(g(x)) = (\sin \sqrt{x})^2$,then

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 $f, g :(1, \infty) \rightarrow \mathbb{R}$ 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

Find $g \circ f$ and $f \circ g,$ if $f(x)=|x|$ and $g(x)=|5x-2|.$

If $f(x) = \frac{(\tan 1^{\circ}) x + \log_{e}(123)}{x \log_{e}(1234) - (\tan 1^{\circ})}$,$x > 0$,then the least value of $f(f(x)) + f(f(4/x))$ is $...........$.

Let $N$ denote the set of all natural numbers,and $Z$ denote the set of all integers. Consider the functions $f: N \rightarrow Z$ and $g: Z \rightarrow N$ defined by $f(n) = \begin{cases} (n+1)/2 & \text{if } n \text{ is odd} \\ (4-n)/2 & \text{if } n \text{ is even} \end{cases}$ and $g(n) = \begin{cases} 3+2n & \text{if } n \geq 0 \\ -2n & \text{if } n < 0 \end{cases}$. Define $(g \circ f)(n) = g(f(n))$ for all $n \in N$,and $(f \circ g)(n) = f(g(n))$ for all $n \in Z$. Then which of the following statements is (are) True?