Let $f(x) = \log_e(\sin x)$ for $0 < x < \pi$ and $g(x) = \sin^{-1}(e^{-x})$ for $x \ge 0$. If $\alpha$ is a positive real number such that $a = (fog)'(\alpha)$ and $b = (fog)(\alpha)$,then which of the following is true?

If $g(x)=x^2+x-2$ and $\frac{1}{2}(g \circ f)(x)=2 x^2-5 x+2$,then one such function $f(x)=$

$f: R \rightarrow R$ and $g: R \rightarrow R$ are two functions such that $f(x)=x^2$ and $g(x)=\frac{1}{x^2}$,then $x^4(f \circ g)(x)$ is equal to

Find the function $g(t)$ if $f(t)=3t-2$ and $(g \circ f)^{-1}(t)=t-2$.

If for two functions $g$ and $f$,the composite function $g \circ f$ is both injective and surjective,then which of the following is true?

Let $f(x) = \sin \left(\frac{\pi}{6} \sin \left(\frac{\pi}{2} \sin x\right)\right)$ for all $x \in R$ and $g(x) = \frac{\pi}{2} \sin x$ for all $x \in R$. Let $(f \circ g)(x)$ denote $f(g(x))$ and $(g \circ f)(x)$ denote $g(f(x))$. Then which of the following is (are) true?
$(A)$ Range of $f$ is $\left[-\frac{1}{2}, \frac{1}{2}\right]$
$(B)$ Range of $f \circ g$ is $\left[-\frac{1}{2}, \frac{1}{2}\right]$
$(C)$ $\lim _{x \rightarrow 0} \frac{f(x)}{g(x)} = \frac{\pi}{6}$
$(D)$ There is an $x \in R$ such that $(g \circ f)(x) = 1$