Let a relation $R$ be defined by $R = \{(4, 5), (1, 4), (4, 6), (7, 6), (3, 7)\}$. Then ${R^{ - 1}}oR$ is

If $f(x) = \begin{cases} \sin x, & x \ne n\pi, n \in \mathbb{Z} \\ 2, & \text{otherwise} \end{cases}$ and $g(x) = \begin{cases} x^2 + 1, & x \ne 0, 2 \\ 4, & x = 0 \\ 5, & x = 2 \end{cases}$,then $\lim_{x \to 0} g(f(x))$ is

Let $f'(x) > 0$ and $g'(x) < 0$ for all $x \in \mathbb{R}$. Then which of the following is true?

Suppose that $g(x) = 1 + \sqrt{x}$ and $f(g(x)) = 3 + 2\sqrt{x} + x$,then $f(x)$ is

If $g(x)=1+\sqrt{x}$ and $f(g(x))=3+2 \sqrt{x}+x$,then $f(f(x))$ is

$f(x) = \begin{cases} 3-x, & -1 \leqslant x < 0 \\ 1+\frac{5x}{3}, & -3 \leqslant x \leqslant 2 \end{cases}$ and $g(x) = \begin{cases} -x, & -2 \leqslant x \leqslant 3 \\ x, & 0 \leqslant x \leqslant 1 \end{cases}$. Find the range of $(f \circ g)(x)$.