For equality of functions $f$ and $g$,which of the following conditions must be satisfied?
$(i)$ $\text{domain of } f = \text{domain of } g$
(ii) $f(x) = g(x)$ for all $x$ in the domain
(iii) $x \in \text{domain of } f$

Let $A = \{1, 2, 3, \ldots, 10\}$ and $f: A \rightarrow A$ be defined as $f(k) = \begin{cases} k + 1 & \text{if } k \text{ is odd} \\ k & \text{if } k \text{ is even} \end{cases}$. Then the number of possible functions $g: A \rightarrow A$ such that $g \circ f = f$ is ...... .

Let $A = \{1, 4, 7\}$ and $B = \{2, 3, 8\}$. Then the number of elements in the relation $R = \{((a_1, b_1), (a_2, b_2)) \in ((A \times B) \times (A \times B)) : a_1 + a_2 \text{ divides } b_2 + b_1\}$ is . . . . . . .

Let $c, k \in R$. If $f(x)=(c+1) x^{2}+(1-c^{2}) x+2 k$ and $f(x+y)=f(x)+f(y)-x y$,for all $x, y \in R$,then the value of $|2(f(1)+f(2)+f(3)+\ldots+f(20))|$ is equal to

Let $f, g$ and $h$ be the real-valued functions defined on $\mathbb{R}$ as $f(x) = \begin{cases} \frac{x}{|x|}, & x \neq 0 \\ 1, & x=0 \end{cases}$,$g(x) = \begin{cases} \frac{\sin(x+1)}{x+1}, & x \neq -1 \\ 1, & x=-1 \end{cases}$ and $h(x) = 2[x] - f(x)$,where $[x]$ is the greatest integer $\leq x$. Then the value of $\lim_{x \rightarrow 1} g(h(x-1))$ is

Let $f(x) = -1 + |x - 2|$ and $g(x) = 1 - |x|$. Then the set of all points where $fog$ is discontinuous is