If the graph of non-constant function is symmetric about the point $(3,4)$ , then the value of $\sum\limits_{r = 0}^6 {f(r) + f(3)} $ is equal to

Let $\quad E_1=\left\{x \in R : x \neq 1\right.$ and $\left.\frac{x}{x-1}>0\right\}$ and $\quad E_2=\left\{x \in E_1: \sin ^{-1}\left(\log _e\left(\frac{x}{x-1}\right)\right)\right.$ is a real number $\}$.

(Here, the inverse trigonometric function $\sin ^{-1} x$ assumes values in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ )

Let $f : E _1 \rightarrow R$ be the function defined by $f(x)=\log _c\left(\frac{x}{x-1}\right)$ and $g: E_2 \rightarrow R$ be the function defined by $g(x)=\sin ^{-1}\left(\log _e\left(\frac{x}{x-1}\right)\right)$

The correct option is:

The function $f(x) = \frac{{{{\sec }^{ – 1}}x}}{{\sqrt {x – [x]} }},$ where $[.]$ denotes the greatest integer less than or equal to $x$  is defined for all  $x$  belonging to

Let $f : R \rightarrow R$ be a function such that $f(x)=\frac{x^2+2 x+1}{x^2+1}$. Then

If $f(x)=\frac{2^{2 x}}{2^{2 x}+2}, x \in R$ then $f\left(\frac{1}{2023}\right)+f\left(\frac{2}{2023}\right)+\ldots \ldots . .+f\left(\frac{2022}{2023}\right)$ is equal to