Algebra of Functions

Let the sets $A$ and $B$ denote the domain and range respectively of the function $f\left(x\right)=\frac{1}{\sqrt{\left[x\right]-x}},$ where $\left[x\right]$ denotes the smallest integer greater than or equal to $x$. Then among the statements

$\left(S1\right): A\cap B=\left(1, \infty \right)-\mathrm{\mathbb{N}}$ and

$\left(S2\right): A\cup B=\left(1, \infty \right)$

If $y=\max \left\{\sqrt{\mathrm{x}}, {\mathrm{x}}^2-4, {\mathrm{x}}^3+2\right\}$, then number of solution(s) of $y=1$ is/are: 

If $3f\left(x\right)+2f\left(\frac{1}{x}\right)=\frac{1}{x}-10$, then $f\left(3\right)=$

Let $c, k\in R$. If $f\left(x\right)=\left(c+1\right)x^2+\left(1-c^2\right)x+2k$ and $f\left(x+y\right)=f\left(x\right)+f\left(y\right)-xy$ for all $x, y\in R$, then the value of $2\left[f\left(1\right)+f\left(2\right)+f\left(3\right)+...+f\left(20\right)\right]$ is equal to ______

If $f\left(x+y\right)=f\left(x\right)f\left(y\right) \forall x,y\in R$ and $f\left(1\right)=\frac{1}{3}$, then the value of $\sum _{n=1}^{\infty }f\left(n\right)$ is

If $a+\alpha =1, b+\beta =2$ and $af\left(x\right)+\alpha f\left(\frac{1}{x}\right)=bx+\frac{\beta }{x}$, $x\ne 0$ then the value of expression $\frac{f\left(x\right)+f\left({\displaystyle \frac{1}{x}}\right)}{x+{\displaystyle \frac{1}{x}}}$ is

If $f\left(x\right)={\mathrm{cosec}}^{-1}\left(x+\frac{1}{x}\right)$, then range of $f\left(x\right)$ is equal to 

If $f:R\to R$ is defined as $f\left(x\right)=\frac{3^x+3^{-x}}{2}$ $\forall x\in R$ and it satisfies $f\left(x+y\right)+f\left(x-y\right)=af\left(x\right)f\left(y\right)$, then $a$ is equal to 

If $f:R\to R$ is defined as $f\left(x+y\right)=f\left(x\right)+f\left(y\right)\forall x,y\in R$  and $f\left(1\right)=5$, then find the value of the following: $\sum _{r=1}^{n}f\left(r\right)=$ ?

The number of real roots of the equation $e^{\left(x-1\right)}+\log x+x-2=0,x\ne 0$,is 

Let an increasing function $f\left(x\right), x>0$ be such that $f\left(x\right)>-\frac{1}{x}\forall x>0$ and $f\left(x\right)f\left(\frac{1}{x}+f\left(x\right)\right)=1\forall x>0$, then the value of $f\left(1\right)$ is

Let $f:R\to R$ be a function such that $f\left(1\right)=1$. Also, $f(x+5)\ge f\left(x\right)+5$ and $f(x+1)\le f\left(x\right)+1\forall x\in R$, then

The domain and range of $f\left(x\right)=\frac{x^2-9}{x-3}$ is

If $f$ is a function such that $2f\left(x\right)+f\left(2-x\right)=x^2$ then the value of $f\left(4\right)$ is

If $f\left(x\right)+2f(1-x)=x^2+1, \forall x\in R$, then the range of $f$ is

If $f^2\left(x\right)·f\left(\frac{1-x}{1+x}\right)=x^3$, $x\notin \left\{-\mathrm{1,1}\right\}$ and $f\left(x\right)\ne 0$, then the value of $\left|\left[f\left(-2\right)\right]\right|$ is (where $\left[.\right]$ is the greatest integer function).

If $f^2\left(x\right)·f\left(\frac{1-x}{1+x}\right)=x^3, x\ne -\mathrm{1,} 1$ and $f\left(x\right)\ne 0$], then find $\left|\left[f\left(-2\right)\right]\right|$ (where $\left[.\right]$ is the greatest integer function).

Let $A$ denote the set of all real numbers $x$ such that $x^3-[x{]}^3=(x-\left[x\right]{)}^3,$ where $\left[x\right]$ is the greatest integer less than or equal to $x.$ Then

If $f\left(x\right)=x^2$ and $g\left(x\right)=2x-1$, then find the value of $\left(f-g\right)\left(41\right)$.

If $f\left(x\right)=x^2$ and $g\left(x\right)=2x+1$, then find the value of $\left(f+g\right)\left(59\right)$.