Basics of Functions

Find the domain for the given function: 

$f\left(x\right)=\frac{1}{\sqrt{\left|x\right|-x}}$

In interval notation (a,b) = {x | a < x < b}, (a,b) represents

If $f\left(x\right)=\frac{x^2-x}{x^2+2x}$, then find the range of $f\left(x\right)$

Let $n$ be a positive integer and define $f\left(n\right)=1!+2!+3!+\dots +n!$. Find polynomials $P\left(x\right)$ and $Q\left(x\right)$ such that $f\left(n+2\right)=Q\left(n\right)f\left(n\right)+P\left(n\right)f\left(n+1\right)$ for all $n\ge 1$.

Consider the function $f\left(x\right)=x^2-x+1$ Find $f\left(0\right)+f\left(3\right)$.

The function $f\left(x\right)=0$ has eight distinct real solution and $f$ also satisfy $f\left(4+x\right)=f\left(4-x\right).$ The sum of all the eight solutions of $f\left(x\right)=0$ is –

$f\left(x\right)=x^2+4x+6$ and $g\left(x\right)=x^2+2x+8$. Prove that addition of functions is commutative using given functions.

$f\left(x\right)=2x^2+4x+5$ and $g\left(x\right)=x^2+3x+4$. Prove that addition of functions is commutative using given functions.

$f\left(x\right)=x^2+2x+2$ and $g\left(x\right)=x+1$. Prove that addition of functions is commutative using given functions.

$f\left(x\right)=x^2+4x+2$ and $g\left(x\right)=x-1$. Prove that addition of functions is commutative using given functions.

$f\left(x\right)=x^2+3$ and $g\left(x\right)=x+4$. Prove that addition of functions is commutative using given functions.

Suppose $h:\sqrt{R}\to \sqrt{R}$be defined as $h\left(x\right)=\sin x$ for $0<x<\frac{\mathrm{\pi }}{2}$, then find the value of $h\left(\frac{1}{2}\right).$

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

If $f\left(2x+3y, 2x-7y\right)=20x,$ then $f\left(x, y\right)$ equals:

The domain of the function $f\left(x\right)=\frac{\sqrt{4-x^2}}{{\cos }^{-1}\left(2-x\right)}$

If $\left|x-1\right|+\left|x-3\right|\le 8,$ then the values of $x$ lie in the interval

The range of the function $f\left(x\right)=\left[\mathrm{s}\mathrm{i}\mathrm{n}x+\mathrm{c}\mathrm{o}\mathrm{s}x\right]$ (where [.] denotes the greatest integer function) is

The number of elements in the domain of the $f\left(x\right)={\sin }^{-1}\left(\frac{x^2-2x}{3}\right)+\sqrt{\left[x\right]+[-x]}$, where $[.]$ denotes greatest integer function, are ______.

Statement $\mathrm{I}$: The maximum value of $\mathrm{s}\mathrm{i}\mathrm{n}\sqrt{2}x+\mathrm{s}\mathrm{i}\mathrm{n}ax$ cannot be $2$.
(where $a$ is positive rational number)
Statement $\mathrm{II}$: $\frac{\sqrt{2}}{a}$ is irrational.

If $f\left(x\right)=a^x+a^{-x}$ and $f\left(x+y\right)+f\left(x-y\right)=kf\left(x\right)·f\left(y\right)$, then $k$ is equal to