Composition of Functions

If  $f:R\to R$  is defined by  $f\left(x\right)=x^2-3x+2,$ The $f\left(f\left(x\right)\right),$ would be $$

Let $f:\left[0, 1\right]\to \left[-1, 1\right]$ and $g:\left[-1, 1\right]\to \left[0, 2\right]$ be two functions such that $g$ is injective and $gof:\left[0, 1\right]\to \left[0, 2\right]$ is surjective. Then,

$g\left(x\right)=1+x-\left[x\right]$ and $f\left(x\right)=\left\{\begin{array}{c}-3; x<0\\ 0; x=0\\ 5; x>0\end{array}\right.$, then $f\left(g\right(x\left)\right)$ is (where $\left[·\right]$ represents greatest integer function)

For $x\in \left(0, \frac{3}{2}\right)$, let $f\left(x\right)=\sqrt{x}$, $g\left(x\right)=\tan x$ and $h\left(x\right)=\frac{1-x^2}{1+x^2}$. If $\varphi \left(x\right)=\left(\left(hof\right)og\right)\left(x\right)$ then $\varphi \left(\frac{\mathrm{\pi }}{3}\right)=$

Let $f\left(x\right)=\left[x\right]$, where $\left[x\right]$ be the greatest integer less than or equal to $x$, $g\left(x\right)=\left\{\begin{array}{cc}0,& x\in Z\\ x^2,& x\in R-Z\end{array}\right.$ where, $Z$ is the set of integers, $\varphi \left(x\right)=\left(fog\right)\left(x\right)$ and $\psi \left(x\right)=\left(gof\right)\left(x\right)$. Then, on the set $R-Z$

Let $f\left(x\right)=x^3$ and $g\left(x\right)=3^x.$ The values of $A$ such that $g\left(f\left(A\right)\right)=f\left(g\left(A\right)\right)$ are

For $x\ne 0, f\left(x\right)=\frac{1}{x}, g\left(x\right)=1-x, h\left(x\right)=\frac{1}{1-x}, P\left(x\right)=\frac{x-1}{x}$ and $Q\left(x\right)=\frac{x}{x-1}$. If $Wog\left(x\right)=Q\left(x\right)$, then $W\left(x\right)$ is equal to

Let $f\left(x\right)$ and $g\left(x\right)$ be two functions defined as $f\left(x\right)=\cos x$ and $g\left(x\right)=\sin x$. Then the maximum value of function defined as $h\left(x\right)=f\left(g\right(x\left)\right)$ is 

If $f\left(x\right)=\frac{x}{\sqrt{1+x^2}}$, then $\left(fofof\right)\left(x\right)=$

The function $f:R\to R$ is defined by$f\left(x\right)=\sin x+\cos x$ is

If $f:R\to R$ and $g:R\to R$ are given by $f\left(x\right)=\cos x$ and $g\left(x\right)=3x^2$. Find $gof$ and $fog$.

If $f\left(x\right)=\frac{x-3}{x+1}$, then $f\left[f\right\{f\left(x\right)\left\}\right]$ equals to

If $f:R\to R, f\left(x\right)=\sin x$ and $g:R\to R, g\left(x\right)=x^2,$ then $g\circ f\left(x\right)=?$

If $f:R\to R, g:R\to R$ defined by $f\left(x\right)=x^2-3x+4$ and $g\left(x\right)=2x+1$, then the value of $x$ for which $f\left(x\right)=\mathit{fog}\left(x\right)$ is

If $f\left(x\right)=e^x$ and $g\left(x\right)=ln\left(x\right)$ for all $x\in [1, \infty )$ then $fog:[1,\infty )\to [1,\infty )$ is

Consider $f\left(x\right)=e^x$ and $g\left(x\right)=2x-5.$ Then ${\left(gof\right)}^{-1}$ equals:

If $R$ be a relation from $A =\mathrm{ }\left\{\mathrm{1,2},\mathrm{ }3,\mathrm{ }4\right\}$ to $B\mathrm{ }=\mathrm{ }\left\{1,\mathrm{ }3,\mathrm{ }5\right\}$ such that, $\left(a,\mathrm{ }b\right)\in R\iff a<b,$ then which one is $Ro{R}^{-1}$

The functions $f, \mathrm{g }\text{and} h$ are defined from the set of real numbers $R$ to $R$ such that $f\left(x\right)=x^2-1, g\left(x\right)=\sqrt{(x^2+1)}$ and $h\left(x\right)=\left\{\begin{array}{c}0, if\mathrm{ x}<0\\ x, if\mathrm{ x}\ge 0\end{array}\right..$Define the following function  $ho\left(fog\right)\left(x\right)$.

For $x\mathit{\in }R\mathit{,}x\ne 0,x\ne 1,$ let $f_{\mathit{0}}\left(x\right)\mathit{=}\frac{1}{1\mathit{-}x}$ and $f_{n+1}\left(x\right)=f_0\left(f_n\left(x\right)\right),n=0,1,2,\dots ..$ Then the value of
$f_{100}\left(3\right)+f_1\left(\frac{2}{3}\right)+f_2\left(\frac{3}{2}\right)$ will be equal to

A function $f:R\to R$ is given by $f\left(x\right)=\left\{\begin{array}{c}-1, x\in Q\\ 1, x\notin Q\end{array} \right.$, then $\left(fof\right)\left(1-\sqrt{3}\right)$ is equal to _____.