Composite Functions

$f\left(x\right)>x;\forall x\in R$. Then the equation $f\left(f\left(x\right)\right)-x=0$ has

Suppose $f$ and $g$ both are linear function with $f\left(x\right)=-2 x+1$ and $f\left(g\right(x\left)\right)=6x-7$, then slope of line $y=g\left(x\right)$ is -

If $f\left(x\right)=-1+|x-1|,-1\le x\le 3;g\left(x\right)=2-|x+1|,-2\le x\le 2$, then calculate $\left(fog\right)\left(x\right)$ & $\left(gof\right)\left(x\right)$. Draw their graph. Discuss the continuity of $\left(fog\right)\left(x\right)$ at $x=-1$ & the differentiability of $\left(gof\right)\left(x\right)$ at $x=1$.

If $f\left(x\right)=-1+\left|x-2\right|, 0\le x\le 4$

$g\left(x\right)=2-\left|x\right|, -1\le x\le 3$

Then find $fog\left(x\right)$ & $gof\left(x\right)$. Draw rough sketch of the graphs of $fog\left(x\right)$ & $gof\left(x\right)$

Consider a function $f:x\to \frac{x+a}{x-1}; x\in R-\left\{1\right\}$ where a is a real constant. If $f$ is not a constant function, then find the following:

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

Let $f\left(x\right)=\left\{\begin{array}{cc}1+x,& 0\le x\le 2\\ 3-x,& 2<x\le 3\end{array}\right.$, then find $\left(fof\right)\left(x\right)$

The function $f\left(x\right)=\sqrt{{\log }_{x^2}\left(x\right)}$ is defined for $x$ belonging to-

If $f_0\left(x\right)=\frac{x}{\left(x+1\right)}$ and $f_{n+1}=f_{0}O f_n\left(x\right)$ , for $n=0,1,2,...,$ then $f_n\left(x\right)$ is-

Given $f\left(x\right)=\log \left(\frac{1+x}{1-x}\right)$ and $g\left(x\right)=\frac{3x-x^3}{1+3x^2}$, then $\left(fog\right)\left(x\right)$ equal-

If $g\left[f\left(x\right)\right]=\left|\sin x\right|$ and $f\left[g\left(x\right)\right]={\left(\sin \sqrt{x}\right)}^2$, then

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