Graphical Representation of Functions

The function $y=f\left(x\right)$ is given graphically 

Sketch the graphs of the function $y=\frac{\left(\left|f\left(x\right)\right|-f\left(x\right)\right)}{2}$

Sketch the graphs of the following functions : $f\left(x\right)=\underset{n\to \infty }{\lim }\sqrt[n]{1+x^n}\left(x\ge 0\right)$

Prove that the graphs of the functions $y={\log }_{a}x$ and $y={\log }_{a^n}x$ can be derived from each other by changing all ordinates in the ratio $1:1/n$.

Knowing the graph of the function $y=f\left(x\right)$, sketch the graphs of the following functions : $y=f\left[\left(x\right)\right]$.

Knowing the graph of the function $y=f\left(x\right)$, sketch the graphs of the following functions : $y=\sqrt{f\left(x\right)}$

Knowing the graph of the function $y=f\left(x\right)$, sketch the graphs of the following functions : $y=f^2\left(x\right)$

Construct the graph of the function $y=f\left(x-1\right)+f\left(x+1\right)$, where $f\left(x\right)=\left\{\begin{array}{l}k\left(1-\left|x\right|\right) \mathrm{at} \left|x\right|\le 1\\ 0 \mathrm{at} \left|x\right|\ge 1\end{array}\right.$.

Sketch the graph of a periodic function with period $T=1$ defined on the half-open interval $\left(0,1\right]$ by the formula $y=x^2$.

Sketch the graph of the function $y=x\left(x^2-1\right)$ by multiplying the ordinates of the graphs $y_1=x$ and $y_2=x^2-1$.

Graph the following functions : $y=x^{2}sign\left(x\right)$, where $\mathrm{sign}\left(x\right)=\left\{\begin{array}{ccc}1& \mathrm{at} & x>0\\ 0& \mathrm{at} & x=0\\ -1& \mathrm{at} & x<0\end{array}\right.$

Graph the following functions : $y=\left|\sin x\right|+\sin x$ on the interval $[0,3\pi ]$;

Graph the following functions : $y=\cos \left(\sin x\right)$;

Graph the following functions : $y=\left|\left|x\right|-1\right|$;

Graph the following functions : $y=\left|x^2-2x-1\right|$;

Graph the function : $y=2\sqrt{-3\left(x+1.5\right)}-1.2$;

Graph the function : $y=3\sin \left(2x-4\right)$;

Graph the function : $y=\frac{1}{\cos x}$;

Graph the following functions : $y=x+\sin x$;

Graph the following functions :$y=x^2-x^3$  

Graph the following functions :$y=x+\frac{1}{x}$