Special Functions and Their Graphs

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 $$

If the function $f:R\to R \text{and} g:R\to R$ are given by $f\left(x\right)=\left|x\right|$ and $g\left(x\right)=\left[x\right]. \left(\left[x\right]\text{ is greatest integer}\right).$ Find $fog\left(-\frac{1}{2}\right) \text{and} gof\left(-\frac{1}{2}\right).$

If [t] denotes the greatest integer ≤ t, then number of points, at which the function $f\left(x\right) = 4 \left|2x + 3\right| + 9 \left[x + \frac{1}{2}\right]- 12 [x + 20]$ is not differentiable in the open interval (-20, 20), is___.

Solve: $(x-3{)}^2-4\left|x-3\right|+3=0$

The number of real roots of ${\left(7+4\sqrt{3}\right)}^{\left|x\right|-8}+{\left(7-4\sqrt{3}\right)}^{\left|x\right|-8}=14$ is:

Solve $\left[\sin \left(\frac{\mathrm{\pi }}{10}\right)+\cos \left(\frac{\mathrm{\pi }}{10}\right)\right]$ where $\left[·\right]$ indicates greatest integer function

Find the total number of values of $x$ satisfying $x^2-3\left|x\right|+2=0.$

Solve $\frac{1}{\left|2x-1\right|}<6$, and express the solution using the interval notation.

Solve $\left|x+1\right|+\left|x+4\right|=1$

Solve $\left|3x-1\right|+\left|2x-3\right|-4=0$

Solve $\left|x-2\right|=6-2x$

FInd the sum of the greatest and least values of $\left|x-2\right|+\left|x-4\right|-\left|2x-8\right| \text{where}, 2⩽x<8 is$

Draw the graph of $e^{-\left|x\right|}.$

Minimum value of $f\left(x\right)=\sqrt{x^2-6x+9}+\sqrt{x^2+10x+25}$ is 

Let $f=\left\{\left(1, 1\right), \left(2, 3\right), \left(0, -1\right), \left(-1, -3\right)\right\}$ be a linear function from $Z$ into $Z$. Find $f\left(x\right)$.

Find the output for the following values of $x$, using the signum function $f\left(x\right)=\left[\begin{array}{ccc}+1& \text{ if }& x>0\\ -1& \text{ if }& x<0\\ 0& \text{ if }& x=0\end{array}\right]$. $x=\{4.93,7.66,0,11,-4.2,3.33333,-5.10\}$

If $f\left(x\right)=\left\{\begin{array}{l}{x}^2,\text{ when }x<0\\ x,\text{ when }0\le x<1\\ \frac{1}{x},\text{ when }x>1\end{array}\right.$, then find $f\left(\sqrt{-3}\right)$.

If $f\left(x\right)=\left\{\begin{array}{l}{x}^2,\text{ when }x<0\\ x,\text{ when }0\le x<1\\ \frac{1}{x},\text{ when }x>1\end{array}\right.$, then find $f\left(1\right)$.

Define greatest integer function. Also draw its graph and write the domain and range of this function.

Identify the graph of signum function $f\left(x\right)=5\frac{\left|x\right|}{x}$ and find whether the given signum function is odd or even.