Embibe Experts Solutions for Chapter: Functions, Exercise 3: Level 3

For any real number $x,$ let $\left[x\right]$ denote the integer part of $x\mathit{;}\mathit{ }\left\{x\right\}$ be the fractional part of $x\mathit{.}$ $\left(\left\{x\right\}=x-\left[x\right]\right)$. Let $A$ denote the set of all real numbers $x$ satisfying $\left\{x\right\}=\frac{x+\left[x\right]+\left[x+\left({\displaystyle \frac{1}{2}}\right)\right]}{20}$. If $S$ is the sum of all numbers in $A,$ find $\left[S\right]$.

The period of the function $c^{\left\{{\sin }^{2}x+{\sin }^2\left(x+\frac{\pi }{3}\right)+\cos x\cos \left(x+\frac{\pi }{3}\right)\right\}}$ is (where $c$ is constant)

Let $f$ and $g$ be functions defined by $f\left(x\right)=\frac{1}{x+1}, x\in R, x\ne -1$ and $g\left(x\right)=x^2+1,x\in R.$ Then, $gof$ is

If $f\left(x\right)$ and $g\left(x\right)$ are periodic functions with periods $7$ and $11,$ respectively. Then, the period of $F\left(x\right)=f\left(x\right)·g\left(\frac{x}{5}\right)+g\left(x\right)·f\left(\frac{x}{3}\right)$ is

The number of real solutions of the equation $1-x=\left[\cos x\right]$ is:

The number of solutions of $[\sin x+\cos x]=3+[-\sin x]+[-\cos x]$ in the interval $[0,2\mathrm{\pi }]$ is (where $[ . ]$ denotes the greatest integer function.)

If graph of $y=f\left(x\right)$, where $f\left(x\right)$ is differentiable in $(-3, 1)$, is as shown in the following figure

and $g\left(x\right)=\ln \left(f\right(x\left)\right),$ then the graph of $y=g(-|x\left|\right)$ is

If the function $f : [1, \infty )\to [1, \infty )$ is defined by $f\left(x\right)=3^{x\left(x-1\right)}$ then $f^{-1}\left(x\right)$ is