Embibe Experts Solutions for Chapter: Function, Exercise 1: EXERCISE

If $f\left(3x+2\right)+f\left(3x+29\right)=0 \forall x\in R$, then the period of $f\left(x\right)$ is

The domain of $f\left(x\right)$ is $\left(0,1\right)$, then domain of $f\left(e^x\right)+f\left(\ln \left|x\right|\right)$ is

If $f\left(x\right)={\left(-1\right)}^{\left[\frac{2x}{\mathrm{\pi }}\right]}, g\left(x\right)=\left|\sin x\right|-\left|\cos x\right|$ and $\varphi \left(x\right)=f\left(x\right)g\left(x\right)$ (where  $\left[.\right]$ denotes the greatest integer function), then the respective fundamental periods of $f\left(x\right), g\left(x\right)$ and $\varphi \left(x\right)$ are

Let $f\left(x\right)=e^{\left\{e^{\left|x\right|}\mathit{sgn}x\right\}}$ and $g\left(x\right)=e^{\left[e^{\left|x\right|}\mathit{sgn}x\right]}, x\in R$, where $\left\{.\right\}$ and $\left[.\right]$ denote the fractional and integral part functions, respectively. Also, $h\left(x\right)=\log \left(f\left(x\right)\right)+\log \left(g\left(x\right)\right)$, then for real $x, h\left(x\right)$ is

Let $f_1\left(x\right)=\left\{\begin{array}{ll}x,& 0\le x\le 1\\ 1,& x>1\\ 0,& \text{otherwise}\end{array}\right.$ and $f_2\left(x\right)=f_1\left(-x\right)$ for all $x$, $f_3\left(x\right)=-f_2\left(x\right)$ for all $x$, $f_4\left(x\right)=f_3\left(-x\right)$ for all$x$. Then which of the following is necessarily true?

The range of $f\left(x\right)=\sqrt{\left(1-\cos x\right)\sqrt{\left(1-\cos x\right)\sqrt{\left(1-\cos x\right)\sqrt{..........\infty }}}}$ is

The exhaustive domain of $f\left(x\right)=\sqrt{x^{12}-x^9+x^4-x+1}$ is

A real-valued function $f\left(x\right)$ satisfies the functional equation $f\left(x-y\right)=f\left(x\right)f\left(y\right)-f\left(a-x\right)f\left(a+y\right),$ where $\text{'}a\text{'}$ is a given constant and $f\left(0\right)=1$. $f\left(a\right)+f\left(2a-x\right)$ is equal to