Embibe Experts Solutions for Chapter: Functions, Exercise 4: EXERCISE-4

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 $x=1$.

Let $f\left(x\right)$ be a real value function not identically zero satisfies the equation,
$f\left(x+y^n\right)=f\left(x\right)+\left(f\right(y){)}^n$ for all real$x\&y$ and $f^{\text{'}}\left(0\right)\ge 0$ where $n(>1)$ is an odd natural number. Find $f\left(10\right)$

Find the domains of definitions of the following function:

$f\left(x\right)=\frac{\sqrt{\cos x-\frac{1}{2}}}{\sqrt{6+35x-6x^2}}$

Prove that if the function $f\left(x\right)=\sin x+\cos px$ is periodic, then $p$ is a rational number.

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 $fog\left(x\right)$ & $gof\left(x\right)$

If for all real values of $u$ & $v$, $2f\left(u\right)\cos v=f\left(u+v\right)+f\left(u-v\right)$, prove that, for all real values of $x$.

$f\left(x\right)+f\left(-x\right)=2a\cos x$

If for all real values of $u$ & $v$, $2f\left(u\right)\cos v=f\left(u+v\right)+f\left(u-v\right)$, prove that, for all real values of $x$.

$f\left(\pi -x\right)+f\left(-x\right)=0$

If for all real values of $u$ & $v$, $2f\left(u\right)\cos v=f\left(u+v\right)+f\left(u-v\right)$, prove that, for all real values of $x$.

$f\left(\pi -x\right)+f\left(x\right)=2b\sin x$ 

Deduce that $f\left(x\right)=a\cos x-b\sin x$, $a,b$ are arbitrary constant.