Amit M Agarwal Solutions for Chapter: Functions, Exercise 4: Target Exercise 3.4

Determine whether the function is even or odd $f\left(x\right)={\left\{{\left(\mathrm{sgn}x\right)}^{\mathrm{sgn}x}\right\}}^n; n$ is an odd integer.

Determine whether the function is even or odd: $f\left(x\right)=\mathrm{sgn}\left(x\right)+x^2$

Determine whether the function is even or odd $f\left(x+y\right)+f\left(x-y\right)=2 f\left(x\right)·f\left(y\right);$ where $f\left(0\right)\ne 0$ and $x,y\in R$.

Determine whether function; $f\left(x\right)={\left(-1\right)}^{\left[x\right]}$ is even, odd or neither of two (where $\left[.\right]$ denotes the greatest integer functions).

A function, defined for all real numbers is defined for $x\ge 0$ as follows:

$f\left(x\right)=\left\{\begin{array}{l}x\left|x\right|, 0\le x<1\\ 2x, x\ge 1\end{array}\right.$

How, is $f$ defined for $x\le 0$, if $f$ is even?

A function defined for all real numbers is defined for $x\ge 0$ as follows: $f\left(x\right)=\left\{\begin{array}{l}x\left|x\right|, 0\le x<1\\ 2x, x\ge 1\end{array}\right.$. How is $f$ defined for $x\le 0$, if  $\left(\mathrm{i}\right) f$is odd ? $\left(\mathrm{ii}\right) f$ is even ?

Show that the function $f\left(x\right)=\frac{2x\left(\sin x+\tan x\right)}{2\left[{\displaystyle \frac{x+21\pi }{\pi }}\right]-41}$, is symmetric about the origin.

If $f:\left[-20, 20\right]\to R$ defined by $f\left(x\right)=\left[\frac{x^2}{a}\right]\sin x+\cos x$, is an even function, then find the set of values of $\text{'}a\text{'}$. (Where $\left[·\right]$ denotes the greatest integral function).