Even and Odd Functions

Prove that the function $f\left(x\right)=x^{2n}$, where $n$ is an integer is an even function

$f\left(x\right)=x^2$ is an even function.

If $f:\left[- 4,0\right]\to R$ is defined by $f\left(x\right)=e^x+\sin x$, its even extension to $\left[- 4,4\right]$ is given by:

If $f\left(x\right)=2x^6+3x^4+4x^2$, then $f^{\text{'}}\left(x\right)$ is

Show that, $f(x)=\log \left(x+\sqrt{1+x^{2}}\right)$ is an odd function.

Prove that $\phi(x)=x^{3} e^{\tan ^{2} x}$ is an odd function.

Which of the following functions is an even function?

[a] $f\left(x\right)=\left|x\right|$

[b] $f\left(x\right)=\left|x-1\right|$

[c] $f\left(x\right)=\left[x\right]-x$

Let $f$ be a function satisfying $f(x+y)=f(x)+f(y)$ for all real $x$ and $y$ and if $f(1)=k$, then show that $f$ is an odd function.

If $f$ is an odd function such that $0 \in D(f)$, then find $f(0)$.

Show that,

[b] product of an odd function and an even function is odd.

[6] Show that,

[a] product of two even (or odd) function is even.

If $f:R\to R$ be a function satisfying $f\left(x+y^2\right)+1+2x=f\left(x\right)+f\left(y^2\right)+2xf\left(y\right) \forall x\mathit{,}y\mathit{\in }R$, then which of the following statements is true?

Let $y=f\left(x\right)$ be an even function defined on the real number $R$. If $f^{\prime }\left(5\right)=-\frac{1}{\sqrt{3}}$, then the angle made by the tangent to the curve $y=f\left(x\right)$ at $x=-5$ with positive direction of $x-$axis is equal to

Among of the following functions  find an even function?

Statement 1: If $f$ is an even function, $g$ is an odd function, then $\frac{f}{g}(g\ne 0)$ is an even function.

Statement 2: If $f(-x)=-f\left(x\right)$ for every $x$ of its domain, then $f\left(x\right)$ is an odd function and if $f(-x)=f\left(x\right)$ for every $x$ of its domain, then $f\left(x\right)$ is an even function then which of the following is correct$?$

The function $g\left(x\right)=\sin \left({\sin }^{-1}\sqrt{\left\{x\right\}}\right)+\cos \left({\sin }^{-1}\sqrt{\left\{x\right\}}\right)-1$ where $\left\{x\right\}$ denotes the fractional part function, is

Let $f:[-2, 2]\to R$ be an odd function defined as $f\left(x\right)=x^3+\tan x+\left[\frac{x^2+1}{\lambda }\right],$ then $\lambda$ belongs to

(where $\left[·\right]$ denotes the greatest integer function)

Which of the following is a function whose graph is symmetrical about the origin?

Consider the function $f\left(x\right)=\mathrm{c}\mathrm{o}{\mathrm{s}}^{-1}\left(\left[2^x\right]\right)+\mathrm{s}\mathrm{i}{\mathrm{n}}^{-1}\left(\left[2^x\right]-1\right)$, then (where [.] represents the greatest integer part function)

The function $f\left(x\right)=\left|x\right|-x^3$ is