Even and Odd Functions

Check whether the function $f\left(x\right)=\frac{x\left(\sin x+\tan x\right)}{\left[{\displaystyle \frac{x+\mathrm{\pi }}{\mathrm{\pi }}}\right]-{\displaystyle \frac{1}{2}}},$ where $\left[.\right]$ denotes greatest integer function is even or odd function.

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.

The below function is even, odd, or neither even nor odd?

$y=x·\frac{a^x-1}{a^x+1}$

The below function is even, odd, or neither even nor odd?

$y=\frac{a^x-a^{-x}}{2}$

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

State whether the following statements are true/false:

State whether the following statements are true/false:

State whether the following statements are true/false:

[v] If a function is not even, then it must be 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?

The derivative of an even function is always an odd function.

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$?$