Even and Odd Functions

Check whether the function $f\left(x\right)=\frac{{\left(1+2^x\right)}^7}{2^x}$ is

The function $f\left(x\right)=\frac{x}{e^x-1}+\frac{x}{2}+1$ is

Check the below function is even, odd, or neither even nor odd?

$y=\sin x-\cos x$

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?

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

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$?$

Let $$\begin{gathered} f\left(x\right)=\left\{\begin{array}{ll}\sin x + \cos x, \\ a,& 0<x<\frac{\pi }{2} \\ x=\frac{\pi }{2}\\ {\tan }^2 x+\mathrm{cosec} x,& \frac{\pi }{2}<x<\pi \end{array}\right. \end{gathered}$$.

Then its odd extension is

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

If $f\left(x\right)=x^2+2x\sin x+3,$ then prove that $f\left(x\right)$ is an even function of $x.$

If $f\left(x\right)=x^3+3x-4{\sin }^{3}x,$ then prove that $f\left(x\right)$ is an odd function of $x.$

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

Which one of the following functions is neither even nor odd?

If $f\left(x\right)=2017\tan \left|x\right|+3x^3\sin x+x^{2018}-3x^2+2x^3$ and $f\left(3\right)=116,$ then $f\left(–3\right)$ is equal to

If $P\left(x\right)$ be a polynomial with real coefficients such that $P\left({\mathrm{s}\mathrm{i}\mathrm{n}}^{2}x\right)=P\left({\mathrm{c}\mathrm{o}\mathrm{s}}^{2}x\right),$ for all $x\in \left[0,\frac{\pi }{2}\right].$ Consider the following statements:

I. $P\left(x\right)$ is an even function.

II. $P\left(x\right)$ can be expressed as a polynomial in ${(2x-1)}^2$

III. $P\left(x\right)$ is a polynomial of even degree. Then

If $f\left(x\right)$ is an even function and $f^{\text{'}}\left(x\right)$ exist for all 'x' then $f^{\text{'}}\left(1\right) + f^{\text{'}}(–1)$ is-