Number of solutions of the equation $2^x + x = 2^{\sin x} + \sin x$ in $[0, 10\pi]$ is -

Let $f(n) = 2^{n+1}$ and $g(n) = 1 + (n+1)2^n$ for all $n \in N$. Then:

Let $f(x) = \begin{cases} \frac{1}{2}, & \text{if } 0 \le x \le \frac{1}{2} \\ \frac{1}{3}, & \text{if } \frac{1}{2} < x \le 1 \end{cases}$,then $f$ is

Let $R$ be the set of real numbers. Define the real function $f: R \rightarrow R$ by $f(x) = x + 10$ and sketch the graph of this function.

Which of the following functions are odd?
$I. f(x)=x\left(\frac{e^x-1}{e^x+1}\right)$
$II. f(x)=k^x+k^{-x}+\cos x$
$III. f(x)=\log \left(x+\sqrt{x^2+1}\right)$

Which of the following is an even function?