The range of $f(x) = \cos (x/3)$ is

Let $f (x) = a^x (a > 0)$ be written as $f( x) = f_1( x) + f_2( x)$ , where $f_1( x)$ is an even function and $f_2( x)$ is an odd function. Then $f_1( x + y) + f_1( x – y )$ equals

Consider the identity function $I _{ N }: N \rightarrow N$ defined as $I _{ N }$ $(x)=x$  $\forall $  $x \in N$ Show that although $I _{ N }$ is onto but $I _{ N }+ I _{ N }:$  $ N \rightarrow N$ defined as $\left(I_{N}+I_{N}\right)(x)=$ $I_{N}(x)+I_{N}(x)$ $=x+x=2 x$ is not onto.

If the domain of the function $f(x)=\log _e$ $\left(\frac{2 x+3}{4 x^2+x-3}\right)+\cos ^{-1}\left(\frac{2 x-1}{x+2}\right)$ is $(\alpha, \beta]$, then the value of $5 \beta-4 \alpha$ is equal to

Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ be a function which satisfies $\mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{f}(\mathrm{x})+\mathrm{f}(\mathrm{y}) \forall \mathrm{x}, \mathrm{y} \in \mathrm{R} .$ If $\mathrm{f}(1)=2$ and $g(n)=\sum \limits_{k=1}^{(n-1)} f(k), n \in N$ then the value of $n,$ for which $\mathrm{g}(\mathrm{n})=20,$ is