Question

Let $f\left(x\right)=2^x=g\left(x\right)+h\left(x\right)$ where $g\left(x\right)$ is an even function and $h\left(x\right)$ is an odd function. If $[.]$ represents the greatest integer function, then the value of $\left[h\right(3\left)\right]$ is equal to

Answer 1

$f\left(x\right)=2^x$
$g\left(x\right)=\frac{f\left(x\right)+f(-x)}{2}=\frac{2^x+2^{-x}}{2}$
$h\left(x\right)=\frac{f\left(x\right)-f(-x)}{2}=\frac{2^x-2^{-x}}{2}$
$h\left(3\right)=\frac{2^3-2^{-3}}{2}=\frac{63}{16}$
$\therefore \left[h\right(3\left)\right]=3$​​​​