$(3)(8$ pts$)$ Let $f(n)=\sqrt[\phantom{2}]{1}+\sqrt[\phantom{2}]{2}+$cdots$+\sqrt[\phantom{2}]{n-1}+\sqrt[\phantom{2}]{n}.$

Find $g(n)$ so that $f(n)$ is $(g(n)).$ Justify your answer by presenting clear arguments for the $O$ and $$ bounds.