JEE Main · 2009 · 12 Apr 2009 · Offline · Sets Relations and Functions

Differentiating gives $f^{\prime }(x)=3x^2+5$, which is positive for every real $x$, so $f$ is strictly increasing and therefore one-one. Being a continuous increasing function whose values run from $-\infty$ to $+\infty$, it takes every real number exactly once, so it is also onto. Thus $f$ is both one-one and onto on the reals.