For real $x$, let $f(x) = x^3 + 5x + 1$, then

To check whether $f$ is one-one, compute the derivative. $$f'(x)=3x^2+5$$ Since $3x^2\ge 0$ for all real $x$, we have $f'(x)>0$ for every real $x$. Therefore $f$ is strictly increasing on $\mathbb{R}$, so it is one-one. To check onto, examine the end behavior. $$\lim_{x\to \infty} f(x)=\infty$$ $$\lim_{x\to -\infty} f(x)=-\infty$$ Because $f$ is a polynomial, it is continuous on $\mathbb{R}$. A continuous function taking values from $-\infty$ to $\infty$ must attain every real value, so $f$ is onto $\mathbb{R}$. Thus $f$ is both one-one and onto. Re-reading the options, this matches option $\text{C}$.