If $f:[1,\infty )\rightarrow\:[2,\infty)$ is defined as $f(x)=x+\large\frac{1}{x}$ then find $f^{-1}(x)$.

Question

(A) $\large\frac{x+\sqrt {x^2-4}}{2}$

(B) $\large\frac{x-\sqrt {x^2-4}}{2}$

(C) $1+\sqrt{x^2-4}$

(D) $\large\frac{x}{1+x^2}$