Find the domain of the difference of the functions $f(x)=\sqrt{x}$ and $g(x)=\frac{1}{x-2}$

Step 1 ：The domain of a function is the set of all possible input values (typically, the set of x-values) which will produce a valid output from a particular function. We need to consider where both functions are defined.

Step 2 ：The function $f(x)=\sqrt{x}$ is defined for all $x\ge 0$. Therefore, the domain of $f(x)$ is $[0,+\mathrm{\infty })$.

Step 3 ：The function $g(x)=\frac{1}{x-2}$ is defined for all $x\ne 2$. Therefore, the domain of $g(x)$ is $(-\mathrm{\infty },2)\cup (2,+\mathrm{\infty })$.

Step 4 ：When we find the difference of the functions $f(x)-g(x)$, the domain will be the intersection of the domains of $f(x)$ and $g(x)$.

Step 5 ：The intersection of $[0,+\mathrm{\infty })$ and $(-\mathrm{\infty },2)\cup (2,+\mathrm{\infty })$ is $[0,2)\cup (2,+\mathrm{\infty })$.