Example 5 Find the range of the function $y=\frac{x^{2}-x+3}{x}(x>0)$.

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Example 5 Find the range of the function $y=\frac{x^{2}-x+3}{x}(x>0)$.
The original function can be transformed into
$$
x^{2}-(y+1) x+3=0
$$

Let $f(x)=x^{2}-(y+1) x+3(x>0)$, noting that $f(0)=3>0$, so the condition for equation (5) to have a solution when $x>0$ is $\Delta \geqslant 0$, and $\frac{y+1}{2}>0$, solving this yields $y \geqslant 2 \sqrt{3}-1$. Therefore, the range of the original function is $\{y \mid y \geqslant 2 \sqrt{3}-1\}$.