Example 6 If the function $f(x)=\sqrt{k x^{2}+k x+3}$ has the domain $\mathbf{R}$, then the range of values for $k$ is
If $k=0$, then $f(x)=\sqrt{3}$, and its domain is $\mathbf{R}$.
If $k \neq 0$, to ensure that $k x^{2}+k x+3 \geqslant 0$ for any $x \in \mathbf{R}$, we should have $\left\{\begin{array}{l}k>0, \\ k^{2}-4 \times 3 k \leqslant 0,\end{array}\right.$ solving this yields $0<k \leqslant 12$, so the range of $k$ is $0 \leqslant k \leqslant 12$.

Key Points for Solving This Problem: The purpose of this problem is to reinforce the understanding of the concept of quadratic functions. Since the problem does not specify that $k \neq 0$, we cannot only consider $k x^{2}+k x+3$ from the perspective of a quadratic function; we must also specifically consider the case where $k=0$.