1. **State the problem:** We want to analyze and understand the function $$y = \sqrt{3x^2 + 9}$$. 2. **Formula and rules:** The function involves a square root, so the expression inside the root, called the radicand, must be non-negative: $$3x^2 + 9 \geq 0$$. Since $$3x^2 \geq 0$$ for all real $$x$$ and 9 is positive, the radicand is always positive, so the domain is all real numbers. 3. **Simplify the expression inside the root:** $$3x^2 + 9 = 3(x^2 + 3)$$ 4. **Analyze the function:** - The square root function outputs only non-negative values, so $$y \geq 0$$. - As $$|x|$$ increases, $$3x^2$$ dominates, so $$y$$ grows approximately like $$\sqrt{3} |x|$$. - The minimum value of $$y$$ occurs at $$x=0$$: $$y = \sqrt{3(0)^2 + 9} = \sqrt{9} = 3$$. 5. **Summary:** - Domain: all real numbers $$(-\infty, \infty)$$. - Range: $$[3, \infty)$$. - The graph is a curve starting at $$y=3$$ when $$x=0$$ and increasing symmetrically as $$x$$ moves away from zero. **Final answer:** The function $$y = \sqrt{3x^2 + 9}$$ is defined for all real $$x$$ and has minimum value 3 at $$x=0$$.