1. **State the problem:** We are given a piecewise function defined as $$y=\begin{cases} x+6 & \text{if } -6 \leq x \leq -2 \\ x^{2} & \text{if } -2 < x < 3 \\ -x+12 & \text{if } x \geq 3 \end{cases}$$ 2. **Understand the function:** This function has three parts depending on the value of $x$: - For $x$ between $-6$ and $-2$ inclusive, $y = x + 6$. - For $x$ between $-2$ and $3$ (not including $-2$ and $3$), $y = x^2$. - For $x$ greater than or equal to $3$, $y = -x + 12$. 3. **Evaluate the function at key points:** - At $x = -6$, $y = -6 + 6 = 0$. - At $x = -2$, from the first piece, $y = -2 + 6 = 4$. - At $x = -2$ from the second piece, $y = (-2)^2 = 4$ (consistent). - At $x = 3$, from the second piece, $y = 3^2 = 9$ (not included since $x<3$), from the third piece, $y = -3 + 12 = 9$ (included). 4. **Summary:** The function is continuous at the boundaries $x = -2$ and $x = 3$. 5. **Graphing:** The function is linear on $[-6,-2]$, quadratic on $(-2,3)$, and linear again on $[3,\infty)$. Final answer: The piecewise function is correctly defined as above with continuity at the boundary points.