Select the correct answer. What is the range of \( g(x)=-2|x+3|+2 \) ? \[ \begin{array}{l}\text { A. }(-\infty, 2] \\ \text { B. }(-3, \infty) \\ \text { C. }(2, \infty) \\ \text { D. }(-\infty, \infty)\end{array} \]

To find the range of the function \( g(x) = -2|x+3| + 2 \), we first note that the term \( |x+3| \) is always non-negative, making its maximum value at \( |x+3| = 0 \) when \( x = -3 \). When substituting \( x = -3 \), we find: \( g(-3) = -2(0) + 2 = 2 \). As \( |x + 3| \) increases, \( g(x) \) decreases without bound due to the negative coefficient (−2). Therefore, there is no lower limit to the values \( g(x) \) takes as \( x \) moves away from -3. This leads us to the conclusion that the maximum value of \( g(x) \) is 2, and it approaches negative infinity. Thus, the range is: \(\text{A. }(-\infty, 2].\) For extra insights: The vertex of this absolute value function occurs at its peak, which is a great visual feature to consider when analyzing similar equations! If you're sketching the graph, it’s super helpful to remember that the \( |x + 3| \) creates a “V” shape; in this case, flipped upside down because of the negative sign—showing how it impacts the range!