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Understanding the domain and range of a piecewise function is crucial for grasping its behavior. The domain refers to all possible input values (x-values) for which the function is defined. For the given function \( g(x) \), the domain is all real numbers \((-\infty, \infty)\) because there is an expression for both \(x \leq 0\) and \(x > 0\). This means you can plug any real number into \(g(x)\) and get a valid output.

On the other hand, the range is about the outputs (y-values) the function can produce. For \(g(x)\), when \(x \leq 0\), it usually gives non-positive values because \(g(x) = -3x\). When \(x > 0\), it produces outputs that start at \(2\) and increase without bound, due to \(g(x) = 3x + 2\). Therefore, the range of \( g(x) \) is the set of all non-positive numbers together with all numbers starting from \(2\) upward, expressed as \((-\infty, 0] \cup [2, \infty)\). This understanding helps us visualize the behavior and scope of the function more clearly.

Graphing a piecewise function involves plotting each segment separately on the same coordinate plane, carefully considering their conditions. To graph \(g(x)\), handle two sections:

• For \(x \leq 0\), use \(g(x) = -3x\). Choose a few values where \(x \leq 0\), such as \(x = 0\) (where \(g(x) = 0\)) and \(x = -1\) (resulting in \(g(x) = 3\)). Connect these points to extend a line downward, which will feature a negative slope.
• For \(x > 0\), use \(g(x) = 3x + 2\). Begin with points like \(x = 1\) (where \(g(x) = 5\)) and \(x = 2\) (giving \(g(x) = 8\)). Draw a line starting just after \((0, 2)\) with a positive slope extending to the right.

These steps generate a graph with two distinct but connected sections - a decreasing line for negative values and an increasing one for positive parts, clearly showcasing how a piecewise function behaves over its domain.

Algebraic expressions are the building blocks of piecewise functions. Each segment of the piecewise function uses a different algebraic expression depending on the range of \(x\). In the function \(g(x)\), these segments are crucial:

• For \(x \leq 0\), the expression is \(-3x\), a simple linear expression indicating a line with a negative slope, which suggests a decrease in value as \(x\) moves left on the x-axis.
• For \(x > 0\), the expression is \(3x + 2\). It also describes a line but with a positive slope and a y-intercept of \(2\), making it shift upwards on the graph as \(x\) increases.

These expressions define the unique outputs for any given \(x\). Understanding them helps predict the overall shape and behavior of the function. Algebraic expressions in piecewise functions allow us to precisely capture real-world situations where conditions change, demonstrating the importance of structure in algebra.