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A linear equation is a mathematical statement that describes a straight line when graphed on a coordinate plane. It has the general form of \( y = mx + b \), where \( m \) denotes the slope, and \( b \) represents the y-intercept. The slope \( m \) indicates how steep the line is and determines the direction, whether it's increasing or decreasing. A positive slope means the line ascends from left to right, while a negative slope means it descends. The y-intercept \( b \) is the point where the line crosses the y-axis.
For instance, in the piecewise function provided, two segments are linear equations: \( \frac{1}{2}x - 1 \) for \( x < 2 \) and \( x - 3 \) for \( x > 2 \). The first has a slope of \( \frac{1}{2} \) and a y-intercept of \( -1 \), showing a gradual rise. The second has a slope of \( 1 \) and crosses the y-axis at \( -3 \). In graphing these pieces, each forms a line portion in their respective domains.

Graphing functions is a visual representation technique that helps understand how functions behave. When you graph piecewise functions, each piece is graphed over its specified domain.

• Start with one segment of the function, like \( \frac{1}{2}x - 1 \), for \( x < 2 \). Sketch the line, ensuring you do not include the point \( (2, -1) \), indicating this with an open circle.


• Next, plot the distinct value at \( x = 2 \), placing a closed circle at \( (2, -4) \) to show this value is included in the function.


• Finally, graph the segment \( x - 3 \) for \( x > 2 \), starting just after \( x = 2 \) with an open circle at \( (2, -1) \) and extending right.

The result is a graph consisting of line portions and points, precisely representing where each piece of the function applies.

Understanding the domain and range is crucial in working with piecewise functions. The **domain** of a function is all the possible input values \( x \), while the **range** is all the potential output values \( y \).
In piecewise functions, the domain can be broken into distinct parts, each corresponding to a different rule. For \( g(x) \), the domain is all real numbers, but it splits at \( x = 2 \):

• For \( x < 2 \), the domain includes values leading up to, but not including, \( 2 \).
• The exact value at \( x = 2 \) is specifically defined as \( -4 \).
• For \( x > 2 \), it includes values starting just after \( 2 \).

Similarly, the **range** is affected by the form of each rule. Each segment describes a portion of the potential output values. Therefore, when analyzing such a function, check each part's start and stop values. This determines how they combine to form the complete output possibilities of the function.