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Graphing functions helps us visualize how a function behaves. For a piecewise function like our example, graphing involves plotting two separate linear segments on the same set of axes. Each segment corresponds to a different formula that applies to specific intervals of the domain. This function is defined in two parts: \(g(x) = 1-x\) for \(0 \leq x \leq 1\) and \(g(x) = 2-x\) for \(1 < x \leq 2\).

Here's how to approach graphing each part:

• Start by identifying the interval for each piece and use these to frame the respective sections of the graph.
• Plot the start and end points of each interval. For \(g(x) = 1-x\), plot points \((0,1)\) and \((1,0)\).
• Draw the slope connecting these points. Ensure the end points are correctly represented: filled if included in the range and open if not.

The second part \(g(x) = 2-x\) involves similar steps. Plot the points \((1,1)\) and \((2,0)\), and connect them ensuring the open and closed circles are properly placed. When you follow these steps, the graph will clearly illustrate how the function changes over its domain.

The domain of a function refers to all possible input values (x-values) for which the function is defined, while the range is the set of possible output values (y-values). For our piecewise function, we have distinct domains for each part:

**Part 1**:

• Domain: \(0 \leq x \leq 1\)
• Range: Obtained by evaluating the function at \(x=0\) and \(x=1\), resulting in values \(1\) and \(0\), respectively. Thus, range is \([0, 1]\).

**Part 2**:

• Domain: \(1 < x \leq 2\)
• Range: Here, evaluate at the boundary \(x=2\) (which gives \(0\)) and at just over 1 (giving \(1\)), so range is \([0, 1)\).

Understanding how each range is determined helps in plotting the graph and understanding the function's behavior. When you're defining the graph, remember that closed circles indicate values that are part of the range, while open circles suggest that the endpoint value is not included in this particular segment of the function.

Linear functions graph as straight lines and have the form \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept. In the piecewise function from the exercise, we have two linear functions:

• For \(g(x) = 1-x\), the function represents a line with a slope of \(-1\), starting at y-intercept 1.
• For \(g(x) = 2-x\), it also has a slope of \(-1\) but starts higher, at y-intercept 2.

A negative slope means the line declines as \(x\) increases, indicating that y decreases when x increases. This decline is evident in both sections of the piecewise function. Understanding linear functions and their characteristics allows you to predict how the graph should look before plotting and confirm the linearity by checking that the plotted points lie on a straight path for the given intervals.