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When working with piecewise functions, identifying extreme values helps us understand the function's behavior over a given interval. Extreme values are the highest or lowest values that a function can reach on its domain. In our example with the function \( g(x) \), these are known as absolute extremum values. For the function \( g(x) \), the absolute minimum value is \(-1\) at \( x = 1^- \) and the absolute maximum value is \(1\) at \( x = 2 \).

This indicates that the function reaches its lowest and highest values at the specified points, even though the function is not continuous throughout the domain.

To determine these extrema:

• Evaluate the function at key points such as endpoints and where the function switches its definition.
• Check points close to boundaries for parts of the piecewise function.
• Take note of changes in function behavior within the defined intervals.

Looking at these points gives insight into how the extremes are consistent with the extreme value principles in piecewise functions.

Sketching the graph of a piecewise function involves piecing together segments that behave differently depending on their defined intervals. For the function \( g(x) \), it changes its definition at \( x = 1 \). In these cases, visualizing the behavior of each interval separately and then combining them can help in understanding the overall shape of the graph.

Here's how you can sketch the graph step by step:

• Identify the equations that define each piece for specific intervals. For \( g(x) \), these are \( -x \) for \( 0 \leq x < 1 \) and \( x-1 \) for \( 1 \leq x \leq 2 \).
• Draw each segment on a separate layer. Start with \( -x \) as it covers \( 0 \leq x < 1 \), resulting in a line with a negative slope passing from (0,0) to just before (1,-1).
• Next, graph \( x-1 \), which starts from (1,0) to (2,1), creating a positive slope.
• Ensure the piecewise segments connect smoothly if they meet at any point in their domain, indicating their transitional behavior.

By carefully plotting these sections, one can easily visualize the graph's V-shape structure, conveying the behavior within these intervals.

Continuity in a function occurs when there are no sudden jumps or breaks in its graph within a domain. When parts of the function suddenly switch, it results in discontinuity at that point. For piecewise functions like \( g(x) \), understanding where and why a function is discontinuous is crucial.

The function \( g(x) \) experiences a change at \( x = 1 \):

• As you approach \( x = 1 \) from the left (denoted as \( 1^- \)), the function value tends to \(-1\).
• However, directly at \( x = 1 \), the value is defined as \( 0 \) by the second piece of the function.

This sudden change at \( x = 1 \) indicates a discontinuity at this point.

Although \( g(x) \) is not continuous throughout \([0, 2]\), it still conforms to the basic principles surrounding extrema in closed intervals. Evaluating the value at, and around, each critical point provides the information needed to deal with discontinuities.

Understanding and highlighting these changes are key to analyzing and interpreting piecewise functions effectively in broader mathematical contexts.