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When it comes to sketching the graph of a piecewise function, you start by identifying the intervals for each piece of the function. For our specific function, we have three parts:
The first part is for when \(x < 0\). Here, the function is simply \(y = 1\), represented by a horizontal line at \(y = 1\) extending to the left.Next, from \(0 \leq x \leq 2\), the function changes to \(y = |x-1|\). This absolute value equation creates a V-shaped graph that has a vertex where \(x = 1\). You plot points such as \( (0, 1)\), \( (1, 0)\), and \( (2, 1)\).Finally, for \(x > 2\), the function returns to \(y = 1\), sketching another horizontal line along \(y = 1\), this time extending to the right from \(x = 2\).Combining these segments, you sketch a graph with a central V shape flanked by horizontal lines. Plotting these clearly shows how the function behaves across different intervals.

Intercepts are the points where the graph crosses the axes. For the x-intercept, we look for where the graph meets the x-axis, or where \(y = 0\).

• For the interval \(0 \leq x \leq 2\), you find the x-intercept at point \(x = 1\) since \(y = 0\) at this location. Thus, the x-intercept is at \( (1, 0)\).
• The y-intercept is located where the graph intersects the y-axis, which is when \(x = 0\). Evaluating with the piece \(0 \leq x \leq 2\) gives \(y = 1\) at \(x = 0\). Thus, the y-intercept is at point \( (0, 1)\).

These intercepts provide critical reference points that aid in sketching the graph accurately and understanding the function's overall behavior.

Discontinuities in a function occur where the function's graph has an abrupt change or "jump". For this specific piecewise function, we assess the points where the function switches between pieces, namely \(x = 0\) and \(x = 2\).
At \(x = 0\), although there is a shift in the function's rule from \(y = 1\) to \(y = |x-1|\), the function itself remains continuous. The left limit (approaching from values of \(x < 0\)) is 1, and the function value at \(x = 0\) is also 1.Similarly, at \(x = 2\), the function goes from \(y = |x-1|\) back to \(y = 1\). Again, the right limit (coming from values of \(x > 2\)) is 1, matching the actual function value at that point.Since these transitions uphold continuity at \(x = 0\) and \(x = 2\), this piecewise function is continuous throughout all plotted intervals, with no discontinuities.