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Sketch the graph of the piecewise function F(x) = { 3 - x, for x <= 1; 2x, for x > 1 }. Which description of the graph is correct?

1. A line of slope -1 (value 2 at x=1) for x <= 1, and a line of slope 2 for x > 1 starting just above (1, 2) with a jump up at x = 1.
2. A line of slope -1 for x <= 1 and a line of slope 2 for x > 1, continuous at x = 1.
3. A line of slope 1 for x <= 1 and a line of slope 2 for x > 1, with a jump down at x = 1.
4. A single straight line of slope 2 for all x.

Correct answer: A line of slope -1 (value 2 at x=1) for x <= 1, and a line of slope 2 for x > 1 starting just above (1, 2) with a jump up at x = 1.

Solution

For x <= 1 the graph is the line y = 3 - x (slope -1), giving the point (1, 2) included. For x > 1 the graph is y = 2x (slope 2); as x -> 1+ it approaches y = 2 but that point is not included (open), and it increases steeply. At x = 1 the left piece ends at (1, 2) and the right piece begins just above 2 for x slightly greater than 1 — actually both branches meet the value 2 at x=1, so the function value is 2 (from the left branch) and the right branch limit is also 2, making it continuous in value but with a corner (slope changes from -1 to 2). The correct description is the decreasing line of slope -1 reaching (1, 2), then a line of slope 2 for x > 1 continuing upward from near (1, 2).

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