Answer:
- upper right
- see 2nd attachment
- upper left
Explanation:
1. The answer choices leave you with two decisions:
- where does the solid dot go?
- what is the slope of the curve to the right of x = 3?
The first expression in the piecewise definition says ≤ 1, so the solid dot goes on that piece. (The key being "or equal to".) This narrows the choices to the top two diagrams.
The second expression tells you y=x for x > 1. The graph of this is a line with a positive slope, not a horizontal line. This narrows the choice to the upper-right diagram.
2. The boundary of each segment of the piecewise function is at x=3. Both parts of the function definition agree that y=-6 when x=3. When you plot this, you can start at that point (3, -6) and plot rays in both directions.
To the left of that point, the ray will have a slope of -1/2. That is, it will rise 1 unit for each 2 units to the left.
To the right of that point, the ray will have a slope of -2. It will fall 2 units for each 1 unit to the right. The second attachment shows the graph.
3. Here, you're asked to identify the graph of the ceiling function shifted one unit to the left. (The +1 in x+1 means the parent function is shifted one unit left.) The ceiling function (ceiling(x)) gives the smallest integer equal to or larger than a given number. That is, for 0, the value is 0, but for 0.0001, the value is 1. The graph is a stair-step with solid dots on the right ends of the steps.
So, the graph of ceiling(x+1) will have a solid dot at (0, 1) and an open dot at (0, 2) with a line extending to the right to a solid dot at (1, 2). This matches the upper left graph.