Problem 1
Answers:
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Explanation:
The function is continuous despite the fact it's not one whole curve, but rather two pieces. A discrete function would be a function that looks like a scatter plot (see problem 5) since we would have a set of discrete dots or points.
The domain is the set of allowed inputs, which in this case is only x = -3 and x = 2. Since we have so few elements in the domain, we just list them as a set of items.
In contrast, the range has infinitely many items. So we just say "set of all real numbers". The range is the set of all allowed y outputs.
We do not have a function because this graph fails the vertical line test. The vertical line test is where you ask yourself "is it possible to draw a vertical line through more than one point on the curve?". If the answer is "yes", then the graph fails the vertical line test and it's consequently not a function. In this graph, such an event occurs.
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Problem 2
Answers:
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Explanation:
This graph is continuous for similar reasons as problem 1.
The domain stretches from x = -5 to x = 5. We exclude x = -5 because of the open hole at the left endpoint. We include x = 5 because of the closed filled in hole at the right endpoint.
The range is from y = -2 to y = 2, including both endpoints. Despite the open hole at the left endpoint, other points like (5,-2) will help include y = -2 be part of the range.
This graph passes the vertical line test, so we have a function.
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Problem 3
Answers:
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Explanation:
This graph is continuous. The only discrete graph is problem 5.
The domain is the set of all real numbers since there are no gaps, holes, jumps, etc and because the graph stretches to the left and right forever. Any x input works.
The range is the set of all real numbers for similar reasons but the graph stretches forever upward and downward.
We have a function because this graph passes the vertical line test.
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Problem 4
Answers:
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Explanation:
This graph is continuous since it's one single line.
The domain is the set of all real numbers because any x input works. There are no restrictions on x.
The only output possible is y = 3 since all points on this line have y coordinate 3. Therefore the range is the set {3}. It is a singleton set because it only has 1 element in it.
This graph passes the vertical line test, so we have a function.
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Problem 5
Answers:
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Explanation:
We have a discrete graph because we have a set of points rather than a continuous curve connecting infinitely many points.
The domain is the set of x coordinates of each point shown. So that's how we get to {-3, -2, 1, 2, 5}
Similarly, the range is a finite set of values. The range is the set of y coordinates of each point shown. The range is {-5, 0, 1, 4}. Duplicate y values are ignored.
Since none of the points are vertically stacked or lined up, we can say this graph passes the vertical line test. Therefore, we have a discrete function.
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Problem 6
Answers:
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Explanation:
This curve is continuous since we don't have a set of discrete points like the graph in problem 5 shows.
The domain is the set of x values smaller than 4 or equal to 4. We simply say the domain is [tex]x \le 4[/tex]
The range is the set of y values y = 0 or larger, so [tex]y \ge 0[/tex] is the shorthand way of saying it.
This graph passes the vertical line test, so we have a function.
