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Domain and Range from a Graph To determine the domain of a relation, such as the first graph, we need to determine the smallest and largest possible first members of the ordered pairs. In the graph, are we concerned with values of x or of y? |
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We are concerned with x. Will the smallest x be associated with the part of the curve which is furthest left or right? |
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The part which is furthest left. Are there any points on the graph with |
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No. Are there any points on the graph with an x-coordinate of -4.1? |
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No. For this figure, which of the points has the smallest x-coordinate? |
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(-4, -1) What is the x-coordinate of this point? |
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The x-coordinate is -4. What partial information does this give us about the domain? |
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At least, x ³ -4 is a requirement on the domain.Use similar reasoning to determine the point with the largest x-coordinate. |
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The point (5, 4) has the largest x-coordinate. What is the x-coordinate of this point? |
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The x-coordinate is 5. Use this information to describe the domain fully. |
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Domain: {x | -4 £ x £ 5}Now use similar reasoning to determine which points determine the boundaries of the range. |
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The points (-1, -2) and (5, 4) have the extreme values of the y-coordinate, and determine the boundaries of the range. Describe the range. |
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Range: {y | -2 £ y £ 4}Now scroll to Figure 2 in the right hand frame. Let's determine the domain and range for this ellipse. First decide on the smallest value of x for any point on the graph. |
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The left vertex ("end") of the ellipse has the smallest x-coordinate. Read that coordinate. |
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The smallest x-coordinate is -1. Why isn't it 0? |
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x = 0 is the smallest x-coordinate where the graph crosses the x-axis, but we are looking for the smallest x-coordinate anywhere on the graph. The point at (-1, -2) has the smallest x-coordinate. Find the largest x-coordinate for any point on the graph. |
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The point at the right vertex ("end") has an x-coordinate of 7. No points are farther to the right, or have a larger x-coordinate. Why isn't x = 6 the largest value of x? |
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x = 6 is the largest value of x for a point on the x-axis, but we want the largest x-coordinate anywhere. State the domain for this graph. |
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{x | -1 £ x £ 7}Let's do a similar analysis to find the Range. What is the smallest y-coordinate of any point on the graph? |
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The point (3, -5) is on the graph, so the smallest y-coordinate is -5. Find the largest y-coordinate of any point on the graph. |
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The point (3, 1) is on the graph, so the largest y-coordinate is 1. State the range for the ellipse. |
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The range is { y | -5 £ y £ 1 }Scroll to Figure 3 in the right hand frame. Here's another graph. We'll find the domain and range. First, though, let's review: what do the arrow points on the "ends" of the graph indicate? |
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The arrow points indicate that the graph continues without limit in the indicated direction. What does the arrow point on the left end of the graph tell us about the smallest value of the x-coordinate of points on the graph? |
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The arrow indicates that points on that part of the curve are located further and further to the left (and higher). The x-coordinates of these points will be more and more negative. Does this mean that the x-coordinates get smaller and smaller? |
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Yes. What about the arrow point at the right end of the graph? What does it indicate about the x-coordinates of points on that part of the graph? |
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The x-coordinates of points on the graph increase without limit. Use this information to describe the domain. |
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The domain is all the real numbers, or Notice that we use only the < sign to indicate the behavior near - ¥ and ¥ .Now consider the range. Determine the smallest y-coordinate of any point on the graph. |
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At the low point (2, -1), the y-coordinate is -1. What about the largest y-coordinate? |
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The arrows on the ends of the graph indicate that y increases without limit. Use this information to describe the range of this function. |
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The range is { y | -1 £ y < ¥ }. Notice that we use only the < sign indicate the growth without limit.Now scroll to Figure 4 in the right hand frame. |
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The dots mean that the point is included in the graph. Determine the smallest value of x on this graph. |
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The smallest value of x is -8. Find the largest value of x on the graph. |
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The largest value of x is +11. Is this enough information to specify the domain? |
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No. It was before. Why not now? |
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This graph is different. It has a gap in it. For example, there is no point on the graph with an x-coordinate of -0.5. Use this hint to specify the domain. |
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The domain is { x | -8 £ x £ -2 or 1 £ x £ 11 }.Now let's find the range. What is the smallest value of y-coordinate of any point on the graph? |
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The smallest y-coordinate is y = -2. Find the largest y-coordinate of any point on the graph. |
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The largest y-coordinate is y = 9. Is there a gap to account for? |
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Yes, between 2 and 6. Express the range. |
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The range is { y | -2 £ y £ 2 or 6 £ y £ 9 }. |
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