224 CHAPTER 2 Graphs and Functions SOLUTION (a) The domain is the set of x-values, 5-1, 0, 1, 4, 56. The range is the set of y-values, 5-3, -1, 1, 26. (b) The x-values of the points on the graph include all numbers between -4 and 4, inclusive. The y-values include all numbers between -6 and 6, inclusive. The domain is 3-4, 44. Use interval notation. The range is 3-6, 64. (c) The arrowheads indicate that the line extends indefinitely left and right, as well as up and down. Therefore, both the domain and the range include all real numbers, which is written 1-∞, ∞2. (d) The arrowheads indicate that the graph extends indefinitely left and right, as well as upward. The domain is 1-∞, ∞2. Because there is a least y-value, -3, the range includes all numbers greater than or equal to -3, written 3-3, ∞2. S Now Try Exercises 27 and 29. Relations are often defined by equations, such as y = 2x + 3 and y2 = x, so we must sometimes determine the domain of a relation from its equation. In this text, we assume the following agreement on the domain of a relation. Agreement on Domain Unless specified otherwise, the domain of a relation is assumed to be all real numbers that produce real numbers when substituted for the independent variable. To illustrate this agreement, because any real number can be used as a replacement for x in y = 2x + 3, the domain of this function is the set of all real numbers. x y 0 (0, –1) (4, –3) (–1, 1) (1, 2) (5, 2) x y 0 –4 –6 6 4 Range Domain x y 0 x y 0 2 –3 EXAMPLE 3 Finding Domains and Ranges from Graphs Give the domain and range of each relation. (a) (b) (c) (d)
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