226 CHAPTER 2 Graphs and Functions SOLUTION (a) In the defining equation (or rule), y = x + 4, y is always found by adding 4 to x. Thus, each value of x corresponds to just one value of y, and the relation defines a function. The variable x can represent any real number, so the domain is 5x x is a real number6, or 1-∞, ∞2. Because y is always 4 more than x, y also may be any real number, and so the range is 1-∞, ∞2. (b) For any choice of x in the domain of y = 22x - 1, there is exactly one corresponding value for y (the radical is a nonnegative number), so this equation defines a function. The quantity under the radical sign cannot be negative—that is, 2x - 1 must be greater than or equal to 0. 2x - 1 Ú 0 Solve the inequality. 2x Ú 1 Add 1. x Ú 1 2 Divide by 2. The domain of the function is C 1 2 , ∞B. Because the radical must represent a nonnegative number, as x takes values greater than or equal to 1 2 , the range is 5y y Ú 06, or 30, ∞2. See Figure 22. (c) The ordered pairs 116, 42 and 116, -42 both satisfy the equation y2 = x. There exists at least one value of x—for example, 16—that corresponds to two values of y, 4 and -4, so this equation does not define a function. Because x is equal to the square of y, the values of x must always be nonnegative. The domain of the relation is 30, ∞2. Any real number can be squared, so the range of the relation is 1-∞, ∞2. See Figure 23. (d) By definition, y is a function of x if every value of x leads to exactly one value of y. Substituting a particular value of x, say 1, into y … x - 1 corresponds to many values of y. The ordered pairs 11, 02, 11, -12, 11, -22, 11, -32, and so on all satisfy the inequality, so y is not a function of x here. Any number can be used for x or for y, so the domain and the range of this relation are both the set of real numbers, 1-∞, ∞2. 2 4 6 3 x y 1 2 0 Domain Range y = !2x – 1 Figure 22 4 8 12 16 –4 –2 2 4 x y y2 = x Domain Range 0 Figure 23 The vertical line test is a simple method for identifying a function defined by a graph. Deciding whether a relation defined by an equation or an inequality is a function, as well as determining the domain and range, is more difficult. The next example gives some hints that may help. EXAMPLE 5 Identifying Functions, Domains, and Ranges Determine whether each relation defines y as a function of x, and give the domain and range. (a) y = x + 4 (b) y = 22x - 1 (c) y2 = x (d) y … x - 1 (e) y = 5 x - 1
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