5 5 www.mhhe.com/messersmith Definition/Procedure Example The range of a relation is the set of all values of the second coordinates in the set of ordered pairs. (p. 640) A function is a relation in which each element of the domain corresponds to exactly one element of the range. (p. 640) Alternative defi nition: A relation is a function if each x-value corresponds to one y-value. (p. 641) In a), the domain is {4, 1, 3, 5}, and the range is {12, 3, 9, 15}. In b), the domain is {4, 9, 11}, and the range is {1, 6, 17}. The relation in a) is a function. The relation in b) is not a function. The Vertical Line Test If no vertical line can be drawn through a graph that intersects the graph more than once, then the graph represents a function. If a vertical line can be drawn that intersects the graph more than once, then the graph does not represent a function. (p. 642) y f (x) is called function notation and it is read as “y equals f of x.” Finding a function value means evaluating the function for the given value of the variable. (p. 645) This graph represents a function. Anywhere a vertical line is drawn, it will intersect the graph only once. This graph is not the graph of a function. A vertical line can be drawn so that it intersects the graph more than once. If f (x) 2x 9, fi nd f (4). Substitute 4 for x and evaluate. f (4) 2(4) 9 8 9 1 Therefore, f (4) 1. When determining the domain of a relation, it can be helpful to keep these tips in mind. 1) Ask yourself, “Is there any number that cannot be substituted for x?” 2) If x is in the denominator of a fraction, determine what value of x will make the denominator equal 0 by setting the expression equal to zero. Solve for x. This x-value is not in the domain. (p. 644) Determine the domain of f (x) 6 x 3 . x 3 0 Set the denominator 0. x 3 Solve. When x 3, the denominator of f (x) 6 x 3 equals zero. The domain contains all real numbers except 3. The domain of the function is (q, 3) ´ (3, q). x y 5 5 5 x y 5 5 5 A relation is any set of ordered pairs. A relation can also be represented as a correspondence or mapping from one set to another. (p. 640) The domain of a relation is the set of values of the fi rst coordinates in the set of ordered pairs. Relations: a) {(4, 12), (1, 3), (3, 9), (5, 15)} b) 49 11 16 17 10.5 Introduction to Functions 656 CHAPTER 10 Quadratic Equations
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