SECTION 2.1 Functions 61 OBJECTIVES 1 Describe a Relation (p. 61) 2 Determine Whether a Relation Represents a Function (p. 63) 3 Use Function Notation; Find the Value of a Function (p. 65) 4 Find the Difference Quotient of a Function (p. 68) 5 Find the Domain of a Function Defined by an Equation (p. 69) 6 Form the Sum, Difference, Product, and Quotient of Two Functions (p. 71) 2.1 Functions Now Work the ‘Are You Prepared?’ problems on page 73. • Intervals (Section A.9, pp. A72–A74) • Solving Inequalities (Section A.9, pp. A76–A79) • Evaluating Algebraic Expressions, Domain of a Variable (Section A.1, pp. A6–A7) • Rationalizing Denominators and Numerators (Section A.10 , pp. A85 – A86 ) PREPARING FOR THIS SECTION Before getting started, review the following: 1 Describe a Relation Often there are situations where one variable is somehow linked to another variable. For example, the price of a gallon of gas is linked to the price of a barrel of oil. A person can be associated to her telephone number(s). The volume V of a sphere depends on its radius R. The force F exerted by an object corresponds to its acceleration a. These are examples of a relation , a correspondence between two sets called the domain and the range . DEFINITION Relation A relation is a correspondence between two sets: a set X, called the domain , and a set Y, called the range . In a relation, each element from the domain corresponds to at least one element from the range. If x is an element of the domain and y is an element of the range, and if a relation exists from x to y, then we say that y corresponds to x or that y depends on x, and we write →x y . It is often helpful to think of x as the input and y as the output of the relation. See Figure 1. Suppose an astronaut standing on the Moon throws a rock 20 meters up and starts a stopwatch as the rock begins to fall back down. The astronaut measures the height of the rock at 1, 2, 2.5, 3, 4, and 5 seconds and obtains heights of 19.2, 16.8, 15, 12.8, 7.2, and 0 meters, respectively. This is an example of a relation expressed verbally . The domain of the relation is the set { } 0, 1, 2, 2.5, 3, 4, 5 and the range of the relation is the set { } 20, 19.2, 16.8, 15, 12.8, 7.2, 0 . The astronaut could also express this relation numerically , graphically , or algebraically . The relation can be expressed numerically using a table of numbers, as in Table 1, or by using a set of ordered pairs . Using ordered pairs, the relation is ( ) ( ) ( ) ( ) ( ) ( ) ( ) { } 0, 20 , 1, 19.2 , 2, 16.8 , 2.5, 15 , 3, 12.8 , 4, 7.2 , 5, 0 where the first element of each pair denotes the time and the second element denotes the height. Suppose x represents the number of seconds on the stopwatch and y represents the height of the rock in meters. Then the relation can be expressed graphically by plotting the points ( ) x y , . See Figure 2 on the next page. The relation can be represented as a mapping by drawing an arrow from an element in the domain to the corresponding element in the range. See Figure 3 on the next page. Figure 1 Relation x S y Output y4 y1 y2 y3 x4 x1 x2 Input x3 Time (in seconds) Height (in meters) 0 20 1 19.2 2 16.8 2.5 15 3 12.8 4 7.2 5 0 Table 1
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