223 2.3 Functions On this particular day, an input of pumping 7.870 gallons of gasoline led to an output of $29.58 from the purchaser’s wallet. This is an example of a function whose domain consists of numbers of gallons pumped, and whose range consists of amounts from the purchaser’s wallet. Dividing the dollar amount by the number of gallons pumped gives the exact price of gasoline that day. Use a calculator to check this. Was this pump fair? (Later we will see that this price is an example of the slope m of a linear function of the form y = mx.) Domain and Range We now consider two important concepts concerning relations. Domain and Range For every relation consisting of a set of ordered pairs 1x, y2, there are two important sets of elements. • The set of all values of the independent variable x is the domain. • The set of all values of the dependent variable y is the range. EXAMPLE 2 Finding Domains and Ranges of Relations Give the domain and range of each relation. Determine whether the relation defines a function. (a) 513, -12, 14, 22, 14, 52, 16, 826 (b) 4 100 6 200 7 300 –3 x y -5 2 0 2 5 2 (c) SOLUTION (a) The domain is the set of x-values, 53, 4, 66. The range is the set of y-values, 5-1, 2, 5, 86. This relation is not a function because the same x-value, 4, is paired with two different y-values, 2 and 5. (b) The domain is 54, 6, 7, -36 and the range is 5100, 200, 3006. This mapping defines a function. Each x-value corresponds to exactly one y-value. (c) This relation is a set of ordered pairs, so the domain is the set of x-values 5-5, 0, 56 and the range is the set of y-values 526. The table defines a function because each distinct x-value corresponds to exactly one y-value (even though it is the same y-value). S Now Try Exercises 19, 21, and 23. In a function, there is exactly one value of the dependent variable, the second component, for each value of the independent variable, the first component. y = 2x 4 (Input x) 8 (Output y) Function machine NOTE Another way to think of a function relationship is to think of the independent variable as an input and of the dependent variable as an output. This is illustrated by the input-output (function) machine for the function y = 2x.
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