62 CHAPTER 2 Functions and Their Graphs Finally, from physics, the relation can be expressed algebraically using the equation = − y x 20 0.8 2 Figure 2 2 4 6 x y 15 20 5 10 (0, 20) (1, 19.2) (2, 16.8) (2.5, 15) (3, 12.8) (4, 7.2) (5, 0) Time (in seconds) Height (in meters) Figure 3 1 2 2.5 3 4 5 x 20 19.2 16.8 15 12.8 7.2 0 y Time (in seconds) Height (in meters) 0 Solution The relation establishes a correspondence between the input, year, and the output, price of a First Class U.S. postage stamp. (a) The domain of the relation is { } 2019, 2020, 2021, 2022, 2023, 2024 . The range of the relation is { } $0.55, $0.58, $0.60, $0.63, $0.66, $0.68, $0.73 . (b) The relation expressed as a set of ordered pairs is ( ) ( ) { 2019, $0.55 , 2020, $0.55 , ( ) ( ) ( ) ( )}( ) 2021, $0.55 , 2021, $0.58 , 2022, $0.60 , 2023, $0.63 2023, $0.66 , ( ) ( ) 2024, $0.68 , 2024, $0.73 . (c) See Figure 4 for the relation expressed as a mapping, using t for year and p for price. (d) Figure 5 shows a graph of the relation. Describing a Relation A verbal description of a relation is given below. The price of First Class U.S. postage stamps has changed over the years. To mail a letter in 2019 and 2020 the cost was $0.55. In 2021, the cost was $0.58 and in 2022, the cost was $0.60. For the first part of 2023, the cost was $0.63 and $0.66 for the rest of the year. For the first part of 2024, the cost was $0.68 and $0.73 for the rest of the year. Using year as input and price as output, (a) What are the domain and the range of the relation? (b) Express the relation as a set of ordered pairs. (c) Express the relation as a mapping. (d) Express the relation as a graph. EXAMPLE 1 Figure 4 First Class stamp price 2019 Year, t Price, p 2020 2021 2022 2023 $0.55 $0.58 $0.60 $0.63 $0.66 2024 $0.68 $0.73 Figure 5 First Class stamp price (2019—2024) 0 Year Price (in dollars) 0.55 0.60 0.65 0.70 0.75 2019 2020 2021 2022 2023 2024 Now Work PROBLEM 17 Katherine Johnson (1918–2020) Katherine Johnson was an American mathematician whose calculations of orbital mechanics as a NASA employee were critical to the success of the U.S. space program. Without the aid of any computers, Katherine used geometry and advanced calculus along with physics to figure out the paths for a spacecraft to orbit Earth and to land crewed missions on the Moon. She was the main character in the movie Hidden Figures Credit: Science History Images/Alamy Stock Photo
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