SECTION 2.1 Functions 63 2 Determine Whether a Relation Represents a Function Look back at the relation involving the height of a rock on the Moon described at the beginning of the section. Notice that each input, time, corresponds to exactly one output, height. Given a time, you could tell the exact height of the rock. But that is not the case with the price of stamps. Given the year 2024, you cannot determine the price of a stamp with certainty. It could be $0.68, or it could be $0.73. Consider the mapping of the relation in Figure 6. It shows a correspondence between a substance and its specific heat. Notice that for each substance you can tell its specific heat with certainty. The relation associating the time to the height of the rock is a function , and the relation associating a given substance to its specific heat is a function. But the relation associating the year to the price of a First Class postage stamp is not a function. To be a function, each input must correspond to exactly one output. DEFINITION Function Let X and Y be two nonempty sets.* A function from X into Y is a relation that associates with each element of X exactly one element of Y. The set X is called the domain of the function. The set Y is called the codomain of the function. For each element x in X, the corresponding element y in Y is called the value of the function at x, or the image of x.The set of all images of the elements in the domain is called the range of the function. See Figure 7. Since there may be some elements in Y that are not the image of some x in X, it follows that the range of a function may be a proper subset of the codomain Y, as shown in Figure 7. The idea behind a function is its certainty. If an input is given, we can use the function to determine the output. This is not always possible if a relation is not a function. The requirement of “one output” provides a predictable behavior that is important when using mathematics to model or analyze the real world. It allows doctors to know exactly how much medicine to give a patient, an engineer to determine the material to use in construction, a store manager to choose how many units to keep in stock, etc. Determining Whether a Relation Given by a Mapping Is a Function For each relation, state the domain and range. Then determine whether the relation is a function. (a) See Figure 8. This relation shows a correspondence between an item on McDonald’s $1-$2-$3 menu and its price. Source: McDonald’s Corporation, 2024. (b) See Figure 9. This relation shows a correspondence between activity level and daily calories needed for an individual with a basal metabolic rate (BMR) of 1975 kilocalories per day (kcal/day). (c) See Figure 10. This relation shows a correspondence between gestation period (in days) and life expectancy (in years) for five animal species. EXAMPLE 2 Figure 9 Sedentary Activity Level Light Intense Moderate 2370 2716 3407 3061 Daily Calories Figure 10 Gestation (days) 122 201 284 240 5 8 12 20 15 Life Expectancy (years) Figure 8 Sweet tea Menu Item Hamburger, Decaf coffee Double Cheeseburger, McChicken $1 $3 $2 Price *The sets X and Y will usually be sets of real numbers, in which case a (real) function results.The two sets can also be sets of complex numbers, and then we have defined a complex function. In the broad definition (proposed by Lejeune Dirichlet), X and Y can be any two sets. Figure 6 Specific heat of some common substances Air Substance Lead Graphite Water Copper 0.128 0.387 0.711 4.18 1.00 Specific Heat (J/g5C) Figure 7 Domain Y y Range X x (continued)
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