ANSWERS TO STUDENT CHECKS Student Check 1 a. Domain = {-3, -2, -1, 0, 1} Range = {-27, -8, -1, 0, 1} b. Domain = {1869, 1919, 1969, 1989, 1999, 2005, 2006, 2007} Range = {563, 1041, 2525, 3535, 4084, 4276, 4314, 4352} c. Domain = {AirTran, Alaska, American, Continental, Delta, JetBlue, Northwest, Southwest, United, US Airways}, Range = {2.5, 3.2, 3.4, 4.3, 7.5, 7.9, 10.3, 13.8, 13.9, 14.2} Student Check 2 a. Domain = (-∞, ∞) and Range = {-2} b. Domain = (-∞, ∞) and Range = (-∞, 3 c. Domain = -4, ∞) and Range = 0, ∞) Student Check 3 a. Domain = (-∞, ∞) b. Domain = (-∞, -7) ∪ (-7, ∞) c. Domain = 2, ∞) Student Check 4 a. = {0, 1, 2, 3, . . .} b. (0, 9) SUMMARY OF KEY CONCEPTS 1. The domain of a function is the set of x-values for which the function is defined. The range is the set of corresponding y-values. In a set, table, or mapping, the x- and y-values are listed explicitly. To write the domain and range, list the appropriate values in a set. 2. To find the domain and range from a graph, state the intervals of x and y, respectively, that contain the graph. The domain is found by reading the graph horizontally from left to right and the range is found by reading the graph vertically from the bottom up. 3. The domain of a function that is represented algebraically is the set of real numbers that make the function defined. a. If the function contains a fraction with variables in the denominator, exclude the values that make the denominator zero. b. If the function contains a square root, exclude the values that make the expression inside the square root negative. 4. When dealing with real-world applications, the domain consists not only of the values that make the function defined, but also the values that are reasonable. For example, a negative number may make the function defined but it may not make sense in the context of the application. GRAPHING CALCULATOR SKILLS The graphing calculator can assist us in determining values for the domain of a function that is represented algebraically. We input the function into the calculator and then graph it or examine the table of ordered pairs to determine input values that make the function defi ned or undefi ned. Example 1: Find the domain of f (x)= 2 x-3 . Solution: Y= 2 4 ( X,T,u,n 2 3 ) x 3 GRAPH 2nd GRAPH Note x 5 3 makes the function undefined as evidenced by the error message in the Y1-column of the table and also the gap in the graph. So, the graph and table show us that the domain is (-∞, 3) (3, ∞). Example 2: Find the domain of f (x) =!x 2 4. Solution: Y= 2nd x2 X,T,u,n 2 4 ) GRAPH 2nd GRAPH Note that the graph shows the leftmost x-value is 4 and it extends indefinitely to the right. The table shows an error message for all values less than 4, which means these numbers are not included in the domain. So, the table and graph show us that the domain is 4, ∞). 196 Chapter 3 Graphs, Relations, and Functions
hendricks_intermediate_algebra_1e_ch1_3
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