Determine whether each relation represents y as a function of x. 1. 3x + 7y = 21 This equation represents y as a function of x, because for every x-value there is exactly one corresponding y-value. function 2. This equation represents y as a function of x, because for every x-value there is exactly one corresponding y-value. function 3. This is not a function because the graph does not pass the Vertical Line Test. not a function esolutions Manual - Powered by Cognero Page 1
4. This represents a function because the graph passes the Vertical Line Test. function 5. Evaluate f (2) for. Because 2 2, use f (x) = x + 10. 12 esolutions Manual - Powered by Cognero Page 2
6. SPORTS During a baseball game, a batter pops up the ball to the infield. After t seconds, the height of the ball in feet can be modeled by h(t) = 16t 2 + 50t + 5. a. What is the baseball's height after 3 seconds? b. What is the relevant domain of this function? Explain your reasoning. a. Find h(3). Therefore, the height of the baseball after 3 seconds was 11 feet. b. The relevant domain represents the interval of time beginning when the ball was hit and ending when it reached the ground. Because time cannot be negative, t 0. To determine the time at which the ball landed, set h(t) equal to 0 and use the Quadratic Formula to solve for t. Because the ball was hit at t = 0 and landed at t 3.22, the domain of the function is {t 0 t 3.22, x R} or [0, 3.22]. a. 11 ft b. [0, 3.22]; Sample answer: Because time cannot be negative and the height of the ball is 0 when t = 3.22, 0 t 3.22. esolutions Manual - Powered by Cognero Page 3
Use the graph of each function to find its y-intercept and zero(s). Then find these values algebraically. 7. From the graph, it appears that f (x) intersects the y-axis at approximately (0, 0), so the y-intercept is 0. Find f (0). Therefore, the y-intercept is 0. From the graph, the x-intercepts appear to be at about 4, 0, and 4. Let f (x) = 0 and solve for x. Therefore, the zeros of f are 4, 0, and 4. y-intercept: 0; zeros: 4, 0, 4; esolutions Manual - Powered by Cognero Page 4
8. From the graph, it appears that f (x) intersects the y-axis at approximately (0, 5), so the y-intercept is 5. Find f (0). Therefore, the y-intercept is 5. From the graph, there appears to be an x-intercept at about 25. Let f (x) = 0 and solve for x. Therefore, the zero of f is 25. y-intercept: 5; zero: 25; Use the graph of h to find the domain and range of each function. 9. The arrow on the right side of the graph indicates that the graph will continue without bound to infinity. The graph begins at (0, 0). Therefore, the domain and range of h are (0, ). D = [0, ), R = [0, ) esolutions Manual - Powered by Cognero Page 5
10. The domain includes every value of x and the range includes only integer values of y. Therefore, the domain is and the range is. Determine whether each function is continuous at x = 5. Justify your answer using the continuity test. 11. f (x) = When x = 5, or. The function is not defined when x = 5, and therefore is not continuous at x = 5. Discontinuous at x = 5; f (x) is undefined when x = 5. 12. f (x) = f(5) = 2.5, = 2.5, = 2.5, and = f (5). The function is continuous at x = 5. Continuous at x = 5; f(5) = 2.5,, and. esolutions Manual - Powered by Cognero Page 6
Use the graph of each function to describe its end behavior. 13. From the graph, it appears that f (x) investigate the function as x increases. as x, and f (x) as x. Construct a table of values to x f (x) 1000 1 10 9 100 1.02 10 6 0 4 100 9.8 10 5 1000 1 10 9 From the graph, it appears that f (x) as x, and f (x) as x. 14. From the graph, it appears that f (x) 5 as x investigate the function as x increases., and f (x) 5 as x. Construct a table of values to x f (x) 1000 4.999 100 4.99 0 undefined 100 5.01 1000 5.001 From the graph, it appears that f (x) 5 as x, and f (x) 5 as x. esolutions Manual - Powered by Cognero Page 7
15. MULTIPLE CHOICE The graph of f (x) contains a(n) discontinuity at x = 1.5. A undefined B infinite C jump D removable f(3) =3 and, so f (x) has a removable discontinuity at x = 3. Therefore, the correct answer is D. D esolutions Manual - Powered by Cognero Page 8
Use the graph of each function to estimate intervals to the nearest 0.5 unit on which the function is increasing, decreasing, or constant. 16. When the graph is viewed from left to right, we can estimate that the graph of f is increasing on ( increasing on (3, )., 3) and Create a table of values using x-values in each interval. (, 3): x f (x) 8 117 5 60 3 32 0 5 1 0 (3, ): x f (x) 2 3 5 0 8 21 12 77 The tables support the conjecture that the graph of f is increasing on (, 3) and increasing on (3, ). f is decreasing on (, 3) and increasing on (3, ). esolutions Manual - Powered by Cognero Page 9
17. When the graph is viewed from left to right, we can estimate that the graph of f is increasing on ( decreasing on ( 2, 1.5), and increasing on (1.5, )., 2), Create a table of values using x-values in each interval. (, 2): x f (x) 10 823 8 387 5 63 3 3 1 5 ( 2, 1.5): x f (x) 1.5 7.875 1 5 0 3 1 9 1.25 9.484 (1.5, ): x f (x) 2 7 3 9 5 107 8 509 12 1773 The tables support the conjecture that the graph of f is increasing on ( increasing on (1.5, )., 2), decreasing on ( 2, 1.5), and f is increasing on (, 2), decreasing on ( 2, 1.5), and increasing on (1.5, ). esolutions Manual - Powered by Cognero Page 10
18. PHYSICS The height of an object dropped from 80 feet above the ground after t seconds is f (t) = 16t 2 + 80. What is the average speed for the object during the first 2 seconds after it is dropped? The average speed for the object is 32 ft/s downward. 32 ft/s esolutions Manual - Powered by Cognero Page 11