Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Abuse of notation wikipedia , lookup
Big O notation wikipedia , lookup
Elementary mathematics wikipedia , lookup
History of the function concept wikipedia , lookup
Proofs of Fermat's little theorem wikipedia , lookup
Continuous function wikipedia , lookup
Non-standard calculus wikipedia , lookup
Multiple integral wikipedia , lookup
Function (mathematics) wikipedia , lookup
Name: Mathematical Investigations An Opportunity to Demonstrate Knowledge – Without your calculator 1. Generalize each pattern: a. {(2, 7), (1, 4), (0,1), (1, 2), (2, 5), (3, 8)} (x, y) b. y 3x 1 (3,5),1 , (3,5),1 , (2,5),1 , (2,10), 2 , (3,15),3 , (20,30),10 , ( x, y), z z greatest common factor of x and y Fcn Exam p. 1 Rev. F11 For problems 3, if the representation is a function, give its domain and range. If it is not a function, do not give its domain and range; instead, indicate why it is not a function. Be specific! 3. (3, 2), (7, 4), (9,5), (7,11), (0,13) Function? Yes No X The element 7 in the domain correspond to two different elements in the range(4 and 11). By definition of a function, each element in the domain must correspond to a unique element in the range. 4. 5. If f = { (–4,6), (–1,3), (0,5), (2,7), (7,–4) } and g = { (–4,2), (0,–4) ,(7,–1), (5,7) (6, 9)} a. Determine the function: f g = {(4, 7), (0, 6), (7,3), (5, 4)} b. Determine f g (7) = 3 c. If f g ( x) 4 , then g ( x ) d. Determine the function f 7___ and x = 5 . f = {(2, 4), (7, 6)} x4 . Give your answer using interval notation and graph 2x 6 your answer on the number line. Find the domain of h( x) x4 0. 2x 6 The critical points are: x = 4, 3 . We label these points on the number line using an open circle for x = 3 since this is not in the domain, and a closed circle for x = 4. The critical points divide (partition is a nice word) the number line into three sections. Numbers less than x = 3 work in the inequality, as do numbers greater than (or equal to) x = 4 . We need to solve Domain = (, 3) [4, ) in interval notation. Fcn Exam p. 2 Rev. F11 6. If f ( x) x 2 2 x 48 , a. find all a such that f a 0 . f (a) a 2 2a 48 (a 6)(a 8), so f (a) 0 a 6 or 8 b. find all b such that f b 6 = 0. In order for f b 6 = 0, we must have (from part a), b 6 6 or b 6 8. Therefore, b 0 or b 14. 7. Given the function f (t ) (t 2,3t 5) , a. Find an equation for y in terms of x. (that is, eliminate the parameter t). x t 2 t x 2. therefore, y 3t 5 3( x 2) 5 y 3x 1 b. Write the domain and range of f. Domain = {t t is any real number} 8. Range = {( x, y) x and y are real numbers} Suppose that Rachel is renting calculator batteries where the fixed cost of the rental is $2 plus $0.25 per minute for each minute of use. Since she needs them back by the end of class, batteries can only be rented out for 15 minutes. Note: fractions of a minute generate a charge for an additional full minute. a. Write an equation that gives the cost of rental, y, in terms of the number of minutes the batteries are rented. y .25t 2 , t represents the rental time in minutes. b. Find the domain and range of the function you found in part a. Domain = {t 0 t 15} Range = {2.25, 2.50,....5.75} Fcn Exam p. 3 Rev. F11 9. Given f ( x) x 2 5 and g ( x) x 4 . a. Determine f g ( x) = b. Give the domain of f g ( x) . Domain = {x x 4} c. Use the axes below to sketch the graph of f g ( x) . x4 2 5 x 1 , provided x 4 Fcn Exam p. 4 Rev. F11