1.1 Brutus Bites Back
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1 FUNCTIONS AND THEIR INVERSES Brutus Bites Back A Develop Understanding Task Remember Carlos and Clarita? A couple of years ago, they started earning money by taking care of pets while their owners are away. Due to their amazing mathematical analysis and their loving care of the cats and dogs that they take in, Carlos and Clarita have made their business very successful. To keep the hungry dogs fed, they must regularly buy Brutus Bites, the favorite food of all the dogs. CC BY Franco Vannini Carlos and Clarita have been searching for a new dog food supplier and have identified two possibilities. The Canine Catering Company, located in their town, sells 7 pounds of food for $5. Carlos thought about how much they would pay for a given amount of food and drew this graph: 1. Write the equation of the function that Carlos graphed. 1
2 FUNCTIONS AND THEIR INVERSES 1.1 Clarita thought about how much food they could buy for a given amount of money and drew this graph: 2. Write the equation of the function that Clarita graphed. 3. Write a question that would be most easily answered by Carlos graph. Write a question that would be most easily answered by Clarita s graph. What is the difference between the two questions? 4. What is the relationship between the two functions? How do you know? 5. Use function notation to write the relationship between the functions. 2
3 FUNCTIONS AND THEIR INVERSES 1.1 Looking online, Carlos found a company that will sell 8 pounds of Brutus Bites for $6 plus a flat $5 shipping charge for each order. The company advertises that they will sell any amount of food at the same price per pound. 6. Model the relationship between the price and the amount of food using Carlos approach. 7. Model the relationship between the price and the amount of food using Clarita s approach. 8. What is the relationship between these two functions? How do you know? 9. Use function notation to write the relationship between the functions. 10. Which company should Clarita and Carlos buy their Brutus Bites from? Why? 3
4 FUNCTIONS AND INVERSES 1.1 READY, SET, GO! Name Period Date READY Topic: Inverse operations Inverse operations undo each other. For instance, addition and subtraction are inverse operations. So are multiplication and division. In mathematics, it is often convenient to undo several operations in order to solve for a variable. Solve for x in the following problems. Then complete the statement by identifying the operation you used to undo the equation = 3x Undo multiplication by 3 by x 2. Undo division by 5 by 5 = 2 3. x +17 = 20 Undo add 17 by 4. x = 6 Undo the square root by ( ) 5. 3 x +1 = 2 Undo the cube root by then 6. x 4 = 81 Undo raising x to the 4 th power by ( ) 2 = x 9 Undo squaring by then SET Topic: Linear functions and their inverses Carlos and Clarita have a pet sitting business. When they were trying to decide how many each of dogs and cats they could fit into their yard, they made a table based on the following information. Cat pens require 6 ft 2 of space, while dog runs require 24 ft 2. Carlos and Clarita have up to 360 ft 2 available in the storage shed for pens and runs, while still leaving enough room to move around the cages. They made a table of all of the combinations of cats and dogs they could use to fill the space. They quickly realized that they could fit in 4 cats in the same space as one dog. cats dogs Need help? Visit 4
5 FUNCTIONS AND INVERSES Use the information in the table to write 5 ordered pairs that have cats as the input value and dogs as the output value. 9. Write an explicit equation that shows how many dogs they can accommodate based on how many cats they have. (The number of dogs d will be a function of the number of cats c or! = #(%).) 10. Use the information in the table to write 5 ordered pairs that have dogs as the input value and cats as the output value. 11. Write an explicit equation that shows how many cats they can accommodate based on how many dogs they have. (The number of cats c will be a function of the number of dogs d or % = '(!).) Base your answers in #12 and #13 on the table at the top of the page. 12. Look back at problem 8 and problem 10. Describe how the ordered pairs are different. 13. a) Look back at the equation you wrote in problem 9. Describe the domain for! = #(%). b) Describe the domain for the equation % = '(!) that you wrote in problem 11. c) What is the relationship between them? Need help? Visit 5
6 FUNCTIONS AND INVERSES 1.1 GO Topic: Using function notation to evaluate a function. The functions f ( x), g( x), and h( x) are defined below. f ( x) = x g( x) = 5x 12 h( x) = x 2 + 4x 7 Calculate the indicated function values in the following problems. Simplify your answers. ( ) f ( 2) f ( a) f ( a + b) 14. f ( ) g( 2) g( a) 18. g '() + +) ( ) h( 2) h( a) h( a + b) 22. h Need help? Visit 6
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