Math 1314 Lab Modul Name _____________________ Show all work for each of the following problems. Simplify and clearly e 2indicate all answers. 1. For f(x) = 3×2 + 5 and g(x) = 7x − 2 , a. Verify: g(x + 2) ≠ g(x) + g(2). b. Find (f – g)(x). c. Using the resulting function in (b), show that (f – g)(2) = f(2) – g(2). (The work should be different for each side of the equation.) f d. Is (fg)(0) = (0)? Explain. g e. Find f(x + h) − f(x) , h ≠ 0. h Math 1314 Lab Module 2 page 2 2. A size 36 shoe in France is size 3.5 in England. A function that converts shoe sizes 3x − 94 in France to those in England is g(x) = . A size 6 shoe in the United States 4 is size 3.5 in England. A function that converts shoe sizes in England to those in the 5 United States is f(x) = x + . 2 a. Use composition of functions to find a function that converts shoe sizes in France to those in the United States. Simplify the function. b. Determine the shoe size in the United States for a size 38 shoe in France. Write the answer in a complete sentence using proper grammar and correct spelling. 3. A dance studio has fixed monthly costs of $1500 that include rent, utilities, insurance, and advertising. The studio charges $60 for each private lesson, but has a variable cost for each lesson of $35 to pay the instructor. a. Write a linear cost function representing the cost to the studio C(x) to hold x private lessons for a given month. b. Write a linear revenue function representing the revenue R(x) for holding x private lessons for the month. c. Write a linear profit function representing the profit P(x) for holding x private lessons for the month. d. Determine the number of private lessons that must be held for the studio to break-even. Math 1314 Lab Module 2 page 3 4. In a study using 50 foreign-language vocabulary words, the learning rate L (in words per minute) was found to depend on the number of words already learned, x, according to the equation L(x) = 20 – 0.4x. a. State the coordinates of the x-intercept. b. State the coordinates of the y-intercept. c. State the slope. d. Graph the linear function for x ≥ 0. (Label the axes completely!!) e. Is the learning rate increasing or decreasing? Explain why your answer makes sense in the context of the problem. Use a complete sentence with proper grammar and correct spelling. Math 1314 Lab Module 2 page 4 5. Suppose that 650 lb of coffee are sold when the price is $4 per pound, and 400 lb are sold at $8 per pound. a. List the data points (use price as the independent variable). b. Find the slope of the line joining the points. c. Interpret the meaning of the slope in the context of this problem. Write the interpretation in a complete sentence with proper grammar and correct spelling. d. Use the point-slope form to write a linear equation for this data. Write the answer in function notation. e. Use this function to predict how much consumers would be willing to buy at a price of $6 per pound. 6. For each final matrix, state the solution. 1 0 0 3 0 1 0 −1 0 0 1 8 1 0 −6 11 0 1 4 7 0 0 0 −2 1 0 2 9 0 1 −8 5 0 0 0 0 Math 1314 Lab Module 2 page 5 7. The University of Texas at Austin has three times as many students enrolled as the University of Miami. The University of California, Berkley has 3,000 more than twice the number of students as the University of Miami. If the three schools have a total enrollment of 96,000 students, what is the enrollment at each school? x= a. Describe what the variables represent: y = z= b. Write the system of linear equations: c. Write the Augmented Matrix for the system of linear equations then solve using the Gauss-Jordan Elimination Method. Show all row operations and at least 5 of the resulting matrices. Use proper notation. Write the solution as an ordered triple, if appropriate. d. Now, answer the question in a complete sentence with proper grammar and correct spelling: What is the enrollment at each school? Math 1314 Lab Module 2 page 6 8. The ABC Ink Company is a small family owned company that sells packages of ink cartridge refills for smartpens. The Xavier set contains one blue ink refill and one black ink refill. The Yvonne set includes two blue ink refills, three black ink refills, and one red ink refill. The Zena set includes four blue ink refills, five black ink refills, and one red ink refill. The company has sold most of its stock and has found that it has only 11 blue ink cartridge refills, 14 black ink cartridge refills, and 3 red ink cartridge refills. How many of each set should the company package to sell in order to use all of the remaining ink cartridges so that there will be none left in inventory. x= a. Describe what the variables represent: y = z= b. Write the system of linear equations: c. Solve the system of equations using Gauss-Jordan Elimination. Show all proper row operations and the resulting matrices. Write the solution as an ordered triple, if appropriate. d. Fill out the table to show the possible combinations of sets the company can package: x y z e. In at least one complete sentence with proper grammar and correct spelling, write the solutions in terms of what the variables represent.
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