Other Functions and their Inverses

CHAPTER Other Functions and their Inverses Water tanks have been used throughout human history to store water for consumption. Many municipal water tanks are placed on top of towers so that water drawn from them will be forced by gravity to flow through pipes at high pressure, reducing the need for water pumps. You will use absolute value equations to understand the flow of water in and out of a municipal water tank..1 Pieces and Absolute Absolute Value Functions as Piecewise Functions p. 405.2 Properties of Absolute Value Functions Domain, Range, Vertex, Axis of Symmetry, Zeros, and Intercepts p. 411.3 Taxes and Taxis Properties of Piecewise Functions p. 419.4 Tanks a Lock Solving Absolute Value Equations and Inequalities p. 427.5 We ve Got the Power Power Functions and Inverses p. 433.6 Back and Forth Inverses p. 445 Chapter Other Functions and their Inverses 403

404 Chapter Other Functions and their Inverses

.1 Pieces and Absolute Absolute Value Functions as Piecewise Functions Objectives In this lesson, you will: Graph absolute value functions. Define absolute value functions as piecewise functions. Key Term piecewise functions Problem 1 1. Graph the function f(x) x 3. 2. Write a linear function g(x) whose graph is the same as f for x 3. Graph g(x) on the same grid using a different color. 3. Write a linear function h(x) whose graph is the same as f for x 3. Graph h(x) on the same grid using a different color. 4. Describe f(x) in terms of g(x) and h(x). Lesson.1 Absolute Value Functions as Piecewise Functions 405

5. A piecewise function is a function that has different rules for different parts of its domain. Each element of the domain is associated with one and only one rule. Define f(x) x 3 as a piecewise function using g(x) and h(x). Problem 2 1. Consider the function f(x) 2x 1. a. Graph the function f(x). b. Write a linear function g(x) whose graph is the same as f for the left branch of the function. Over what interval of x is g(x) the same as f(x)? Graph g(x) on the same grid using a different color. c. Write a linear function h(x) whose graph is the same as f for the right branch of the function. Over what interval of x is h(x) the same as f(x)? Graph h(x) on the same grid using a different color. d. Define f(x) 2x 1 as a piecewise function using g(x) and h(x). 406 Chapter Other Functions and their Inverses

2. Consider the function f(x) 3 x 6 1. a. Graph the function f(x). b. Write a linear function g(x) whose graph is the same as f for the left branch of the function. Over what interval of x is g(x) the same as f(x)? Graph g(x) on the same grid using a different color. c. Write a linear function h(x) whose graph is the same as f for the right branch of the function. Over what interval of x is h(x) the same as f(x)? Graph h(x) on the same grid using a different color. d. Define f(x) 3 x 6 1 as a piecewise function using g(x) and h(x). 3. Consider the function f(x) 2 x 2 3. a. Graph the function f(x). Lesson.1 Absolute Value Functions as Piecewise Functions 407

b. Write a linear function g(x) whose graph is the same as f for the left branch of the function. Over what interval of x is g(x) the same as f(x)? Graph g(x) on the same grid using a different color. c. Write a linear function h(x) whose graph is the same as f for the right branch of the function. Over what interval of x is h(x) the same as f(x)? Graph h(x) on the same grid using a different color. d. Define f(x) 2 x 2 3 as a piecewise function using g(x) and h(x). Problem 3 1. Rewrite each absolute value function as a piecewise function. Then, graph the function. a. f(x) 2x 4 1 408 Chapter Other Functions and their Inverses

b. f(x) 3 2x 1 2 2. Graph each piecewise function. Then, rewrite the piecewise function as an absolute value function. g(x) x 2 for x (, 0 a. f(x) h(x) x 2 for x 0, ) Lesson.1 Absolute Value Functions as Piecewise Functions 409

b. h(x) 4x 2 f(x) g(x) 4x 7 for x (, 1 for x 1, ) 3. Write an equation for the absolute value function shown in the graph. y 4 3 2 1 8 7 6 5 4 3 2 1 1 x 2 3 4 Be prepared to share your methods and solutions. 4 Chapter Other Functions and their Inverses

.2 Properties of Absolute Value Functions Domain, Range, Vertex, Axis of Symmetry, Zeros, and Intercepts Objectives In this lesson, you will: Determine the domain and range of absolute value functions. Determine the vertex and axis of symmetry of absolute value functions. Determine the x- and y-intercepts of absolute value functions. Determine extreme points of absolute value functions. Determine rates of change of absolute value functions. Determine intervals of increase and decrease of absolute value functions. Problem 1 Graph each absolute value function. Then, determine the domain, range, vertex, x-intercept(s) and y-intercept. Label each key characteristic on the graph. 1. y x 2 5 a. Graph: b. Domain: c. Range: d. Vertex: Lesson.2 Domain, Range, Vertex, Axis of Symmetry, Zeros, and Intercepts 411

e. x-intercept(s): f. y-intercept: 2. y x 3 1 a. Graph: b. Domain: c. Range: d. Vertex: e. x-intercept(s): 412 Chapter Other Functions and their Inverses

f. y-intercept: 3. y 2 x 3 4 a. Graph: b. Domain: c. Range: d. Vertex: e. x-intercept(s): f. y-intercept: Lesson.2 Domain, Range, Vertex, Axis of Symmetry, Zeros, and Intercepts 413

4. y 2x 4 1 a. Graph: b. Domain: c. Range: d. Vertex: e. x-intercept(s): f. y-intercept: 414 Chapter Other Functions and their Inverses

Problem 2 1. Rewrite each absolute value function as a piecewise function. Then, determine the intervals of x for which the function is increasing and decreasing, and the rate of change within each interval. a. y x 2 5 b. y x 3 1 c. y 2 x 3 4 d. y 2x 4 1 2. Is each function in Question 1 increasing or decreasing at the vertex? Explain. 3. Extreme points can also be defined as points where a graph changes from increasing to decreasing or from decreasing to increasing. Do graphs of absolute value functions have extreme point(s)? If so, which point(s) are extreme points? Lesson.2 Domain, Range, Vertex, Axis of Symmetry, Zeros, and Intercepts 415

Problem 3 1. Determine the following key characteristics for an absolute value function of the form f(x) a bx h k. a. Domain: b. Range: c. Vertex: d. x-intercept(s): e. y-intercept: f. Intervals of increase and rate of change: g. Intervals of decrease and rate of change: 416 Chapter Other Functions and their Inverses

2. How does the value of a affect the graph of the function f(x) a bx h k? 3. How does the value of b affect the graph of the function f(x) a bx h k? 4. How does the value of h affect the graph of the function f(x) a bx h k? 5. How does the value of k affect the graph of the function f(x) a bx h k? Be prepared to share your solutions and methods. Lesson.2 Domain, Range, Vertex, Axis of Symmetry, Zeros, and Intercepts 417

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.3 Taxes and Taxis Properties of Piecewise Functions Objectives In this lesson, you will: Determine the domain and range of piecewise functions. Determine the extrema of piecewise functions. Determine intervals where piecewise functions increase, decrease, and remain constant. Determine rates of change of piecewise functions. Determine points of discontinuity. Key Terms point(s) of discontinuity step function least integer function Problem 1 As of December 31, 2004, Georgia has had 6 income tax brackets, as shown. For incomes more than $0 and up to and including $750, the tax rate on every dollar of income earned is 1%. For incomes more than $750 and up to and including $2250, the tax rate on every dollar of income earned is 2%. For incomes more than $2250 and up to and including $3750, the tax rate on every dollar of income earned is 3%. For incomes more than $3750 and up to and including $5250, the tax rate on every dollar of income earned is 4%. For incomes more than $5250 and up to and including $7000, the tax rate on every dollar of income earned is 5%. For incomes more than $7000, the tax rate on every dollar of income earned is 6%. The brackets define relatively low annual incomes, so many state residents pay the highest rate of 6 percent. Lesson.3 Properties of Piecewise Functions 419

1. Write a piecewise function f(x) for the tax paid in Georgia for income x. 2. What is the domain and range of f(x)? 3. How much tax must be paid on an income of $750? $751? 4. Each interval of the piecewise function is represented by a linear function. Would the lines that represent these two functions be connected on the graph? Explain. 5. Are the endpoints of the line representing the first interval included? Explain. 6. Are the endpoints of the line representing the second interval included? Explain. 7. How can you use the graph to illustrate whether or not the endpoints are included? 420 Chapter Other Functions and their Inverses

8. When graphing intervals on a number line, an included endpoint is indicated by a closed point ( ). An endpoint that is not included is indicated by an open point ( ). Follow the same convention when graphing piecewise functions. Graph f(x) for 0 x 00. 9. What is the rate of change for 0 x 750? For 750 x 00?. What happens to the value of f at x 751? 11. How much tax must be paid on an income of $2250? $2251? 12. How much tax must be paid on an income of $7000? $7001? Lesson.3 Properties of Piecewise Functions 421

13. What happens to the value of f(x) at the end of each income tax bracket? 14. Points on a graph where there are asymptotes or breaks are called points of discontinuity. What are the points of discontinuity for the function f(x)? 15. Graph f(x) for 0 x,000. 16. Does f(x) have any extreme points? Explain. 422 Chapter Other Functions and their Inverses

Problem 2 In 2006, taxicab rates in Macon, Georgia, consisted of $1.20 for the first mile or part of a mile and $1.20 for each additional mile or part of a mile. 1. Define a piecewise function g(x) for the cost of a taxicab ride in Macon, Georgia, of up to 5 miles. g(x) 2. What are the points of discontinuity for g(x)? 3. What is the slope within each interval? Explain. 4. Graph g(x) for x 5 miles. 5. Is the function increasing or decreasing? Lesson.3 Properties of Piecewise Functions 423

6. Within each interval, is the function increasing or decreasing? Explain. 7. Can you write a piecewise definition representing the cost of a taxicab ride for any number of miles? Why or why not? 8. Can you think of other situations which may pose a similar challenge when defining them as piecewise functions? 9. Functions such as the cost of a taxicab ride in Macon, Georgia, can be defined using a function called the step function or least integer function. The least integer function L(x) x is defined as the greatest integer less than or equal to x. Calculate each of the following: a. 2 d. b. 2.34 e. c. 2.99999 f. 0.17 1.2 0.2. Graph L(x) x. 424 Chapter Other Functions and their Inverses

11. Why do you think this function is sometimes called a step function? 12. Write a function for the cost of a taxicab ride in Macon, Georgia, using the least integer function. Problem 3 In 2006, the taxicab rates in Atlanta, Georgia, consisted of $2.50 for the first mile or 1 part of a mile plus $0.25 for each additional of a mile or part thereof. 8 1. What is the fare for riding in a taxicab in Atlanta, Georgia, for 4 miles? For 2.1 miles? 2. Define a piecewise function f(x) for the cost of a taxicab ride in Atlanta, Georgia, for 0 x 2 miles. Lesson.3 Properties of Piecewise Functions 425

3. Graph this function. 4. Write a piecewise function for the cost of a taxicab ride in Atlanta, Georgia, using the step function. Be prepared to share your solutions and methods. 426 Chapter Other Functions and their Inverses

.4 Tanks a Lock Solving Absolute Value Equations and Inequalities Objectives In this lesson, you will: Use absolute value functions to model situations. Solve absolute value equations graphically. Solve absolute value equations analytically. Solve absolute value equations using technology. Problem 1 A river lock works in conjunction with a dam to ensure proper water depth in a river. A lock is a large container in the shape of a rectangular prism. Waterproof gates at each end can be filled and emptied to allow boats to travel around a dam from either direction. The part of the river below the lock is called the lower pool. The part above the lock is the upper pool. To transport a boat from the upper pool to the lower pool, water is added to the lock so that the water level in the lock is equal to the water level of the upper pool. Once the water levels are equal, the upper gates of the lock are opened and the boat enters the lock. Lesson.4 Solving Absolute Value Equations and Inequalities 427

The gates are closed and the water in the lock is slowly drained until the water level in the lock is equal to the water level of the lower pool. Once the water levels are equal, the lower gates of the lock are opened and the boat can proceed down the river. To transport a boat from the lower pool to the upper pool, the process is reversed. The Channelton Locks in Indiana allow boats to navigate the Ohio River. The main lock has a length of 1200 feet and a width of 1 feet. The normal upper pool elevation is 383.0 feet mean sea level. The normal lower pool elevation is 358.0 feet mean sea level. Twenty-five million gallons of water are required to operate the lock. 1. If the water level in the lock is at the lower pool, how much water must be added for the water level to reach the upper pool? 2. It takes approximately 8 minutes to raise the water level in the lock from the lower pool to the upper pool. What is the rate of change in the amount of water in the lock as the lock is being filled? 3. It takes approximately 8 minutes to lower the water level in the lock from the upper pool to the lower pool. What is the rate of change in the amount of water in the lock as the lock is being emptied? 4. During testing, the lock is emptied and filled in a cycle. One cycle is defined as starting at the level of the upper pool, draining to the level of the lower pool, and then returning to the level of the upper pool. Define a piecewise function f(x) for the change in the amount of water in the lock for one complete cycle. 5. What is the domain of f(x)? What is the range of f(x)? 428 Chapter Other Functions and their Inverses

6. Does f(x) have any extreme points? If so, list them. 7. For what interval of time is f(x) decreasing? Increasing? 8. Graph f(x). 9. What other type of function has a graph with the same shape as f(x)?. Define f(x) for 0 x 960 using the function from Question 9. 11. Use Question to calculate the amount of additional water in the lock 600 seconds into a cycle. 12. Graph the function y 1,000,000 on the grid for Question 8. 13. Calculate the points of intersection of y 1,000,000 and f(x) using a graphing calculator. On the grid, label the points of intersection with ordered pairs. Lesson.4 Solving Absolute Value Equations and Inequalities 429

14. How many seconds into a cycle will the lock have an additional 1,000,000 cubic feet of water? 15. How does the algebraic solution from Question 14 compare to the graphical solution in Question 13? 16. Use the graph to determine the times during a cycle when the lock will have less than 1,000,000 cubic feet of additional water. Explain how you determined the solution. 17. Write an inequality to calculate the times during a cycle when the lock will have less than 1,000,000 cubic feet of additional water. Solve the inequality algebraically. 430 Chapter Other Functions and their Inverses

Problem 2 Many communities use raised water tanks to store and distribute water to the residents. Pumps are used to fill the tanks. Gravity is used to distribute water to homes. One community has two water tanks, which are used in a cycle. As one tank is being filled, the other tank is distributing water. One cycle consists of a tank starting full, draining, and then being refilled. Each tank holds 1.5 million gallons of water and distributes water until the water level reaches 0,000 gallons. The tank is then refilled. The community uses an average of 57,000 gallons of water each hour of the day. 1. How long will it take until a full tank requires refilling? 2. The pump fills the water tank at the same rate that water is being distributed. How long will it take for one complete cycle? 3. Define an absolute value function for the amount of water in a tank during a cycle. 4. Graph this function for one cycle. Lesson.4 Solving Absolute Value Equations and Inequalities 431

5. Does the graph have a vertex? If so, what is the vertex? 6. For what intervals is this function increasing? Decreasing? 7. During what time(s) of a cycle will a tank contain 750,000 gallons of water? 8. Write and solve an inequality for the times of a cycle when a tank will contain more than 750,000 gallons of water. Be prepared to share your solutions and methods. 432 Chapter Other Functions and their Inverses

.5 We ve Got the Power Power Functions and Inverses Objectives In this lesson, you will: Define the term power function. Use graphs, tables and equations to compare power functions. Determine the inverse relation of power functions. Key Terms power function even power function odd power function inverse relation one-to-one function Power functions are functions that can be written in the form y ax b where a is a real number and b is a rational number. Problem 1 For each power function in Questions 1 4: a. Complete a table. b. Sketch a graph. c. Describe the graph. 1. y x 1 a. Table: x 2 1 0 y 1 2 Lesson.5 Power Functions and Inverses 433

b. Graph: c. Graph description: 2. y x 2 a. Table: b. Graph: x y 2 1 0 1 2 c. Graph description: 434 Chapter Other Functions and their Inverses

3. y x 3 a. Table: x y 2 1 0 1 2 b. Graph: c. Graph description: Lesson.5 Power Functions and Inverses 435

4. y x 4 a. Table: x y 2 1 0 1 2 b. Graph: c. Graph description: 5. What do you think the power function y x 5 will look like? 436 Chapter Other Functions and their Inverses

6. Sketch a graph of y x 5. Does the graph look like what you predicted? 7. Power functions with exponents that are even numbers are called even power functions. What do you notice about the graphs of even power functions? 8. Power functions with exponents that are odd numbers are called odd power functions. What do you notice about the graphs of odd power functions? 9. Look at the tables for the even power functions. What do you notice about f( 2) and f(2)? What about f( 1) and f(1)?. What conclusion can you make about f( x) and f(x) for even power functions? 11. Look at the tables for the odd power functions. What do you notice about f( 2) and f(2)? What about f( 1) and f(1)? Lesson.5 Power Functions and Inverses 437

12. What conclusion can you make about f( x) and f(x) for odd power functions? 13. The power function f(x) x 2 can be written in factored form as f(x) = (x 0)(x 0). What is the factored form of the power function f(x) x 3? 14. What is the factored form of the power function f(x) x 4? 15. What are the roots of each power function? 16. Where does the graph of each power function intersect the x-axis? Problem 2 Previously you determined the inverse of a linear function. For example, find the inverse of the linear function y 2x 8. Interchange the x and y variables: x 2y 8 Solve for y: y x 8 2 or y 1 2 x 4 The inverse of y 2x 8 is y 1. 2 x 4 1. Complete the tables for the linear function y 2x 8 and its inverse relation y 1. 2 x 4 x y 2x 8 y x y 1 2 x 4 y 1 0 1 2 8 6 4 438 Chapter Other Functions and their Inverses

2. What do you notice about the coordinate pairs of the function and the coordinate pairs of its inverse relation? 3. Graph y 2x 8 and y 1 on the grid. 2 x 4 4. How does the graph of y 2x 8 compare to the graph of y 1? 2 x 4 5. What are the domain and range of y 2x 8? 6. What are the domain and range of y 1? 2 x 4 7. How do the domain and range of y 2x 8 compare to the domain and range of y 1 x 4? 2 Lesson.5 Power Functions and Inverses 439

8. Is the inverse of y 2x 8 also a function? A function is a one-to-one function if both the function and its inverse are functions. The inverse of a one-to-one function f(x) is written as f 1 (x). Problem 3 1. Consider the power function y x 2. Complete the table for y x 2. x 4 3 2 1 0 1 2 3 y 4 2. Complete the table for the inverse of y x 2. x y 4 3 2 1 0 1 2 3 4 440 Chapter Other Functions and their Inverses

3. Using the tables, sketch a graph of y x 2 and its inverse. 4. Is the power function y x 2 a one-to-one function? 5. To write an equation for the inverse of y x 2, interchange x and y and solve for y as shown. y x 2 x y 2 x y Why is it necessary to include the symbol before the radical? Lesson.5 Power Functions and Inverses 441

6. Consider the power function y x 3. Complete the table for y x 3. x y 4 3 2 1 0 1 2 3 4 Complete the table for the inverse of y x 3. x y 4 3 2 1 0 1 2 3 4 442 Chapter Other Functions and their Inverses

7. Using the tables, sketch a graph of y x 3 and its inverse. 8. Is the power function y x 3 a one-to-one function? 9. Write an equation for the inverse of y x 3.. Is it necessary to include the symbol before the radical? Why or why not? 11. Write an equation for the inverse of y x 4. 12. Write an equation for the inverse of y 2x 4. Be prepared to share your solutions and explanations with your class. Lesson.5 Power Functions and Inverses 443

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.6 Back and Forth Inverses Objectives In this lesson, you will: Determine inverses of linear functions. Determine inverses of quadratic functions. Determine inverses of power functions. Problem 1 1. Previously, you explored inverses of several types of functions, including linear, linear absolute value, quadratic, and power functions. Which of these functions have inverses that are functions? Explain why. Determining the inverses of some functions, such as exponential functions, requires defining a special inverse function. Determining the inverses of other functions, such as some polynomial functions, is difficult algebraically. 2. For each linear function, determine its inverse. Then graph the function, its inverse, and the line y x, labeling each. a. f(x) = 4x 5 Lesson.6 Inverses 445

b. y = 2x 5 Problem 2 To determine the inverse of a linear absolute value function, first determine the vertex. Then rewrite the absolute value function as a piecewise function and calculate the inverse of each part of the piecewise function. For each absolute value function, determine its inverse. Then graph the function, its inverse, and the line y x, labeling each. 1. f(x) 2x 446 Chapter Other Functions and their Inverses

2. y 3x 2 3. y 2x 4 3 Lesson.6 Inverses 447

Problem 3 To determine the inverse of a quadratic function, it may be useful to convert the function to vertex form. For each quadratic function, determine its inverse. Then graph the function, its inverse, and the line y x, labeling each. 1. f(x) 2(x 3) 2 4 2. y 3x2 6x 4 448 Chapter Other Functions and their Inverses

Problem 4 For each power function, determine its inverse. Then graph the function, its inverse, and the line y x, labeling each. 1. y 2x 3 4 2. f(x) (x 4) 4 1 Be prepared to share your methods and solutions. Lesson.6 Inverses 449

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