Name: Unit 2 Functions and their Inverses 1
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Linear Review Solve by Graphing. 1. 3x 6y = 12 2x 4y = 8 2. x + y = 2 y = -2x 1 Solve by Substitution. 3. 2x 3y = -1 y = x 1 4. y = -3x + 5 5x 4y = -3 Solve by Elimination. 5. 6. 3
For each question, define your variables, write a system of equations, and solve using any method. Please use a separate sheet of paper to show all work. 7. The length of a rectangle is 3 cm more than twice the width. The perimeter of the rectangle is 42 cm. Find the dimensions of the rectangle. 8. Suppose you have $200 in your account and you save $10 dollars each week. Your friend has $110 in their account and starts saving $15 each week. When will your account balances be the same? 9. The difference of two numbers is 40. Their sum is 66. Find the numbers. 10. A youth group and their leaders visited Mammoth Cave. Two adults and 5 students in one van paid $77. Two adults and 7 students in another van paid $95. Find the adult price and student price of the tour. 11. A winter clothing store had a sale and Cory bought two pairs of gloves and four hats for $43. Mark bought two pairs of gloves and two hats for $30. How much did each pair of gloves and each hat cost? 12. At a recreation and sports facility, 3 members and 3 nonmembers pay a total of $180 to take a yoga class. A group of 5 members and 3 nonmembers pay $210 to take the same class. How much does it cost each member and nonmember to take the yoga class? 13. Joey has $5.75 made up of all dimes and quarters. If Joey has 38 coins, how many of each coin does he have? 4
Solve the system of inequalities. 2x 3y 9 14. x y 4 15. 4x 2y 8 x 3y 6 16. f x x x 2 ( ) 2 6 7 4 x f ( x) 10 17. 2 ( x 2) 7 2x 2y 6 y **Try to graph the quadratic function without a calculator.** 5
18. y x 4 y x6 y 3x 4 19. f x 2 ( ) ( x 3) 8 f x 2 ( ) 2( x 3) 6 20. A sundae requires 3 ice-cream scoops and 4 strawberries, and a milkshake requires 2 ice-cream scoops and 6 strawberries. Ramses wants to make sundaes and milkshakes with at most 25 ice-cream scoops and 37 strawberries. Graph your solution on the following graph. 21. For a person of height h (in inches), a healthy weight W (in pounds) is one that satisfies this system of inequalities: 2 19h w 703 2 25h w 703 Graph the system for 0 h 80 using your graphing calculator. What is the range of healthy weights for a person 67 inches tall? 6
Absolute Value Functions Absolute value variable equations are written as: f(x) = a mx + h + k Graph looks like a right side up or upside down Opens up if the coefficient in front of the absolute value symbols is. f(x) = 4 x + 2 + 3 opens up Opens down if the coefficient in front of the absolute value symbols is. f(x) = -4 x + 2 + 3 opens down The vertex of the graph will be ( h, k). You can use your calculator to find it!! m Let s start with f(x) = x and graph the equation. This is called the parent function. What s the vertex? (, ) Does it open up or down? Domain: Range: You try f(x) = x +2. How is it different from the parent graph? What s the vertex? (, ) Does it open up or down? Domain: Range: 7
Now try: f(x) = x + 2. How is it different from the parent graph? What s the vertex? (, ) Does it open up or down? Domain: Range: Vertical Transformations: A constant added outside the absolute value symbol shifts the graph UP that many units. f(x) = x + 5 moves the parent graph A constant subtracted outside the absolute value symbol shifts the graph DOWN that many units. f(x) = x - 3 moves the parent graph Horizontal Transformations: A constant added inside the absolute value symbols shifts the graph LEFT horizontally. f(x) = x + 2 moves the parent graph A constant subtracted inside the absolute value symbols shifts the graph RIGHT horizontally. f(x) = x - 2 moves the parent graph Reflection over the x-axis: If you have a in front of the absolute value, the graph will be reflected, or, over the x-axis. f(x) = - x moves the parent graph Vertical Stretch/Compression: C f(x), where C is a real number > 0 If C > 1, then f(x) has a vertical by a factor of C units. If 0 < C < 1, then f(x) has a vertical by a factor of C units. f(x) = 2 x How does this compare to the parent? f(x) = 0.5 x How does this compare to the parent? 8
Find the equation of both rays of the absolute value: f(x) = 2 x + 3-2 Solve y = 2 (x+3) -2 and y = 2(-(x+3)) -2 Check intersection point: Practice: In what way would the graph of y = x move according to the following equations? Be specific. 1. y = 4 x + 3-5 2. y =- x - 2 + 7 Application: A rainstorm begins as a drizzle, builds up to a heavy rain, and then drops back to a drizzle. The rate r (in inches per hour) at which it rains is given by the function r 0.5 t1 0.5 and t represents time in hours. Graph the function. How long does it rain? When does it rain the hardest? What is the rate of the rain after 30 minutes? 9
Graphing: Graph y = x -1 + 3. Graph y = -2 x + 3 + 2. Determine the equation of the graph Find the equations of the rays for f(x) = 3 x 2 +5 10
Absolute Value HW 11
Systems and Inequalities of Absolute Value Functions Student Notes Recall Solving Absolute Value Equations and Inequalities Algebraically 2x 4 7 17 x 3 7 10 2x 8 6 0 x 3 7 8 Graphing Absolute Value Inequalities Example 1 Similarly to Linear and Quadratic Equations, you will graph the function using dashed and solid lines. Afterwards, choose a test point to see where the shading needs to take place. y x 2 5 Example 2 Graph the following Absolute Value Inequality. y x 6 4 12
Example 3 Graph the following Absolute Value Inequality. y x 2 7 Solving Systems of Absolute Value Equations by Graphing To solve systems of absolute value equations, graph both equations on the same graph. Find the points of intersection. There can be two, one, or no solutions. Two Solutions One Solution No Solutions Example 1 Graph to Solve the System. y x 3 4 y x 3 6 13
Example 2 Graph to Solve the System. y x 2 4 y x 2 4 Solving Systems of Absolute Value Equations by Graphing Using Technology Steps to Solving Systems of Absolute Value Equations Using Your Calculator 1. Solve equations for y. 2. Enter equations into y1 and y2. 3. Graph. 4. 2 nd Calculate 5. #5 intersect 6. enter, enter, enter Example 1 Graph to Solve the System using the By Hand and with the Calculator. y x 3 5 y x 3 5 14
HW Solve the system of inequalities. y x y x4 y x 6 y x4 f ( x) x 3 6 f ( x) x 4 4 f ( x) 2x 6 5 f ( x) x 3 5 15
f ( x) x 2 6 y x 5 f ( x) x 4 6 f ( x) 2x 2 2 Given the following absolute value inequality, identify a solution that is in the region of possible solutions. y 7 x 4 5 16
Introduction to Piecewise Functions Student Notes Investigation Graph the following two functions on the same graph. f x 2 ( ) ( x 3) 4 g( x) 2x 4 1. Do the equations overlap? 2. State the Domain and Range. f(x) Domain: f(x) Range: g(x) Domain: g(x) Range: 3. Are both equations functions? 4. Find f(0) and g(0). 5. How could we make the two functions continuous? What restrictions needed to be added to our equations? Definition of Piecewise Functions: Example 1 17
Graph the following Piecewise Function. Make sure you restrict your domain for certain pieces of the function. x4, x0 f( x) x 4, x 0 Example 2 Graph the following Piecewise Function. Make sure you restrict your domain for certain pieces of the function. x 2, x2 f( x) x 8, x 2 18
Graphing and Evaluating Piecewise Functions with Context Example 1 Graph the following Piecewise Function. Make sure you restrict your domain for certain pieces of the function. f( x) x 2 x x 2 4, 2 4, x 2 Example 2 Graph the following Piecewise Function. Make sure you restrict your domain for certain pieces of the function. x 4, x 1 f( x) x 8, x 1 Example 3 Graph the following Piecewise Function. Make sure you restrict your domain for certain pieces of the function. x, x 1 f( x) 5, -1 x 1 x, x1 19
Evaluating Piecewise Functions Given the following piecewise function, evaluate the following. Hint: You can use your graph from the previous example if needed. x, x 1 f( x) 5, -1 x 1 x, x1 f(-9)= f(-1)= f(1)= f(-5)= f(0)= f(6)= Student Try Example Given the following piecewise function, evaluate the following. Hint: You can use your graph from the previous example if needed. x 4 7, x 1 f ( x) 2x 3, -1<x<1 2 x x 2, 1 f(-3)= f(-1)= f(1)= f(-5)= f(0)= f(4)= 20
Piecewise Functions in Context Postal charges for mailing packages depend on both weight and destination and this leads to an application of piecewise functions. For example, the rates for a certain destination are shown in the table below. Weight in Pounds (x) Postage Cost (y) Under 1 $0.80 1 or more, but under 2 $1.00 2 or more, but under 4 $1.25 4 or more $1.50 Create a piecewise function using the table above. Graph the piecewise function on the following graph. Piecewise Functions in Context There are two parking garages to choose from when parking in Downtown Raleigh. Parking Garage 1 Parking Garage 2 $6 Dollars for the First Hour $5 Dollars per hour, up to 5 hours $5 Dollars for the Second Hour $4 Dollars per hour after that $4 Dollars for each hour starting on the Third Hour 1. What is the cost to park in both parking garages for 2.75 hours? 2. What is the cost to park in both parking garages for 5.25 hours? 3. What is the best option for parking in Downtown Raleigh? 21
Homework Piecewise functions Evaluate the function for the given value of x. Match the piecewise function with its graph. Graph the function. 19. 20. 21. 22. The admission rates at an amusement park are as follows. Children 5 years old and under: free Children between 5 years and 12 years, inclusive: $10.00 Children between 12 years and 18 years, inclusive: $25.00 Adults: $35.00 a) Write a piecewise function that gives the admission price for a given age. b) Graph the function. 22
Inverses of Functions Kathy and Kevin are sharing their graphs for the same set of data. Both students insist that they are correct, but yet their graphs are different. They have checked and re-checked their data and graphs. Can you explain what might have happened? Has this ever happened to you? Kathy s Graph Kevin s Graph In mathematics, when the independent and dependent variables are reversed, we have what is called the inverse of the function. If you look at Kathy and Kevin s graph again, you might notice that the ordered pairs have been switched. For example in Kathy s graph the ordered pairs of (0,1) and (2,4) are (1,0) and (4,2) in Kevin s graph. The inverse of a function helps to get back to an original value of x. Investigation: Inverses Consider the following functions: 1) f(x) = 6 + 3x 2) f(x) = 1 (x 6) 3 Part 1: Graphs of Inverse Functions What line are the inverses reflected over? 23
What do you notice about the ordered pairs in each function? Is there a relationship between f(x) and g(x)? How do I find inverses? 1. Find the and the of the given equation. 2. Switch and in the equation. 3. Solve for. 4. Make the range become the new of the new equation, and make the domain become the new of the new equation. If you consider half of the parabola, then this function has an inverse. Let s look at a simpler function y = x 2. If you only consider the positive values of x in the original function, the graph is half of a parabola. When you reflect the half over the y = x line, it will pass the vertical line test as shown on the right. Algebraically, let s switch the x and y values and solve for y. Function: y = x 2, where x 0 Switch the x and y values: x = y 2 Solve for y: As shown, the both graphs pass the vertical line tests and are inverse functions of each other. The above example means that if you restrict the domains of some functions, then you will be able to find inverse functions of them. 24
Example 1: y = 1 2 x 7 Example 2: y = 3(x + 2) 2 4 Example 3: y = 3 x 3 + 6 25
You Try! y = 4(x + 8) 2 5 You Try! y = 4 x + 7 + 5 Use the horizontal line test to determine if a function has an inverse function. If ANY horizontal line intersects your original function in ONLY ONE location, your function has an inverse which is also a function. Which of the following functions would have inverses that are a function? Go back and look at the inverses you found in the notes...are these functions?? 26
HW Functions and inverses (Kuta) 27
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Inverse Functions in Context ~ Student Investigation 1) When calibrating a spring scale, you need to know how far the spring stretches based on given weights. Hooke s law states that the length a spring stretches is proportional to the weight attached to the spring. A model for one scale is l = 0.5w + 3 where l is the total length (in inches) of the spring and w is the weight (in pounds) of the object. a. Find the inverse model for the scale. b. If you place a melon on the scale and the spring stretches to a total length of 5.5 inches, how much does the melon weigh? 2) ABC Electric uses this formula, f (x) = 750 10x, to depreciate computers, where f is the value of a computer and x is the number of months since its purchase. Calculate f (36). What is the meaning of f (36)? What is the meaning of b in f (b) = 60? What is the value of b? Write a formula for f 1, and explain what it means in this situation. When will the depreciated value of a computer be less than $400? What is the meaning of c in f 1 (c) = 60? What is the value of c? 29
Function operations Review: Evaluate each function for the given value of x. Let f(x) = 3x + 4. Find f(-2). Let g(x) = 2x 2 3x + 1. Find 3g(a + 2). Part 1: Basic operations with Functions Operation Definition Examples if f(x)= x + 2 and g(x) = 3x Sum (f + g)x = f(x) + g(x) Difference (f g)x = f(x) g(x) Product (f g)x = f(x) g(x) Quotient f f ( x) ( x), g( x) 0 g g( x) Given f(x) = x 2-3x + 1 and g(x) = 4x + 5, find each function. a) (f + g)(x) b. (f g)(x) Given f(x) = x 2 + 5x 1 and g(x) = 3x 2, find each function. f a) (f g)(x) b. ( x) g Composition of Functions taking the output (y-value) of one function and making it the input ( x-value) of another function. Consider this.... You have a coupon for $25 off at. There is also a 30% off sale for the that you want to buy. Does the order in which these discounts are taken matter? Original Price 1 st Discount 2 nd Discount Sale Price So..... it is better for the shopper to. Remember..... Domain: the set of all input values Range: the set of all output values 30
Definition of Composition of Functions: The composition of function f with function g is written f g() x = f(g(x)). START ON THE INSIDE & WORK YOUR WAY OUT!!! Let s Watch a Video: http://www.youtube.com/watch?v=s4aezeltpdo f(x) = x 2 + x g(x) = 4 x Find (f g)(x). Find (g f)(x). I. Finding a value of a composition given a function Given f(x) = x+5 and g(x) = x 2 2. Evaluate each expression. a. f(g(3))= b. g(f(3)) = f REMEMBER!!! g() x = f(g(x)) NOT f ( x) g( x ) II. Finding a composition equation given functions Given f(x) = 3x2 and g(x)= 2x +4 Find () f g x Find g f () x Composing is NOT MULTIPLICATION! 31
Function Composition Practice! 1. Find (f g)(x) and (g f)(x) for f(x) = x + 3 and g(x) = x 2 + x 1. 2. Evaluate (f g)(x) and (g f)(x) for x = 2. 3. Find (g h)(x) and (h g)(x) if g(x) = 2x h(x) = x 3 + x 2 + x + 1 4. If f(x) = x 2 x and g(x) = x 1, what is f(g(x))? 5. A box of laundry detergent sells for $3.25. The price the store pays is determined by the function f(x) = x 1, where x is the selling price of the detergent. The wholesale price is determined by g(x) = x -.75, where x is the price the store pays. What is the wholesale price? APPLICATIONS 6. A store is offering a 10% discount on all items. In addition, employees get a 25% discount. a. Write a composite function to model taking the 10% discount first. b. Write a composite function taking the 25% discount first. c. Suppose you are an employee. Which discount would you prefer taken first? 7. Tyrone Davis has $180 deducted from every paycheck for retirement. He can have these deductions taken before taxes are applied, which reduces his taxable income. His federal income tax rate is 18%. If Tyrone earns $2200 every pay period, find the difference in his net income if he has the retirement deduction taken before taxes or after taxes. 32
Composition of Functions with Context Taken from: https://www.georgiastandards.org/frameworks/gso%20frameworks/acc-math-ii-unit-2-se-inverse-polynomial- Functions.pdf 1. Aisha made a chart of the experimental data for her science project and showed it to her science teacher. The teacher was complimentary of Aisha s work but suggested that, for a science project, it would be better to list the temperature data in degrees Celsius rather than degrees Fahrenheit. a. Aisha found the formula for converting from degrees Fahrenheit to degrees Celsius: 5 C F 32. 9 Use this formula to convert freezing (32 F) and boiling (212 F) to degrees Celsius. b. Later Aisha found a scientific journal article related to her project and planned to use information from the article on her poster for the school science fair. The article included temperature data in degrees Kelvin. Aisha talked to her science teacher again, and they concluded that she should convert her temperature data again this time to degrees Kelvin. The formula for converting degrees Celsius to degrees Kelvin is K C 273. Use this formula and the results of part a to express freezing and boiling in degrees Kelvin. c. Use the formulas from part a and part b to convert the following to K: 238 F, 5000 F. In converting from degrees Fahrenheit to degrees Kelvin, you used two functions, the function for converting from degrees Fahrenheit to degrees Celsius and the function for converting from degrees Celsius to degrees Kelvin, and a procedure that is the key idea in an operation on functions called composition of functions. Composition of functions is defined as follows: If f and g are functions, the composite function f g (read this notation as f composed with g) is the function with the formula f g( x) f g( x), where x is in the domain of g and g(x) is in the domain of f. 33
2. We now explore how the temperature conversions from Item 1, part c, provide an example of a composite function. a. The definition of composition of functions indicates that we start with a value, x, and first use this value as input to the function g. In our temperature conversion, we started with a temperature in degrees Fahrenheit and used the formula to convert to degrees Celsius, so 5 the function g should convert from Fahrenheit to Celsius: g( x) x 32. What is the 9 meaning of x and what is the meaning of g(x) when we use this notation? b. In converting temperature from degrees Fahrenheit to degrees Kelvin, the second step is converting a Celsius temperature to a Kelvin temperature. The function f should give us this conversion; thus, f ( x) x 273. What is the meaning of x and what is the meaning of f (x) when we use this notation? c. Calculate f g(45) f g(45). What is the meaning of this number? d. Calculate f g( x) f g( x) is the meaning of f g( x )?, and simplify the result. What is the meaning of x and what e. Calculate f g(45) f g(45) with your calculation from part c? f. Calculate g f ( x) g f ( x) using the formula from part d. Does your answer agree, and simplify the result. What is the meaning of x? What meaning, if any, relative to temperature conversion can be associated with the value of g f ( x )? 3. We now explore function composition further using the context of converting from one type of currency to another. On the afternoon of August 2, 2016, each Japanese yen (JPY) was worth 0.187996 Mexican pesos (MXN), each Mexican peso was worth 0.0470470 Euro (EUR), and each Euro was worth 1.12269 US dollars (USD). (You can update this information using www.xe.com a. Using the rates above, write a function P such that P(x) is the number of Mexican pesos equivalent to x Japanese yen. b. Using the rates above, write a function E that converts from Mexican pesos to Euros. 34
HW Inverses in context & Operations with functions 1. To make a long-distance call, your phone company charges $1.50 to make the connection, and an additional $0.10 for every minute that you are on the line once connected. a. Write an equation for the price of a long-distance call, p, in terms of the length of the call in minutes, m: b. When you get the phone bill, you see that your sister made a long-distance call that cost $2.75. How long was she on the phone? c. Think about how you solved part (b). Write an equation to determine m in terms of p. (That is, how do you calculate the length of a call based on its price?) 2. ABC Electric company uses the formula, f(x)= 750-10x, to depreciate computers where f(x) is the value of the computer and x is the number of months since its purchase. a) Calculate f(36). What is the meaning of f(36)? b) What is the meaning of b in f(b) = 60. What is the value of b? c) Write the formula f -1, and explain what it means in this situation. d) When will the depreciation of a computer be less than $400? e) What is the meaning of f -1 (c) = 60? What is the value of c? 35
Operations with Functions (Kuta) 36
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Math 3 ~ Unit 1: Functions & Their Invereses Test Review! Section #1: Find the appropriate solution for the assigned function(s) f(x) = 3x 8 and g(x) = -2x 2 + 5x 7 1. f(0) 2. f(g(2)) 3. 2[f(2) g(1)] 4. f(g(x)) 5. f(x) g(x) 6. f(2z) + g(z) Section #2: Find the solution(s) to the following system. y 4x 4 7. 2 y x 2x 4 Solutions: y x3 8. y 2 27 x 5 5 Solutions: Section #3 Sketch the solution for each inequality. y x 2 9. 10. 4 x 3 y 9 y 5x 2 x3y 6 38
5 y x2 11. 2 1 y x2 2 12. 3x 2y 2 x2y2 Section #4: Solve the systems of linear equations through graphing, substitution, or elimination. Use method indicted if possible. 13. The equations 8x4y 28 and 9x 9y 36 represents the amount of money collected from the Stone Creek Movie Theater. If y represents the cost of an adult ticket to get into the movie and x represents the cost of a child ticket to get into a movie then what is the cost of each adult ticket? Use any method! METHOD 2: Elimination 14. 4 x 5 y 17 3x4y5 15. 1 x y 3 2 x 2y 6 39
Solve by GRAPHING x 4y 8 16. 3x2y 6 17. 3 x 2 y 6 2x 4y 12 Method 3: Substitution x y 1 18. x2y 7 19. 5m 9n 21 2m2n14 Section #5: Evaluate each piece-wise function for its given domain. x 20. f x 2 x 6 1 f ( 10) f 2 f f 2 1 x 1 3 x 1 2 x 1 x 0 21. f x 2x 1 0 x 5 3 x 5 5 f 2 f 0 f f (7) 40
22. Looking at #24 find: 2 f (1) 3( f ( 3) f (0)) Section #6: For each equation state the domain and range (using interval notation) then find the inverse and state the domain and range of the inverse. Also, determine if the inverse is a function. 23. f x 3 ( ) 5x 7 Domain: Range: 24. f x 5x 11 Domain: Range: f -1 (x) = Domain of f -1 (x): Range of f -1 (x): 25. f x 2 ( ) 3x 1 Domain: Range: f -1 (x) = Domain of f -1 (x): Range of f -1 (x): 26. f x x 4 Domain: Range: f -1 (x) = Domain of f -1 (x): Range of f -1 (x): f -1 (x) = Domain of f -1 (x): Range of f -1 (x): Section #7 Absolute Value Equations & Functions 27. Solve each of the following. a. x1 4 b. 3 y 5 c. 2 3d 4 d. 2m1 2 41
28. Graph the following equation. Then describe the translate of g(x) = x to f(x). f(x) 3 x 2 29. Determine whether each of the following is a function. Justify your answer. Find the Domain and Range of each. a. f ( x) x 3 b. f(x) - x 2 2x - 27 30. You work forty hours a week at a furniture store. You receive a $720 weekly salary, plus a 3% commission on sales over $5000. Assume that you sell enough this week to get the commission. Given the functions f (x) = 0.03x and g(x) = x 5000, which of (f o g)(x) and (g o f )(x) represents your commission? Also, how much will your salary be if your sales were $14,000? How much will your salary be if your sales were $4999? 42