Chapter 2: Introduction to Functions Lesson 1: Introduction to Functions Lesson 2: Function Notation Lesson 3: Composition of Functions Lesson 4: Domain and Range Lesson 5: Restricted Domain Lesson 6: One-to-One Functions Lesson 7: Inverse Functions Lesson 8: Ke Features of Functions This assignment is a teacher-modified version of Algebra 2 Common Core Copright (c) 2016 emath Instruction, LLC used b permission.
Chapter 2 Lesson 1: Introductions to Functions Introduction to Functions: Definition: A function is an rule that assigns eactl one output value (-value) for each input value (-value). These rules can be epressed in different was, the most common being equations, graphs, and tables of values. We call the input variable independent and output variable dependent. Relations and Functions: A relation is a relationship between sets of information. It is a set of ordered pairs. Recall: Domain: the set of all. ( variable) Range: the set of all. ( variable) A function is a specific tpe of relation. In order for a relation to be a function there must be onl and eactl one that corresponds to a given. Basicall, this means that elements of the domain never repeat. Is it a Function? One wa to test if a relation is a function is b using the vertical line test. This can be used when the relation is graphed. The vertical line test states that if a vertical line is drawn anwhere on the graph so that it hits the graph in more than one spot, then the graph is NOT a function. Eercise #1: Let s tr some (a) Is the graph of a parabola, f() = 2 a function?
(b) What about if we have a parabola on the side? Determine if each of the following graphs are functions.
Relationship B Relationship A Eercise #2: One of the following graphs shows a relationship where is a function of and one does not. (a) Draw the vertical line whose equation is = 3 on both graphs. (b) Give all output values for each graph at an input of 3. Relationship A: Relationship B: (c) Eplain which of these relationships is a function and wh.
Eercise #3: The graph of the function = 2 4 + 1 is shown below. (a) State this function s -intercept. (b) Between what two consecutive integers does the larger -intercept lie? (c) Draw the horizontal line = -2 on this graph. (d) Using these two graphs, find all values of that solve the equation below: 2 4 + 1 = -2
Eercise #4: Determine if the following equations represent a function. Justif our answer. (a) 2 + ( - 1) 2 = 4 (b) = 2 + 2 (c) 2 + 2 = 3 (d) = 3 + 3
INTRODUCTION TO FUNCTIONS CC ALGEBRA II HOMEWORK LESSON 1 FLUENCY 1. Determine for each of the following graphed relationships whether is a function of using the Vertical Line Test. (a) (b) (c) (d) (e) (f)
2. Given the relation {(8,2), (3,6), (7,5), (k,4)}, which value of k will result in the relation not being a function? (1) 1 (2) 2 (3) 3 (4) 4 3.) Which relation is not a function? (1) ( 2) 2 + 2 = 4 (3) + = 4 (2) 2 + 4 + = 4 (4) = 4 4.) Which graph represents a function?
APPLICATIONS 5.) Evan is walking home from the museum. He starts 38 blocks from home and walks 2 blocks each minute. Evan s distance from home is a function of the number of minutes he has been walking. (a) Which variable is independent and which variable is dependent in this scenario? (b) Fill in the table below for a variet of time values. Time, t, in minutes 0 1 5 10 Distance from home, D, in blocks (c) Determine an equation relating the distance, D, that Evan is from home as a function of the number of minutes, t, that he has been walking. (d) Determine the number of minutes, t, that it takes for Evan to reach home.
REASONING 6.) In one of the following tables, the variable is a function of the variable. Eplain which relationship is a function and wh the other is not. -2 11 0 7 2 11 4 23 6 43 Relationship #1 0 0 1-1 1 1 4-2 4 2 Relationship #2
Chapter 2 Lesson 2: Function Notation Function Notation: One method of defining a function is b naming the function, indicating the variable inside of the parentheses, and then defining a rule. It is important not to mistake function notation with multiplication. Recall that function rules commonl come in one of three forms: (1) equations, (2) graphs, and (3) tables. Eercise #1: Evaluate each of the following given the function definitions and input values. (a) f() = 5 2 (b) g() = 2 + 4 f(3) = g(3) = f(-2) = g(0) = (c) h() = 2 h(3) = h(-2) =
Eercise #2: Find each of the following. Answers must be in simplest form. (a) Given: f() = 2-4, find: (1) f(-3) (2) f( 1) (b) Given: g() = 2 2 + 1, find: (1) g(6) (2) g(2) Eercise #3: (a) Given f() = 5-1, find the value of when f() = 29.
(b) Given f() = 2 + 4, find the value(s) of when f() = 20. Eercise #4: Boiling water at 212 degrees Fahrenheit is left in a room that is at 65 degrees Fahrenheit and begins to cool. Temperature readings are taken each hour and are given in the table below. In this scenario, the temperature, T, is a function of the number of hours, h. h (hours) T h F 0 1 2 3 4 5 6 7 8 212 141 104 85 76 70 68 66 65 (a) Evaluate T(2) and T(6). (b) For what value of h is T(h) = 76? (c) Between what two consecutive hours will T(h) = 100?
Eercise #5: The function = f() is defined b the graph shown below. Answer the following questions based on this graph. (a) Evaluate f(-1), f(1) and f(5). (b) Evaluate f(0). What special feature on a graph does f(0) alwas correspond to? (c) What values of solve the equation f() = 0? What special features on a graph does the set of - values that solve f() = 0 correspond to? (d) Between what two consecutive integers does the largest solution to f() = 3 lie?
Eercise #6: For a function = g() it is known that g(-2) = 7. Which of the following points must lie on the graph of g()? (1) (7, -2) (3) (0, 7) (2) (-2, 7) (4) (-2, 0)
Chapter 2 Lesson 2: Function Notation Homework 1.) Given: f() = 2 3 + 12, g() = 2 + 1, and h() = 3 9 evaluate the following: (a) g(4) (b) h(-6) (c) f(-5) 2.) Given the following functions, simplif in terms of : f() = 2 3, g() = 2 + 4, h() = - + 7 (a) h(2 3) (b) g(3 1) (c) f(9 2) 3.) If f() =, what is the value of f(-10)? (1) (2) (3) (4)
4.) Based on the graph of the function = g() shown below, answer the following questions. (a) Evaluate g(-2), g(0), g(3) and g(7). (b) What values of solve the equation g() = 0 (c) Graph the horizontal line = 2 on the grid above and label. (d) How man values of solve the equation g()=2?
APPLICATIONS 5.) Ian invested $2500 in an investment vehicle that is guaranteed to earn 4% interest compounded earl. The amount of mone, A, in his account as a function of the number of ears, t, since creating the account is given b the equation At 2500 1.04 t. (a) Evaluate A 0 and A 10. (b) What do the two values that ou found in part (a) represent?
Chapter 2 Lesson 3: Composite Functions Since functions convert the value of an input variable into the value of an output variable, it stands to reason that this output could then be used as an input to a second function. This process is known as composition of functions, in other words, combining the action or rules of two functions. There are two notations that are used to indicate composition of two functions. Two Was of Writing Composite Functions: (1) (2) ***Alwas work from the inside out or start with the function closest to the *** Eercise #1: Given f() = 2 5 and g() = 2 + 3, find values for each of the following. (a) f(g(1)) = (b) g(f(2)) = (c) g(g(0)) = (d) (f o g)(-2)
(e) (g o f)(3) = (f) (f o f)(-1) = Eercise #2: Given f() = - 2 + 3 and g() = 4-5, find values for each of the following. Answers must be in simplest form. (a) g(f( + 1)) (b) (f o g)(2) Eercise #3: If and, which epression is equivalent to? (1) (2) (3) (4)
Eercise #4: The graphs below are of the functions = f() and = g(). Evaluate each of the following questions based on these two graphs. f g (a) g(f(2)) = (b) f(g(-1)) = (c) g(g(1)) = (d) (g o f)(-2) = (e) (f o g)(0) = (f) (f o f)(0) = Eercise #5: If f() = 2 and g() = - 5 then f(g()) = (1) 2 + 25 (3) 2-5 (2) 2-25 (4) 2 10 + 25
Chapter 2 Lesson 3: Composite Functions Homework FLUENCY 1. Given f() = 3 4 and g() = -2 + 7 evaluate: (a) f(g(0)) (b) (g o f)(6) (c) f(f(3)) 2. Given h() = 2 + 11 and g() = evaluate: (a) h(g(18)) (b) g(h(4)) (c) (g o g)(11) 3. The graphs of = h() and = k() are shown below. Evaluate the following based on these two graphs. h k (a) h(k(-2)) (b) (k o h)(0) (c) h(h(-2)) (d) (k o k)(-2)
4. If g() = 3 5and h() = 2 4 then (g o h)() =? (1) 6-17 (3) 5-9 (2) 6-14 (4) - 1 5. If f() = 2 + 5 and g() = + 4 then f(g()) = (1) 2 + 9 (3) 4 2 + 20 (2) 2 + 8 + 21 (4) 2 + 21 APPLICATIONS 6. Scientists modeled the intensit of the sun, I, as a function of the number of hours since 6:00 a.m., h, using the function. The then model the temperature of the soil, T, as a function of the intensit using the function T(I) = of the soil at 2:00 p.m.?. Which of the following is closest to the temperature (1) 54 (3) 67 (2) 84 (4) 38
REASONING 7. Consider the functions f() = 2 + 9 and g() =. Calculate the following. (a) g(f(15)) (b) g(f(-3)) (c) g(f()) (d) What appears to alwas be true when ou compose these two functions?
Chapter 2 Lesson 4: Domain & Range of Functions Because functions convert values of inputs into values of outputs, it is natural to talk about the sets that represent these inputs and outputs. The set of inputs that result in an output is called the domain (-values) of the function. The set of outputs is called the range (-values). This works for an relation. To find the domain we need to look from the left to the right. To determine the range we need to look down and up.
Eercise #1: Given the function {(0,4), (1,5), (2,6), (4,7), (6,4)}, use roster notation to give the domain and range. Roster Form: Eercise #2: State the range of the function f(n) = 2n + 1 if its domain is the set {1, 3, 5}. Show the domain and range in the mapping diagram below.
Eercise #3: The function = g() is completel defined b the graph shown below. Answer the following questions based on this graph. (a) Determine the minimum and maimum -values represented on this graph. (b) Determine the minimum and maimum -values represented on this graph. (c) State the domain and range of this function using set builder notation. Set Builder Notation:
Some functions, defined with graphs or equations, have domains and ranges that stretch out to infinit. Eercise #4: The function f() = 2 2-3 is graphed on the grid below. Which of the following represents its domain and range written in interval notation? (1) Domain [-2, 4] (3) Domain (-, ) Range [-4, 6] Range [-4, ) (2) Domain [-2, 4] (4) Domain (-2, 4) Range (-4, ) Range (-4, 6) f Interval Notation: Eercise #5: Determine the domain and range of the graphs below.
Eercise #6: Determine the domain of the function. (1) -1 < < 6 (2) -1 < < 6 (3) -2 < < 5 (4) -2 < < 5 Eercise #7: Which graph illustrates a quadratic relation whose domain is all real numbers?
Chapter 2 Lesson 4: Domain & Range of Functions Homework 1.) A function is given b the set of ordered pairs {(2, 5), (4, 9), (6, 13), (8, 17)}. Write its domain and range in roster form. Domain: Range: 2.) The function h() = 2 + 5 maps the domain given b the set {-2, -1, 0, 1, 2}. Which of the following sets represents the range of h()? (1) {0, 6, 10, 12} (3) {5, 6, 9} (2) {5, 6, 7} (4) {1, 4, 5, 6, 9} 3.) The graph below represents the function = f(). State the domain and range of this function.
4.) What are the domain and the range of the function shown in the graph below? (1) { > -4}; { > 2} (2) { > -4}; { > 2} (3) { > 2}; { > -4} (4) { > 2}; { > -4} 5.) What is the range of f() = 3 + 2? (1) { > 3} (3) { real numbers} (2) { > 2} (4) { real numbers} 6.) The function = f() is completel defined b the graph shown below. (a) Evaluate f(-4), f(3) and f(6). (b) State the domain and range of this function using interval notation. Domain: Range:
APPLICATIONS 7.) A child starts a pigg bank with $2. Each da, the child receives 25 cents at the end of the da and puts it in the bank. If A represents the amount of mone and d stands for the number of das then A(d) = 2 + 0.25d gives the amount of mone in the bank as a function of das (think about this formula). (a) Evaluate A(1), A(7) and A(30). (b) For what value of d will A(d) = $10.50. (c) Eplain wh the domain does not contain (d) Eplain wh the range does the value d = 2.5. not include the value A = $3.10.
Chapter 2 Lesson 5: Restricted Domain Do Now: Eercise #1: Which of the following values of would not be in the domain of the function =? Eplain our answer. (1) = 0 (3) = -3 (2) = 5 (4) = -8 Domain of Radical Functions: Eercise #2: Which of the following values of would not be in the domain of the function =? Eplain our answer. (1) = 1 (3) = 5 (2) = 0 (4) = -5 Domain of Rational Functions:
Radical in the Denominator: 3.) Find the domain of the following function:
Practice: Find the domain of each of the following functions. 4.) 3 1 5.) f ( ) 5 10 8 6.) f () 1 2 8 7.) f () 3 9 8.) 1 f ( ) 9.) 2 9 2 10
Chapter 2 Lesson 5: Restricted Domain Homework FLUENCY 1.) What is the domain of the function? (1) { > 0} (3) { > -4} (2) { > 0} (4) { > -4} 2.) Which of the following values of would not be in the domain of the function defined b f() =? (1) = -3 (3) = 3 (2) = 2 (4) = -2 3.) Determine an values of that do not lie in the domain of the function f() =. Justif our response.
4.) Which of the following values of does lie in the domain of the function defined b g() =? (1) = 0 (3) = 3 (2) = 2 (4) = 5 5.) Which of the following would represent the domain of the function =? (1) {: > 3} (3) {: < 3} (2) {: < 3} (4) {: > 3 Find the domain of each of the following functions. 6.) f() = 2 6 7.) f() = 2 2 4 9 8.). h() = 5 + 5
9.) g() = 1 3 5 10.) h() = 5 5 11 11.) g() = 2 + 12 + 20 12.) f() = 2 13.) f() = 1 7 14.) f() = 11 3 2
Chapter 2 Lesson 6: One-to-One Functions Functions as rules can be divided into various categories based on shared characteristics. One categor is comprised of functions known as one-to-one. The following eercise will illustrate the difference between a function that is one-to-one and one that is not. Eercise #1: Consider the two simple functions given b the equations f() = 2 and g() = 2. (a) Map the domain {-2, 0, 2} using each function. Fill in the range and show the mapping arrows. Domain of f Range of f Domain of g Range of g -2-2 0 0 2 2 (b) What is fundamentall different between these two functions in terms of how the elements of this domain get mapped to the elements of the range? Tip: In a one to one function, each element of the range is paired with element from the domain. Therefore, no repeat.
(1) Eercise #2: Of the four tables below, one represents a relationship where is a one-to-one function of. Determine which it is and eplain wh the others are not. (2) (3) (4) 4 2-2 1 1 2-3 10 4-2 -1 0 2 4-2 9 9 3 0 1 3 8-1 7 9-3 1 2 4 16-2 10 Eercise #3: Consider the following four graphs, which show a relationship between the variables and. (1) (2) (3) (4)
Eercise #4: Which of the following represents the graph of a one-to-one function? (1) (2) (3) (4)
Eercise#5: The distance that a number,, lies from the number 5 on a one-dimensional number line is given b the function D() = 5. Show b eample that D() is not a one-to-one function. Eercise #6: Which function is one-to-one? (1) k() = 2 + 2 (2) g() = 3 + 2 (3) f() = + 2 (4) j() = 4 + 2 Eercise #7: Which function is not one-to-one? (1) {(0,1), (1,2), (2,3), (3,4)} (3) {(0,1), (1,0), (2,3), (3,2)} (2) {(0,0), (1,1), (2,2), (3,3)} (4) {(0,1), (1,0), (2,0), (3,2)}
Chapter 2 Lesson 6: One-to-One Functions Homework FLUENCY 1. Which of the following graphs illustrates a one-to-one relationship? (1) (2) (3) (4) 2. Which of the following graphs does not represent that of a one-to-one function? (1) (2) (3) (4)
3. In which of the following graphs is each input not paired with a unique output? (1) (2) (3) (4) 4. In which of the following formulas is the variable a one-to-one function of the variable? (Hint tr generating some values either in our head or using TABLES on our calculator.) (1) = 2 (3) = 2 (2) = (4) = 5 5. Which diagram represents a relation that is a one-to-one function? (1) (2) (3) (4)
APPLICATIONS 6. A recent newspaper gave temperature data for various das of the week in table format. In which of the tables below is the reported temperature a one-to-one function of the da of the week? (1) (2) (3) (4) Mon 75 Mon 75 Mon 58 Mon 56 Tue 68 Tue 72 Tue 52 Tue 58 Wed 65 Wed 68 Mon 81 Mon 85 Thu 74 Thu 72 Tue 76 Tue 85 REASONING 7.) Consider the function f() = round(), which rounds the input,, to the nearest integer. Is this function one-to-one? Eplain or justif our answer.
Chapter 2 Lesson 7: Inverse Functions The inverse of a function, is a relation in which the domain and range of the original function have been echanged. Simpl put, the and have switched places. Inverse functions have their own special notation. It is shown in the bo below. To Find Inverses Algebraicall: 1.) For Ordered Pairs: The inverse of a function is formed b interchanging the and coordinates of each point in the function. 2.) For Equations: (a) Epress f() in terms of and. (b) Switch and to form the inverse. (c) Put back into = form b solving for. (d) Write in inverse notation.
Eercise #1: If the point (-3, 5) lies on the graph of = f(), then which of the following points must lie on the graph of its inverse? (1) (3, -5) (3) (5, -3) (2) (-5, 3) (4) (-1/3, 1/5) Eercise #2: Find the equation for the inverse of each function below. (a) f() = 3 + 5 (b) g() = ½ 2 8 (c) (d)
Eercise #3: f() is a linear function which is graphed below. Use its graph to answer the following questions. (a) Evaluate f -1 (2) and f -1 (-2). (b) Determine the -intercept of f -1 (). (c) On the same set of aes, draw a graph of.
Eercise #4: A table of values for the simple quadratic function f() = 2 is given below along with its graph. -2-1 0 1 2 f() 4 1 0 1 4 a) Graph the inverse b switching the ordered pairs. f 1 ( ) (b) What do ou notice about the graph of this function s inverse?
Eercise #5: Given the relation A: {(3,2), (5,3), (6,2), (7,4)}. Which statement is true? (1) Both A and A -1 are functions (2) Neither A nor A -1 is a function. (3) Onl A is a function. (4) Onl A -1 is a function. Eercise #6: Which equation defines a function whose inverse is not a function? (1) = (2) = - (3) = 3 + 2 (4) = 2
Chapter 2 Lesson 7: Inverse Functions Homework FLUENCY 1. If the point (-7, 5) lies on the graph of = f(), which of the following points must lie on the graph of its inverse? (1) (5, -7) (3) (7, -5) (2) (-1/7, 1/5) (4) (1/7, -1/5) 2. The function = f() has an inverse function = f -1 (). If f(a) = -b then which of the following must be true? (1) f -1 (-b) = -a (3) f -1 (-b) = a (2) f -1 (1/a) = -1/b (4) f -1 (b) = -a 3. The graph of the function = g() is shown below. The value of g -1 (2) is (1) 2.5 (3) 0.4 (2) -4 (4) -1
4. Which of the following functions would have an inverse that is also a function? (1) (2) (3) (4) 5. For a one-to-one function it is known that f(0) = 6 and f(8) = 0. Which of the following must be true about the graph of this function s inverse? (1) its -intercept = 6 (3) its -intercept = -6 (2) its -intercept = 8 (4) its -intercept = -8 6. The function = h() is entirel defined b the graph shown below. (a) Sketch a graph of = h -1 (). Create a table of values if needed. (b) Write the domain and range of = h() and = h -1 () using interval notation. = h() = h -1 () Domain: Domain: Range: Range:
APPLICATIONS 7. The function = A(r) = πr 2 is a one-to-one function that uses a circle s radius as an input and gives the circle s area as its output. Selected values of this function are shown in the table below. r 1 2 3 4 5 6 Ar 4 9 16 25 36 (a) Determine the values of A -1 (9π) and A -1 (36π) from using the table. (b) Determine the values of A -1 (100π) and A -1 (225π). (c) The original function = A(r) converted an input, the circle s radius, to an output, the circle s area. What are the inputs and outputs of the inverse function? Input: Output:
REASONING 8. The domain and range of a one-to-one function, = f(), are given below in set-builder notation. Give the domain and range of this function s inverse also in set-builder notation. = f() Domain { -3 < < 5} = f -1 () Domain: Range { > -2} Range:
Chapter 2 Lesson 8: Ke Features of Functions The graphs of functions have man ke features whose terminolog we will be using all ear. It is important to master this terminolog, most of which ou learned in Common Core Algebra I. Eercise #1: The function = f() is shown graphed to the right. Answer the following questions based on this graph. (a) State the -intercept of the function. f (b) State the -intercepts of the function. What is the alternative name that we give the -intercepts? (c) Over the interval -1 < < 2 is f() increasing or decreasing? How can ou tell?
(d) Give the interval over which f() > 0. What is a quick wa of seeing this visuall? (e) State all the -coordinates of the relative maimums and relative minimums. Label each. Relative Minimum Relative Maimum: (f) What are the absolute maimum and minimum values of the function? Where do the occur? (g) State the domain and range of f() using interval notation. Absolute Minimum: Absolute Maimum: (h) If a second function g() is defined b the formula of g?, then what is the -intercept
Eercise#2: Consider the function g() = 2 1-8 defined over the domain -4 < < 7. (a) Sketch a graph of the function to the right. (b) State the domain interval over which this function is decreasing. (c) State zeroes of the function on this interval. (d) State the interval over which g() < 0. (e) Evaluate g(0) b using the algebraic definition of the function. What point does this correspond to on the graph? (f) Are there an relative maimums or minimums on the graph? If so, which and what are their coordinates?
You need to be able to think about functions in all of their forms, including equations, graphs, and tables. Tables can be quick to use, but sometimes hard to understand. Eercise #3: A continuous function f() has a domain of -6 < < 1 with selected values shown below. The function has eactl two zeroes and has eactl two turning points, one at (3, -4) and one at (9, 3). -6-1 0 3 5 8 9 13 f() 5 0-2 -4-1 0 3 1 (a) State the interval over which f() < 0. (b)state the interval over which f() is increasing
We can sketch the graph of functions based on certain characteristics. When sketching graphs, it is helpful to plot the -intercepts and an absolute etrema first. Eercise 4: Draw a graph of = f() that matches the following characteristics. Decreasing on: -10 < < -2 and 6 < < 10 Increasing on: -2 < < 6 Zeros at = -4, 0, and 8; f(-2) = -3 Absolute minimum of -3 and absolute maimum of 5
KEY FEATURES OF FUNCTIONS CC ALGEBRA II HOMEWORK LESSON 8 FLUENCY 1. The piecewise linear function f() is shown to the right. Answer the following questions based on its graph. (a) Evaluate each of the following based on the graph: (i) f(4) (ii) f(-3) (b) State the zeroes of f(). (c) Over which of the following intervals is f() alwas increasing? (1) -7 < < -3 (3) -5 < < 5 (2) -3 < < 5 (4) -5 < < 3
(d) State the coordinates of the relative maimum and the relative minimum of this function. Relative Maimum: (e) Over which of the following intervals is f() < 0? (1) -7 < < -3 (3) -5 < < 2 (2) 2 < < 7 (4) -5 < < 2 Relative Minimum: (f)a second function g() is defined using the rule g() = 2f() + 5. Evaluate g(0) using this rule. What does this correspond to on the graph of g? (g) A third function h() is defined b the formula h() = 3-3. What is the value of g(h(2))? Show how ou arrived at our answer.
2. For the function g() = 9 ( + 1) 2 do the following. (a) Sketch the graph of g on the aes provided. (b) State the zeroes of g. (c) Over what interval is g() decreasing? (d) Over what interval is g() > 0? (e) State the range of g.
3. Draw a graph of = f() that matches the following characteristics. Increasing on: -8 < < -4 and -1 < < 5 Decreasing on: -4 < < -1 f(-8) = -5 and zeroes at = -6, -2 and 3 Absolute maimum of 7 and absolute minimum of -5 4. A continuous function has a domain of -7 < < 10 and has selected values shown in the table below. The function has eactl two zeroes and a relative maimum at (-4, 12) and a relative minimum at(5, -6). -7-4 -1 0 2 5 7 10 f() 8 12 0-2 -5-6 0 4 (a) State the interval on which f() is decreasing. (b) State the interval over which f() < 0.