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Lesson 5.1: Introduction to FunctionsLearning Goals:What is a function and how do you determine if a relation is a function?What is a one-to-one function and how do you determine if a function is one-to-one?How do you evaluate functions?What is a function?A relation is a set of ordered pairs or collection of points (i.e.: coordinates, equations, graphs, circle diagrams)Based on your observations of the above table, what is a function?A function is a relation in which each element of the domain corresponds to one and only one element in the range or a relation in which no two ordered pairs have the same first element. It can also be called a relation with no repeating x- values.To determine if a relation is a function given its graph we use the vertical line test.Be sure students understand the definition of onto. A function is onto if each element of B is mapped by at least one element of the domain A.Examples: 1. Consider the correspondence between a set of people in a room and his/her height in centimeters.Barbara→165Keith→176Robert→165Maria→168Notice that two different people may have the same height, but no person can have 2 different heights.This is an example of a function.2. Decide if the following relations are functions or not and explain why.(a) {1, 2, 3, -1, 5, 7, -2, 1}Yes, no repeating x-values.(b) 0, 2, 3,2, -5, 1, 6, 2Yes, no repeating x-values.(c) {(-1, 1), (2, 3), (2, -3), (4, 7)}No, x-value of 2 repeats.(d) {6, 1, -2, 3, 7, 2, -2, 1}No, x-value of -2 repeats.3. Determine if each of the following graphs are functions.yesnonoyesOne-to-One FunctionsWhat is the difference between the functions in column A and those in column B?A One-to-One Function: is a function with no repeating x or y-values.To see if a relation is a one-to-one function use both the vertical line test (x-values) and the horizontal line test (y-values)4. Determine if each of the following relations are one-to-one and explain.a) 2, 1, 3, 2, 4, 2, 5, 1Not 1-1, but is a function.b) 1, -1, 2, -2, 3, -3, 4, -41-1c) 4, 1, 5, 0, 4, 2, 3, -1Not 1-1, and not a function.Determine if the following relations are one-to-one:No (fails HLT), No (fails VLT), Yes, Yes3876675-33718500Evaluating FunctionsThe f(x) notation can be thought of as another way of representing the y-value in a function, especially when graphing. The y-axis is even labeled as the f(x) axis, when graphing. Oftentimes, mathematics has a variety of ways to say the same thing. Examine these notations that are all equivalent:5. Write each of the following in function notation:(a) The population of a certain city is represented by the equation y=500,000(2)t.ft=500,000(2)t(b) The amount of money a school fundraises is represented by the equation y=500+2n.gn=500+2n or gn=500+2n(c) The speed of a baseball hit by a batter is represented by y=9.8v2-16v+32.h(v)=9.8v2-16v+32Practice:6) If fx=2x2+6x, find f(-2).fx=2-22+6-2fx=24-12fx=8-12f-2=-47) If gx=2x+22, find g(-3).gx=2-3+22gx=-6+22gx=16g-3=48) If fx=4x-7, find x when fx=13.13=4x-720=4x5=x9) If gx=3x+52 and gx=4, find x.4=3x+528=3x+53=3x1=x10) If fx=kx2 and f2=3, find k3=k223=4k34=.75=k11) Find k when fx=3x2+kx-1 and f1=9.9=312+k1-19=3+k-19=2+k7=kHomework 5.1: Introduction to Functions1. Explain whether the pairing is a function.2. Which graph does not represent a function.3. Which graph represents a function?4. Which relation is not a function?1) (x-2)2+y2=42) x2+4x+y=43) x+y=44) xy=439719250005. Which diagram is not a function?6. Explain whether or not each of the following is a one-to-one function.7. If gx=-3x+2 and gx=-1, find x.8. If fx=6x2-10x+2, find f(2).9. If gx=x2+2 and gx=11, find the positive value of x.10. Find k if y=3k2x when f6=2.11. Given this graph of the function fx:40767009652000Find:(a) f-4=(b) f2=(c) What is x when fx=4?(d) What is x when fx=-1?Lesson 5.2: Domain and Range of a FunctionLearning Goals:What is domain and range of a function?How do we determine the domain and range of a function?Do Now:Write an inequality that represents the solution set given by the shaded region of the number line.x>-7 and x<3 x≤13 or x≥2-7<x<3 Or= ∪And= ∩ Review of Interval Notation:1. Write each of the following solutions in set builder notation and interval notation.Set Builder Notation (inequality)Interval Notation [ or )x≥22, ∞x<8(-∞, 8)x≤-3 or x≥-1(-∞, -3] or [-1, ∞)-4≤x≤-2[-4, -2]x<2 or x>3-∞,2 or (3, ∞)The domain of a function is the set consisting of all first elements of the ordered pairs. (Abscissas or x- values.)“input” of the relationThe range of a function is the set consisting of all second elements of the ordered pairs. (Ordinates or y-values.)“output” of the relationFinding the domain and range from the GRAPH2. Find the domain and range of each of the following in interval notation.a) D= -1, 6 & R=[-2, 5]b) D= -6, 5 & R=[-8, 8] c) D=-∞, ∞ & R=-8, ∞d) D=[0, ∞) & R=(-∞, ∞) e) D=-∞, ∞ & R=-4, ∞ Finding the domain and range from the EQUATIONUse the calculator to graph and sketch!3. Find the domain and range of each of the following.a) y=-(x+5)2b) y=x+3D=(-∞, ∞) Absolute Value!R=[0, -∞) Math→NUM→1:abs(1714500-1270000-127000D=-∞, ∞ & R=[0, ∞)2nd→0:catalog→abs(c) y=x+1-2d) y=x2-2x-3D=-1,∞ D=-∞, -1 & [3, ∞)R=[-2, ∞) R=[0, ∞)f) If the domain of fx=2x+3 is {-3<x≤0}, which number is not in the range? (a) -1 (b) 0 (c) 3 (d) 6Homework 5.2: Domain and Range of a Function1. What is the domain of the function fx=x-46-x?(1) x≠4, x≠6 (2) xx≤6 (3) xx≥6 (4) xx<62. Function m(x) is defined as mx=x+5x2+3x-10. State the domain of function m(x).3. What is the domain of the function hx=x2-4x-5?(1) xx≥1 or x≤-5(2) xx≥5 or x≤-1(3) x-1≤x≤5(4) x-5≤x≤14. Which relation is not a function?(1) y=2x+4 (2) y=x2-4x+3 (3) x=3y-2 (4) x=y2+2x-35. Which graph does not represent a function?7505700401320006. State the domain and range of each of the following in interval notation:7. If hx=2x+3, find h(-1).8. If fx=-3x+10 and fx=13, find x.9. If hx=-kx+14 and h2=18, find the value of kLesson 5.3: Evaluating Functions and Operations with FunctionsLearning Goals:How do you evaluate functions?How do you perform operations with functions?Function notation:We have seen linear functions written in the form y=mx+b. By naming a function f, you can write it in function notation.fx=mx+bThe symbol f(x) is another name for y and is read as “the value of f at x” or simply as “f of x". It does not mean f times x.Other letters used to represent functions are g or h. fx=yDirections: Evaluate the following functions:1. fx=x-7, when x=32. gx=x2, when x=-5f3=3-7g-5=(-5)2f3=-4g(-5)=253. fx=x2-x+3; f(-2) 4. If fx=kx+5 and f3=36,f-2=(-2)2-(-2)+3find the value of k. x=3 & y=36f-2=4+2+336=k3+5f-2=931=3k313=k5. fx=2x2-3x+4; f(-x)6. fx=2x+5; fx-8. f-x=2-x2-3-x+4fx-8=2x-8+5f-x=2x2+3x+4fx-8=2x-16+5Don’t try to solve/factor!fx-8=2x-117. fx=2x2+4; f(a-2)8. fx=x2-2x+4; f(x+2)fa-2=2a-22+4fx+2=x+22-2x+2+4fa-2=2a-2a-2+4fx+2=x+2x+2-2x-4+4fa-2=2a2-4a+4+4fx+2=x2+4x+4-2xfa-2=2a2-8a+8+4fx+2=x2+2x+4fa-2=2a2-8a+12Definition of Operations of Functions:Sumf+gx=fx+g(x)Difference (order matters)f-gx=fx-g(x)Productf?gx=f(x)?g(x)Quotient (order matters)fgx=f(x)g(x), g(x)≠0For each new function, the domain consists of those values of x common to the domains of f and g. The domain of the quotient function is further restricted by excluding any values that make the denominator, g(x), zero.9. Given fx=x2-4 and gx=x+2, find each function.(a) f+gx(b) f-gxfx+g(x) fx-g(x)(x2-4)+(x+2)x2-4-(x+2)x2+x-2x2-4-x-2x2-x-6(c) (f?g)(x)(d) fg(x)f(x)?g(x) f(x)g(x)x2-4x+2x2-4x+2 DOPsx3+2x2-4x-8(x-2)(x+2)x+2x-210. Given fx=2x2+5x-3 and gx=x+3, find each function.(a) f+gx(b) f-gx2x2+5x-3+(x+3)2x2+5x-3-(x+3)2x2+6x2x2+5x-3-x-32x2+4x-6(c) (f?g)(x)(d) fg(x)2x2+5x-3x+32x2+5x-3x+32x3+6x2+5x2+15x-3x-9(2x-1)(x+3)x+32x3+11x2+12x-92x-111. Given ga=-3a2-a and ha=-2a-4, find each of the following:(a) g+h-5 (b) gh4-3a2-a+-2a-4-3a2-a-2a-4-3a2-3a-4-342-4-24-4-3-52-3-5-4-316-4-8-4-325+15-4-48-4-12-75+15-4-52-12-64133Find the indicated value:(a) f+g1=(b) f-g1=-2+2-2-20-4(c) f?g2=(d) (f+g)(-3)0*2-2-20-4Homework 5.3: Evaluating Functions and Operations with FunctionsEvaluate each of the following functions:1. fx=2x-3; f-52. fx=x2-4; fx+13. Given fx=x2-5x+6 and gx=x-3, find each function(a) f+gx(b) f-gx(c) (f?g)(x)(d) fg(x)7. Find the indicated value: (a) (f-g)(-2) (b) (f÷g)(2)Lesson 5.4: Composition of FunctionsLesson Goals:What is the composition of functions?How do we perform the composition of functions?Warm-Up: You are shopping for sneakers. Foot locker is having a sale – everything in the store is 20% off. You find a pair of sneakers that are originally priced for $90. When you get to the register, you find out that there is an additional 15% off.a) Find the cost of the sneakers with 20% off.90*.20=18 off so 90-18=72 or 90.80=72 b) Using your answer to part (a) find the cost of the sneakers with the additional 15% off. 72*.15=10.8 off so 72-10.8=$61.20 In this example we see that the output of the 20% off is used as the input to the 15% off function. This is known as the composition of functions and can be generalized for any functions f and g. $90 → 20% → 72 → 15% → $61.20There are 2 notations we use for composition of functions: Order matters! Always work right to left!f°gx=f(gx)g°fx=g(fx)Example 1: Given fx=x2-5 and gx=2x+3, find the values for each of the following:a) f(g1)g1=21+3 =2+3 =5Then f5=52-5 =25-5 =20b) (g°f)(3)f3=32-5 =9-5 =4Then g4=2(4)+3 =8+3 =11c) (g°g)(0)g0=20+3 =3g3=23+3 =6+3 =9d) (g(ff-1)f-1=-12-5 =1-5 =-4f-4=-42-5 =16-5 =11g11=211+3 =22+3 =25Example 2: The graphs below are the functions y=fx and y=g(x). Evaluate each of the following questions based on these two graphs.a) gf2b) fg-1c) gg1f2=3g-1=-5g1=3g3=3f-5=-0.5g3=3d) g°f-2e) f°g0f) f°f0f-2=1g0=0f0=2g1=3f0=2f2=3Example 3: Given the function fx=3x-2 and gx=5x+4, determine formulas in simplest y=ax+b form for:a) fgxb) g°fxgx=5x+4fx=3x-2f5x+4=35x+4-2g3x-2=53x-2+4 =15x+12-2 =15x-10+4 =15x+10 =15x-6Example 4: Given the function fx=x2 and gx=x-5, determine formulas in simplest form for:a) (f°g)(x)b) g(fx)gx=x-5fx=x2fx-5=x-52gx=x2-5 =x-5x-5 =x2-10x+25PUSH YOURSELF!Example 5: For each function h below, find two functions f and g such that hx=f(gx). There are many correct answers.a) hx=(3x+7)2b) hx=3x2-8gx=3x+7gx=x2-8fx=x2fx=3xc) hx=4(2x-3)3d) hx=(x+1)2+2(x+1)gx=2x-3gx=x+1fx=4x3fx=x2+2xHomework 5.4: Composition of Functions1. If fx=x2+4 and gx=1-x, what is the value of f(g-3)?2. Using fx=x2 and gx=x+1, find (g°f)(-1).3. If fx=3x+1 and gx=x2-1, find (f°g)(2).4. If gx=3x-5 and hx=2x-4, find the formula for (g°h)(x).5. If fx=x2+5 and gx=x+4, find the formula for f(gx).6. The graphs of y=h(x) and y=k(x) are shown below. Evaluate the following based on these two graphs.a) h(k-2) b) (k°h)(0) c) h(h-2) d) (k°k)(-2)7. Consider the functions fx=2x+9 and gx=x-92. Calculate the following:a) gf15b) gf-3c) g(fx)d) What appears to always be true when you compose these two functions?Lesson 5.5: Inverse FunctionsLearning Goals:What is the inverse of a function and how do we find the inverse of a function?Graphically, what is the relationship between a function and its inverse?Warm-Up: Consider the two linear functions given by the formulas fx=3x+72 and gx=2x-73.(a) Calculate fg-1(b) Calculate fg5g-1=2-1-73=-2-73=-93=-3g5=25-73=10-73=33=1f-3=3-3+72=-9+72=-22=-1f1=31+72=3+72=102=5(c) Without calculation, determine the value of f(gπ).π Because fx and g(x) are inverses!Inverse FunctionsTwo functions are inverses because they literally “undo” one another. The general idea of inverses, fx and g(x), is shown below in the mapping diagram.In other words, two functions are inverses if their x and y-values are switched.If a function y=f(x) has an inverse that is also a function we represent it as y=f-1(x).The inverse does not mean reciprocal so f-1(x)≠1f(x)How to Find the Inverse of a FunctionThe process of finding an inverse is simply swapping the x and y coordinates.1. Find the inverse of hx=1, 2, -3, 4, 5, -6h-1x=-6, 5, 2, 1, (4, -3) 2857500209550002. Graph and label f-1(x), the inverse of f(x) on the set of axes below.333375000 What type of reflection are the graphs of f(x) and its inverse?Reflection over the line y=xSolving for an inverse relation algebraically is a three step process:Set the function =ySwap the x and y variablesSolve for y3. What is the inverse of the function4. What is the inverse of the function gx=3x+5?fx=-12x-2?y=3x+5y=-12x-2 x=3y+5x=-12y-2 x-5=3yx+2=-12y y=x-53=g-1(x)-2x-4=y=f-1(x) 5. What is the inverse of the function mx=5+x6-2x?y=5+x6-2x x=5+y6-2y 6x-2xy=5+y 6x-5=2xy+y 6x-5=y2x+1 GCF 6x-52x+1=y=m-1(x) 6. When fx=x-72, what is the value of (f°f-1)(3)?y=x-72 x=y-72 2x+7=y=f-1(x) f°f-13= f-13=23+7=6+7=13 f13=13-72=62=3 How to Determine if the Inverse of a Graph is a FunctionTo find the inverse of a function, simply switch the x and y coordinates (values).Graph: The graph of an inverse relation is the reflection of the original graph over the identity line, y=x. It may be necessary to restrict the domain on certain functions to guarantee that the inverse relation is also a function. If the original function is a one-to-one function, the inverse will be a function. Use the horizontal line test to determine if a function is a one-to-one function.True or False: The inverse of the graph shown below will be a function. Justify your answer.a and b pass the HLT so the inverse is a function. C fails the HLT so the inverse will not be a function.Homework 5.5: Inverse Functions1. Write the set of input-output pairs for the functions of f and g by filling in the blanks below. (The set F for the function f has been done for you.)F=1, 3, 2, 15, 3, 8, 4, -2, 5, 9G={-2, 4, , , , , , , , }2. Cindy thinks that the inverse of fx=x-2 is gx=2-x. To justify her answer, she calculates f2=0 and then substitutes the output 0 into g to get g0=2, which gives back the original input. Show that Cindy is incorrect by using other examples from the domain and range of f.Find the inverse for these functions:3. fx=4x+124. fx=x+3x5. True or False: Since f(x) is a reflection 6. Graph the inverse of the function below: 33718508445500of gx, g(x) is also the inverse of f(x). Justify your answer. Lesson 5.6: Average Rate of ChangeLearning Goal: In geometry you learned how to find the slope of a line over a given interval (2 points)After today’s lesson you should be able to:Find the average rate of change of a function over a given interval [a,b]Apply average rate of change to word problemsAverage Rate of Change:The average rate of change of the function y=f(x) between x=a and x=b is average rate of change=change in ychange in x=y2-y1x2-x1=fb-f(a)b-aWarm up: Find the slope of the function over the given intervals1. from x=1 to x=42. from -2, 8 to (15, -22)-11-14-1=-123=-4-22-815-(-2)=-30173. from point A to point Briserun=21=2Discuss with a partner:1. Given the graph above, how do you find the average rate of change over the interval [3, 6]? Find the slope of points 3, 23 & (6, 27)2. Which interval has a larger average rate of change 3, 6 or [6, 8]? Why?[3, 6] has a steeper slope!The average rate of change is the slope of the secant line between 2 points of a graph. It will tell you if the graph is generally increasing or decreasing over a specific interval!Example 1: Given the function fx=-x2+4x-1, find the average rate of change from x=1 to x=5. What does this tell you about the graph of the function over the interval [1, 5]?average rate of change=change in ychange in x=fb-f(a)b-af5-f(1)5-1 (-52+45-1)--12+41-15-1 (-25+20-1)-(-1+4-1)4 -25+20-1+1-4+14 -84 -2 The graph is mostly decreasing between [1, 5]Example 2: Suppose an object is thrown upward with an initial velocity of 52 feet per second from a height of 125 feet. The height of the object t seconds after it is thrown is given by ht=-16t2+52t+125Find the average velocity in the first three seconds after the object is thrown.How do you know this is asking for the average rate of change? Average Velocity4048125113538000First three seconds means: Interval = [0, 3]average rate of change=change in ychange in x=fb-f(a)b-ah3-h(0)3-0 (-163)2+523+125-(-1602+520+125)3-0 -169+156+125-(-160+0+125)3 -144+156+125-(0+0+125)3 137-1253 123 4 Example 3: An object travels such that its distance, d, away from its starting point is shown as a function of time, t, in seconds, in the graph below.Is the average speed of this object greater on the interval 0≤t≤5 or 11≤t≤14? Justify your answer. Means find slope! 0≤t≤511≤t≤1411≤t≤14 is greater becausef5-f05-0f14-f(11)14-11the slope 5>4.20-0579-64320515345Example 4: Forrest Gump went running for several hours. The table below represents the number of miles he had run, m, after t hours. Is the average speed constant for this function? Justify your pare the slopes over the intervals 2, 4 & [4, 7]18-104-2=82=4 and 30-187-4=123=4 and 30-107-2=205=4All slopes are the same so it is constant.Extend your thinking: The table below represents a linear function. Fill in the missing entries.Because it is a linear function the slope “average rate of change” will be constant1, 5f5-f15-1=1-(-5)5-1=1+54=64=325, 11 11, x[19,45]f11-f511-5=32 fx-f11)x-11=32f45-f(19)45-19=32y-16=32 fx-f11)x-11=32y-2226=3218=2y-2 22-10x-11=3278=2y-44 20=2y 12x-11=32122=2y10=y 24=3x-3361=y57=3x19=xHomework 5.6: Average Rate of ChangeFind the average rate of change for the following functions over the given interval. What does the average rate of change tell you about the function on the interval?1. From x=0 to x=32. From x=-1 to x=13. fx=-x-2+5 in the interval [3, 11]3181350-58420004. Which of the two functions has the largest average rate of change from x=-2 to x=4? ................
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