Module 1: Relations
Contents
1. Step 8 Your answer appears X 1 920589771 which we round to 1 921 If your answer is different look carefully at your equation for mistakes Use the cursor keys to locate and correct them Step 9 Go back to Step 7 Change your bound line to 2 3 and your X guess to 2 5 Then press ALPHA SOLVE Ensure you get the answer 2 07815274 which rounds to 2 078 If you got a BAD GUESS message it means your guess is outside your bounds Step 10 Go back to Step 7 and use your third guess of 4 5 and bounds of 4and5 Then press ALPHA SOLVE Ensure you get the answer 4 175770562 which rounds to 4 176 So the three solutions for the equation 3x3 13x2 10x 50 from least to greatest are 1 921 2 078 and 4 176 Module 1 12 Section 1 Lesson 1 Principles of Mathematics 12 Adjusting the Viewing Window Finally we adjust the viewing window on the calculator so that more of the graph is visible Recall that the graph you saw on your calculator see page 9 went off the top and bottom of the screen you could not see it all To fix that you adjust the Viewing Window Press the WINDOW button at the top of the keyboard Y ou see the list of values at the right From the top these values tell you that the x axis in your window ranges from 10 to 10 with a ticmark i Co ordinate values for the Standard toe y number PNeSaine ts Vi2. Module 1 44 Section 1 Lesson 4 Principles of Mathematics 12 vii viii 4 Find i the zeros of the following functions ii the maximum number of turns iii the left right behaviour iv the domain and range and v sketch the graph of the function a f x 4x x b g x x 2 c h x x x 6x Check your answers in the Module 1 Answer Key Module 1 Principles of Mathematics 12 Section 1 Review 45 Review 1 Solve the following equations using your graphing calculator a x 4 2x 4x 7 b 2x 3x 4 gt _ 4x3 x 3x xX d 0 4x 0 78x 2 4x 7 0 2 If f x 3x 3x 1 and g x V2x 1 find a f 2 g 1 sie EE OS OS aang ree 25 x 7S x 3 i Determine the x and y intercepts of these functions ii Find the domain and range a f x VJ1 2x b g x 2 x 2 4 c A x x 2 x 4 x 2 Module 1 46 Section 1 Review Principles of Mathematics 12 4 Find the inverse of the following functions a f x 5x 2 b g x x 1 3 5 Sketch the graphs of the following functions Label the intercepts and give the domain and range of each a f x x x 2 b g x 2 x 9 x 2 2x 1 Check your answers In the Module 1 Answer Key Now do the section assignment which follows this section E ZX Module 1 Principles of Mathematics 12 Section Assignment 1 1 47 PRINCIPLES OF MATHEMATICS 12 Section Assignm
3. x 2 0 yields x 2 sox 4 2 Module 1 Principles of Mathematics 12 Section 1 Lesson 4 39 The maximum number of turns is 6 1 5 but because of the x2 2 factor we only get 3 turns domain Using the graphing calculator as in Example 1 we get range 9 3 0792014 c kisa fifth order polynomial or quintic which means it is an odd polynomial It has a negative leading coefficient so it falls from Quadrant II It has at most 5 1 4 turns domain range When we plot it using the graphing calculator we find it has only one maximum and one minimum with a kink near the y axis called a stationary point which you don t have to remember The local maximum is located 1 54591952 0 0510976 In Lesson 1 to solve an equation with the graphing calculator we used the O Solver function from the MATH menu and typed in the equation to be solved That method works but we could miss a solution for a higher order polynomial because there can be several solutions close to each other Module 1 40 Section 1 Lesson 4 Principles of Mathematics 12 To find the zero x intercept of a graphed equation press 2nd CALC then 2 zero J ust as you did for a minimum or maximum mark the left bound right bound and a first guess for each crossing point on the x axis Include only one crossing point between your left and right bounds Then write down the d
4. Module 1 Principles of Mathematics 12 Section 1 Lesson 1 5 Lesson 1 GRAPHING CALCULATOR REVIEW Outcomes Upon completing this lesson you will be able to carry out these operations on a graphing calculator e Enter and edit polynomial equations e Graph the equations and adjust the viewing window e Solve the equations Overview If you used a graphing calculator for Principles of Math 11 then consider this lesson optional You should do just a few of the Bes exercises to make sure you remember how the graphing and solving functions work The directions here are specific to the T1I 83 or T1 83Plus models from Texas Instruments You may be using a different TI model or one made by Hewlett Packard Sharp Casio or another manufacturer Any graphing calculator will get you through the provincial exam except the HP 48 which is not allowed But if you use another brand of calculator you will need to refer to its user manual to find out how to do what these instructions tell you Module 1 6 Section 1 Lesson 1 Principles of Mathematics 12 Solving Polynomial Equations Using the Graphing Calculator You have developed considerable skill at finding the rational which includes integral roots of given polynomial equations But the roots of a polynomial equation are not necessarily rational They might well be irrational numbers non periodic non terminating decimals This possibility can make the algebraic solution of
5. y y One to one function Not one to one Has an inverse function Horizontal line cuts the graph at two points Inverse is not a function Not a function Vertical line cuts the graph at two points Inverse will not be a function either because a horizontal line does the same Sometimes in Principles of M athematics 12 we will get around this restriction by considering only a portion of the original function say a piece of the graph which passes both line tests even if the whole graph passes only the vertical test In the inverse function range and domain get reversed The domain of f x becomes the range of f 1 x The range of f x becomes the domain of f 1 x To find the inverse of a function y f x replace x and y with each other and solve for y Module 1 Principles of Mathematics 12 Section 1 Lesson 3 27 Example 1 Find the inverse of f x 2x 5 x 2y 5 replace x and y with each other 2y x 5 solve for y _X 5 2 5 f 1 _ X x Check that your answer is correct by showing that f f 1 x x and f 1 f x x sma s x a ee en ee Fen ge 1 x 5 nls LOG is indeed the inverse of f x 2x 5 Example 2 a Find the inverse of the function x aes The domain of f x x 2 is 0 b State the domain and range of f x and f x Solution a y replace f x by eae N swap x and y 2 x x y 2 1 solve for y ee X FS X Module 1 28 Section 1 Lesson 3 Principles of Mathe
6. 2 x 1 x 2 0 x2 0 o x 1 0 or x 2 0 x 1 0 2 The term with the highest power is x4 found by multiplying together the x terms in the equation x2 xx xx x so the function is a quartic or fourth order polynomial The coefficient of x4 is positive so the graph opens up x2 is a double root so the graph touches the x axis at x 0 Module 1 Principles of Mathematics 12 Section 1 Lesson 4 37 menu of a graphing calculator Press Y X T 0 x X T 0 1 X T 0 The interval 1 0 between the roots is less than the interval 0 2 so the graph goes down farther between 0 and 2 Note that we have a relative minimum in the interval 1 0 a relative maximum at 0 0 and an absolute minimum in the interval 0 2 To determine range we need to find the value of the absolute minimum We can do this using the minimum choice from the CALC At the flashing cursor carefully type 2 Your display should look likethis Y x x 1 x 2 Y Y etc Press GRAPH to see your display Press WINDOW to obtain the WINDOW or WINDOW FORMAT menu Change the Ymin setting to 5 in order to better see the minimum value To calculate the minimum value press 2nd TRACE to get the CALC ulate menu Scroll down to 3 minimum or press 3 You will notice the bli
7. 9 x Similarly ae Be 2x 1 1 3x 2 Ja fx ax RA a 2x 1 1 1 y3x 2 TRONS ae PrI Module 1 Principles of Mathematics 12 Section 1 Lesson 2 19 Rule When functions are combined arithmetically like that the domain of the result is the intersection of domains from the two functions it s the set of all points that belong in both the original domains The same is true of the combined range it s the intersection of the two separate ranges That intersection of domains rule applies for all four arithmetic operations between any two functions But the division operation f 9 has an additional rule the combined domain and range cannot include any value that makes the new denominator go to zero Note Whenever this course uses a plain square root sign it refers only to the positive square root This definition is common in most modern mathematics Examples If you solve x V4 then the answer is x 2 but not x 2 If it wants the negative root this course will ask you to solve x V4 If this course asks you for both roots it will ask you to solve x 4 l Example 5 Using the above two functions f and g find the domain and range of f 9 f g fxg and f 9 Solution By inspection domain of f x 4 and the range of f f y 0 x 5 and y 0 are not allowed because f cannot have a zero in the denominator D 2 Similarly domain of g 3 0 The square root of a negat
8. Principles of Mathematics 12 Contents Section 1 Section 2 Module 1 Module 1 Relations Review of Mathematics 11 Lesson1 Graphing Calculator Review Lesson 2 Functions and Interval Notation Lesson3 Inverse Functions Lesson4 Polynomial Functions and Their Graphs Review Section Assignment 1 1 Transformations Lesson1l Translations Lesson2 Reflections Lesson3 Absolute Value Functions Lesson4 Stretches and Compressions Lesson5 Reciprocal Relations Lesson6 Composition of Transformations Summary Review Section Assignment 1 2 Answer Key 15 24 31 45 47 59 69 81 89 103 113 121 123 127 137 Module 1 2 Contents Principles of Mathematics 12 Module 1 Principles of Mathematics 12 Section 1 Introduction 3 Module 1 Section 1 Review of Mathematics 11 Introduction In this section you will review a number of the concepts and processes encountered at the end of Principles of Math 11 Most of these concepts deal with functions and coordinate geometry A solid foundation of polynomials and rational functions is necessary to master transformations which you will encounter in Sections 2 and 3 of this module and also in Modules 2 and 3 Section 1 Outline Lesson 1 Graphing Calculator Review Lesson 2 Functions and Interval Notation Lesson 3 Inverse Functions Lesson 4 Polynomial Functions and Their Graphs Review Module 1 4 Section 1 Introduction Principles of Mathematics 12 Notes
9. Principles of Mathematics 12 Section 1 Lesson 2 23 Guided Practice 1 Given that f x 4x2 x 3 and g x 1 2x find a f 0 g f 9 x b 0 h g f x c f 2 i fxg 2 d g 1 4 j f 9 0 e f a k g f b 1 f gb 1 2 Determine the x and y intercepts for the following functions a f x 2x2 8 b g x V2x 5 c k x 5 x 3 Using the information from your answers to question 2 write the domain and range for each function using i set notation ii interval notation 4 Given that p x x 4 and q x 3x l a determine each of the following i p q x ii p q 3 iii q p a b find the domain and range of i p q x ii q p x Check your answers in the Module 1 Answer Key Module 1 24 Section 1 Lesson 3 Principles of Mathematics 12 Lesson 3 Inverse Functions Outcomes Upon completing this lesson you will be able to e verify that two functions are inverses of each other e identify one to one functions e find inverse functions Overview Many of the important new functions you will learn about in Principles of Mathenatics 12 are the inverses of other functions The concept of an inverse is essential to solving mathematics problems Definition Inverse Function Let f and g be two functions such that f g x x for every x in the domain of g and g f x x for every x in the domain of f Then the function g is the inverse of the function f denoted by
10. a a3 2a Module 1 16 Section 1 Lesson 2 Principles of Mathematics 12 Example 2 State the domain and range of the function p x 2 x 1 4 3 Y Thereis no restriction on the 1 3 values that x can take Domain R x The parabola has a maximum value at 1 3 Range fy y lt 3 ye R Turn to Appendix 1 for some background information on how to read set notation Informal Rules for Finding a Function s Domain and Range There is no sure fire formula to determine the extent of a function along the X axis its domain or the Y axis its range The domain and range are often obvious if you just look at the graph here are some guidelines for finding the domain and range of a function by looking at the equation not the graph 1 Linear functions f x mx b always have infinite domain and range in both directions i e out to and also to The only exception is when m O then y b and the domain remains infinite but the range is simply b 2 Parabolic functions f x a x h 2 b have infinite domain but a limited range the range goes to infinity in one direction on the Y axis but not in the other direction This rule applies to any even powered function x2 x4 x6 and so on If the exact range boundary is not obvious from the equation your graphing calculator can identify it for you this is taught in Lesson 4 3 In square root functions with x somewhere under the root sign both the doma
11. alls to the left and rises to the right Another way to express this is the graph rises from the third quadrant y ae gt x If the leading coefficient is negative y lt 0 then the graph rises to the left and falls to the right Or it falls from the second i quadrant 2 When the degree n of a polynomial is even i e of degree 2 4 6 If the leading coefficient is positive y gt 0 then the graph rises to the left and right or opens up X If the leading coefficient is negative lt 0 then the graph falls to the left and right or opens down Module 1 Principles of Mathematics 12 Section 1 Lesson 4 33 4 Thefunction f has at most n real roots If you have a cubic 5 function you can expect three roots at most When you have a quartic function you can expect it to have at most four roots and so on X f x X 4x cubi c at most three roots y f x x 5x 4 quartic at most four roots In general the graph of a cubic function is shaped like a sideways S as shown Graphs of f x ax bx cx d oF a gt 0O Module 1 34 Section 1 Lesson 4 Principles of Mathematics 12 In general the graph of a quartic equation has a W shape or an M shape Graphs of f x ax b o dx e y y a gt 0 a lt 0 6 If a polynomial f x has a squared factor such as x c then x c is a double root of f x 0 In th
12. e same way as it is used on a regular calculator in that it performs the function shown above the keys The X 1 6 n Key The X T 0 n key is the one you use to insert a variableinto the equation you type To type sin 6 you hit the sin key then the X T 0 n key Then close the parenthesis This key also inserts the x into polynomial equations Braces To type a brace use the 2nd key and the corresponding parenthesis Most users don t bother with braces since nested parentheses work just as well Equations 1 and 2 mean the same thing on a graphing calculator Hes 45 Equation 1 is easier for humans to read but equation 2 is easier to enter on the calculator fewer keys to press Step 1 To solve for the roots of this equation we will solve for the zeros of the corresponding polynomial function Y 3x3 13x2 10x 50 We begin by typing in the function as follows Module 1 Principles of Mathematics 12 Section 1 Lesson 1 9 Press Y This bring up a flashing cursor to the right of Y in your display window Step 2 At the flashing cursor we begin typing in our function carefully 3 X T 0 n 7 3 13 X T 0 n 72 10 X T 6 50 Although spaces have been used between the above numbers for clarity don t type spaces on the graph
13. ematics 12 3 For the the function q x below sketch the graph of q 1 x 1 4 If f x x 2 and x x 0 find a f 9 x 1 b f xg 2 1 Module 1 Principles of Mathematics 12 Section Assignment 1 1 53 1 c f g x 2 e f x wherex lt 0 5 5 Without using your graphing calculator sketch the graph of g x 2 x 1 2 x 3 Label the intercepts and give the domain and range Module 1 54 Section Assignment 1 1 Principles of Mathematics 12 6 Given f x pom find f x 4 X 1 7 Find the maximum value of the function f x x4 3x2 x 7and 5 use it to determine the range of f Module 1 Principles of Mathematics 12 Section Assignment 1 1 55 8 Solve using your graphing calculator 1 a 2x4 3x3 x 2 E 2x x 3 Module 1 56 Section Assignment 1 1 Principles of Mathematics 12 9 Given f x 24 x g x V x 7 h x 2x 5 Determine a g h x 1 b f h 3 1 c h g f 1 2 Total 37 marks Module 1
14. ent 1 1 Version 06 Module 1 48 Section Assignment 1 1 Principles of Mathematics 12 General Instructions for Assignments These instructions apply to all the section assignments but will not be reprinted each time Remember them for future sections 1 Treat this assignment as a test so do not refer to your module or notes or other materials A scientific calculator and graphing calculator are permitted 2 Where questions require computations or have several steps showing these can result in part marks for some exercises Steps must be neat and well organized however or the instructor will only consider the answer 3 Always read the question carefully to ensure you answer what is asked Often unnecessary work is done because a question has not been read correctly 4 Always clearly underline your final answer so that it is not confused with your work Module 1 Principles of Mathematics 12 Section Assignment 1 1 49 Section Assignment 1 1 Review of Mathematics 11 Total Value 40 marks Mark values in margins ip hiss And 2x 1 1 a h 2 Module 1 50 Section Assignment 1 1 Principles of Mathematics 12 c h a 1 1 d ht x 2 Module 1 Principles of Mathematics 12 Section Assignment 1 1 51 4 2 For thethe function g x J2x 3_ find a thedomain of g b therange of g c they intercept d the x intercept Module 1 52 Section Assignment 1 1 Principles of Math
15. ewing Window on the T1 83 true for the Y axis For your graph to look right adjust the scale so that there are fewer values displayed along the X axis from 5 to 5 say and more displayed along the Y axis As a first guess use your cursor to set Xmin 5 and Xmax 5 The graph with the WINDOW Likewise set Ymin 30 and parameters at Ymin 30 and Ymax 30 Then press GRAPH Ymax 30 again Now the graph looks like the one at right This is better but we re still missing the upper loop of the graph Our final adjustment is to change Y max to 60 That yields the more appropriate display to the lower right We could change X 5 5 Y 30 60 the Xmin value from 5 to 3 for a more balanced look but what we have is good enough When you answer graphing calculator questions on the Provincial Exam you will hand sketch the graph in your calculator s viewing window then write in the Xmin and Xmax and Ymin and Y max values which you set for your window The example at right shows the correct way to write your window settings Module 1 Principles of Mathematics 12 Section 1 Lesson 1 13 Guided Practice Solve each of the equations below using your graphing calculator and showing the following detail Sketch the display as you see it For each of the possible solutions a state your guess b indicate the upper and lower bounds for x using interval notation For example 7 wou
16. f Thus f f x x and f 1 f x x The domain of f must be equal to the range of f 1 and vice versa The graphs of f and f t are related to each other in this way If the point a b lies on the graph f then the point b a lies on the graph of f1 and vice versa This means that the graph of f is a reflection of the graph of f t in the line y x Module 1 Principles of Mathematics 12 Section 1 Lesson 3 25 From Principles of Mathematics 11 you may remember that 1 Theinverse of a function is its reflection in the line y x Each point in the inverse function is the same distance away from the line but on opposite sides of the line Every point a b is transformed to b a 2 For an inverse function to exist the original function must be one to one Every x value must have only one y value and vice versa Thus the graph of the function must pass both the vertical and horizontal line tests Passing the vertical line test means that the original function is truly a function passing the horizontal linetest means that the inverse will also be a function Failing the vertical line test means that we have a relation but not a function like the circle relation in the illustration on the next page Module 1 26 Section 1 Lesson 3 Principles of Mathematics 12 gt lt
17. in and range are infinite in one direction but not in the other 4 In any function where x appears in the denominator of a fraction watch out for specific values of x where the denominator becomes zero At those points the equation has no meaning and the graph shoots off to infinity along an asymptote That x value must be excluded from the domain and the corresponding f x or y value if there is one must also be excluded from the range Module 1 Principles of Mathematics 12 Section 1 Lesson 2 17 You may find these guidelines helpful as you begin Math 12 and work through the course Most students find that domains and ranges become obvious enough that they don t need these guidelines for very long Example 3 Interval Notation One way of reading the set y y lt 3 ye R is All the real numbers between and 3 On a number line it would look like this 0 3 We can write this interval from up to and including 3 as 3 The means that the set doesn t include because infinity is unreachable and the means that the point 3 is included in the set So we see that another way of writing Range fy y lt 3 ye R is Range 3 In this way we can rewrite R as 0 00 4 lt x lt 3 as 4 3 amp 10 lt x lt I2 as 10 12 x 2 as 2 An interval where the end points are both included is called a closed interval and shown as An interval where both end poin
18. ing calculator Here s how your display should look Y4 3X43 13X 2 10 X 50 Y Ya Our font is a little different from that of the graphing calculator but hopefully you get the picture Notice the use of the N key It indicates to the graphing calculator that the operation is a power exponent Step 3 Now wegraph the polynomial function defined in Step 2 by pressing GRAPH You should see a display similar to the one shown here Graph 1 3x 13x 10x 50 Step 4 We identify the approximate value s of the zero s by inspection of the graph Sometimes we need to adjust the viewing window a little so that we can see where if the x intercepts or zeros occur but in this case all three are visible A cubic polynomial can have at most three Real zeros so we need not worry that some are not visible Module 1 10 Section 1 Lesson 1 Principles of Mathematics 12 a The least zero is in the interval 3 1 i e between 3 and 1 A guess might be 2 0 b The middle zero is in the interval 2 3 between 2 and 3 A guess might be 2 5 c The greatest zerois in the interval 4 5 between 4 and 5 A guess might be 4 5 Step 5 Now we will solve for the actual zeros one by one The calculator needs a few details The T1 83 will expect to receive them in this very specific order Function Variable Guess Lower bound Upper bound At this point different calcula
19. is case the graph of y f x is tangent to the x axis at x c as shown in Figures 1 2 and 3 y Figure 1 Cubic y x 1 x 3 rig in Pe x X Figure 2 Quartic y x 1 x 3 x 4 X Module 1 Principles of Mathematics 12 Section 1 Lesson 4 35 y Figure 3 Quintic 5th degree y xX x 1 x 3 which is a more efficient way of writing x y x 0 x 1 x 3 2 If a polynomial P x has a cubed factor such as x c then x c is a triple root of P x 0 In this case the graph of y P x flattens out or plateaus around c 0 and crosses the x axis at this point as shown in Figures 4 5 and 6 y Figure 4 Cubic y x 2 PENES eee X Figure 5 Cubic y x 27 Module 1 36 Section 1 Lesson 4 Principles of Mathematics 12 y Figure 6 Quartic y x 1 x 3 7 Polynomial functions may have relative maxima relative minima absolute maxima or absolute minima or a combination See the above Figure 6 for an example of absolute and relative minima The absolute minimum at 2 25 1 75 and a relative minimum at 1 0 Absolute and relative maxima would exist if the graph opened down Example 1 Sketch the graph of the function f x x2 x 1 x 2 Find the maximum and minimum points Solution Find the y intercept Set x 0 f 0 02 0 1 0 2 0 Therefore 0 0 is the y intercept for this function Find the x intercepts Set y 0 x
20. isplayed x value as the solution Repeat for each crossing point on the x axis In this case c you ll find a zero at 2 146348 which you round to 2 146 If you try the same thing for the relative maximum near x 1 5 which appears to touch the x axis you ll get an error message That s because the graph does not quite reach the x axis there the y value reaches 0 051 but not 0 You can solve equations using either MATH 0 Solver or CALC 2 zero Because the 2 zero function works off a graph it gives you a better chance of noticing and reporting every solution Module 1 Principles of Mathematics 12 Section 1 Lesson 4 41 Guided Practice 1 Rewrite each polynomial in descending order Determine i the degree of the polynomial ii the maximum number of turns in the graph a f x 8x xt b g x x 5x 2x c Alx 3x x x 2 State the left right behavior of the graphs in question 1 as to whether they open up open down rise from Quadrant III or fall from Quadrant II 3 Match each polynomials function with the correct graph a f x 3x 5 f x x 2x c f x 2x 9x 9 d f x 3x3 9x 1 iy e f x ZP x f f x 5x 2x o f x 3x 4x3 h f x x 5X 4x Module 1 42 Section 1 Lesson 4 Principles of Mathematics 12 iii Module 1 Principles of Mathematics 12 Section 1 Lesson 4 43 iv
21. ive number is not real so g cannot be less than Z But the value Range of g 0 4 is in the domain of g Now for the intersections Remember that is the symbol for intersection Domain of f 9 domain of f g domain of fxg amp x S301 3 0 Module 1 20 Section 1 Lesson 2 Principles of Mathematics 12 Range of f g range of f g range of fxg y y O v 0 0 0 c The domain of f g 4 co instead of 3 00 That s because g x is in the denominator and g 0 So 4 must be deleted from the combined domain The range of f g is 0 9 just as it is for f g f g and fxg The value 0 was excluded from the range of f already so it s not going to appear in the range of the combined function Sometimes when you write the range or domain of a combined function it may be simpler to leave the intersection symbol in your answer Finding Intercepts For more complex functions we often need to know the x and y intercepts in order to find the domain and range Rule To find the y intercept set x 0 To find the x intercept set y 0 Example 6 Find the x and y intercepts for the function f x x2 x 1 x 2 Solution y intercept Set x 0 f 0 02 0 1 0 2 0 x intercepts Set y 0 Solve x2 x 1 x 2 0 x2 0 o x 1 0 or x 2 0 x 0 1 or 2 Composition of Functions A composition of two functions is when they are arranged so that one is a function
22. ld indicate that x falls between 5 and 7 c state the actual solutions correct to three decimal places Your graph sketch goes here 1 xX x2 12x 3 2 x3 2x2 x 1 0 Module 1 14 Section 1 Lesson 1 Principles of Mathematics 12 3 x3 6x2 3x 5 0 XE oa IYE 4 X 3x2 9x 9 XE ge IYE 5 0 25x3 0 5x2 6x 2 0 XE cae IYE Check your answers in the Module 1 Answer Key Module 1 Principles of Mathematics 12 Section 1 Lesson 2 15 Lesson 2 Functions and Interval Notation Outcomes Upon completing this lesson you will be able to e identify the domain and range of various functions using set notation and interval notation e find the x and y intercepts of any function e perform operations on functions Overview The concepts of domain and range are necessary to describe functions Interval notation is a most convenient way to describe domain and range We will also review combinations of functions and composition of functions Definitions A function is a relation where each x value has only one y value For any function y f x the domain is the set of possible x values and the rangeis the set of possible y values We can evaluate a function at a particular point by substituting either numbers or algebraic constants into the f x expression and simplifying the result Example 1 If f x x 2x find f 0 f 2 and f a f 0 03 2 0 0 f 2 2 3 2 2 8 4 4 f
23. many repeats of Newton s method in the time it takes to press one button that your first guess need not be all that close toa true solution J ust make sure that your first guess is clearly closer to one solution or crossing point than it is to any other solutions Better yet set the Lower and Upper Bound so that they contain only one crossing point If your guess is sort of midway between two crossing points you can t control which one Newton s method will find Module 1 Principles of Mathematics 12 Section 1 Lesson 1 11 Do type in the commas and do not use spaces TI 83 Step 6 Press MATH scroll down to option 0 Solver Select it with ENTER You should now see EQUATION SOLVER Eqn 0 If you do not see EQUATION SOLVER scroll up If you see an equation already written in use the cursor up up arrow key to place the cursor on the equation Then use the CLEAR key to remove it Step 7 Type the following equation exactly 3X43 13X42 10X 50 Then press ENTER Next to the X on the next line type your first guess 2 0 remember to use the negation key not minus On the bound line type your lower and upper bound for the first guess between braces like this 3 1 Finally place your cursor on the X line and press ALPHA SOLVE ALPHA is the green key near 2nd and SOLVE is on the ENTER key
24. matics 12 Check F F aaa 1 x xie z 1 f F x 4 2 X 2 X 2 2 X b domain of f 0 given range of f 0 4 the maximum value of f is 4 when x 0 domain of f 1 range of f 0 3 range of f 1 domain of f 0 0 Example 3 Given f x 2 find Xx x 1 Step 1 ee Replace f x with y to make manipulation J x 1 easier Step 2 a 2y Switch x and y variables y l Step 3 x y D 2y To isolate y you need to eliminate fractions by multiplying both sides by y 1 Step 4 xy x 2y Expand bracket Step 5 xy 2y x Collect terms with y variable on one side Step 6 y x 2 2 Factor left hand side Step 7 Pe x Divide both sides by x 2 x 2 Module 1 Principles of Mathematics 12 Section 1 Lesson 3 29 Guided Practice 1 For each of the following functions f i find its inverse f 1 ii check your answer by showing that f f 1 x x and f 1 f x x iii find the domain and range of f and f 1 2 Find f 1 x given o x 1 f x 3x Check your answers in the Module 1 Answer Key Module 1 30 Section 1 Lesson 3 Principles of Mathematics 12 Notes Module 1 Principles of Mathematics 12 Section 1 Lesson 4 31 Lesson 4 Polynomial Functions and Their Graphs Outcomes Upon completing this lesson you will be able to e identify a polynomial function e relate its factors to its zeroes e graph a polyno
25. mial function Overview A polynomial function is an expression that can be written in he form a x PE ax a ag wheren is a non negative integer In the above polynomial each of the apx parts is called a term Terms are always added together or subtracted in polynomials never multiplied The a values are called coefficients of the terms Notes 1 The graph of a polynomial function is continuous This means the graph has no breaks you could sketch the graph without lifting your pencil from the paper 2 The graph of a polynomial function has only smooth turns The graph of f has at most n 1 turning points Turning points are points at which the graph changes from increasing as we move to the right to decreasing or vice versa Module 1 32 Section 1 Lesson 4 Principles of Mathematics 12 The function shown above is decreasing in the interval from D to E it remains constant from E toA and it is increasing in the interval from A to B Incidentally because of its sharp corners and its flat section it cannot be the graph of a polynomial function For the graphs that you investigated the cubic equation will have at most 3 1 turns or two turns For the graphs that you investigated the quartic equations will have at most 4 1 turns or 3 turns 3 a When the degree n of a polynomial is odd i e of degree 1 3 5 If the leading coefficient is positive gt 0 then the graph f
26. nking cursor on the graph if not use left arrow key to activate Use the arrows to move the cursor to the left of the lowest minimum value and press ENTER Use the right arrow to move the cursor to the right of the lowest minimum value and press guess ENTER again Move the cursor back close to the minimum as your Module 1 38 Section 1 Lesson 4 Principles of Mathematics 12 Press ENTER again The minimum occurs at 1 443001 2 833422 Therefore range 2 833 0 or y y 2 2 833 We use a closed interval because the graph reaches down to approximately 2 833 Example 2 Inspect the following polynomial functions and determine their basic shape and orientation Determine the domain and range of each a f x x3 4x b g x x6 4x2 c k x x 4x3 6 Hint The absolute minimum or absolute maximum always defines pe the range Solutions a f x is a third order cubic polynomial with a positive leading coefficient The graph rises from Quadrant III in an S shape Furthermore when factored f x x x 2 x 2 we can easily find its zeroes Solving x x 2 x 2 0 we get x 2 0 or 2 Because it has 3 roots it has 3 1 2 turns domain and range R b gisa sixth order polynomial with a negative leading coefficient so it opens down When factored we see that g x x2 x2 2 x2 2 x2 is a repeated factor x2 2 has no real factors
27. of the other Composition is not the same as combination using arithmetic operations between functions Module 1 Principles of Mathematics 12 Section 1 Lesson 2 21 The composition is written as fog x f g x or as gof x 9 f x either with a small hollow circle for the operation or with one function nested inside the other In this course f g x is the usual notation but we ll start by using both forms Example 7 If f x x 3 and g x 2x 1 find fog x and gof x fog x Ff g x f 2x 1 Substitute formula for g x 2x 1 3 Apply formula for f x 2x 2 Simplify gof x 9 f x Q x 3 Substitute formula for f x 2 x 3 1 Apply formula for g x 2x 5 Simplify Notes to remember 1 As Example 7 suggests gef x fog x except in special cases 2 For fog x f g x the range of g becomes the domain of f Example 8 If f x 2 and h x 2 x 1 write an equation for feh x Specify the domain and range Solution foh x f h x 7 3 2 x 1 To find the restrictions on the domain remember that the denominator cannot be zero 2 x 1 0 x l1 Module 1 22 Section 1 Lesson 2 Principles of Mathematics 12 To find the restrictions in the range write the function as ee ee rearrange and solve for 2 x 1 2y x 1 3 2xy 2y 3 2xy 3 2y _ 3 2y 2y Restriction y 0 Domain of f oh x 1 Range of f oh y y 0 Module 1
28. t If you make typing errors at any time you can always scroll to your error using the four cursor arrowhead keys and then 1 type over 2 usethe DEL key to delete or 3 use the nsert function by pressing 2nd and then DEL and then type more characters in the same space Note U pon first turning on your graphing calculator you should see a blank display if you don t press CLEAR In this mode your graphing calculator functions as any scientific calculator does thus enabling you to solve such equations as 2 2 or sin 25 Example 1 Solve the equation 3x3 13x2 10x 50 Solution First we rearrange the equation on paper so that we have zero on one side of the equation 3x3 13x2 10x 50 0 Module 1 8 Section 1 Lesson 1 Principles of Mathematics 12 Turn on your calculator and ensure that the memory is cleared by pressing 2nd then then scroll down to option 3 Clear Entries using the down arrowhead key and select that option by pressing ENTER Now you will see a confirmation screen so you press ENTER while the cursor is next to the words Clear entries Y ou will see the word Done Press CLEAR to get a blank screen Shortcuts You can select the menu options simply by pressing their number if you prefer not to scroll through the other options The 2nd key is used in th
29. the equation very tedious The graphing calculator can simplify the process A word of caution as we begin enter values and follow the steps slowly and carefully The calculator has no tolerance for entry errors no matter how small Also negative numbers must be signed using the negation Module 1 Principles of Mathematics 12 Section 1 Lesson 1 7 button just to the left of the ENTER button not the subtract operation button otherwise you ll get a syntax error message On other brands of calculators the key may bea or key In this lesson we use two sizes of hyphens to distinguish between the negation key and the subtract or minus key just as the TI calculators do For negation we use a short hyphen and for subtraction we use a longer one After this lesson we ll use just the longer dash in all equations you will know the rule by then for choosing the correct key Q What do do when the calculator says Syntax error A Choose option 2 Goto from your screen The blinking cursor will go directly to the error you made so that you can fix it As you go through the following examples perform the steps on your calculator rather than just reading the text You might want to go over the example a number of times until you feel comfortable with the functions As with any skill practice makes perfec
30. tors use different key sequences to solve equations The remaining steps 6 10 are for the T1 83 If you are using a different calculator look in the index of its user manual under Solving equations Besureto e usethe X 1T 6 n key for X e use the minus key within the equation itself e use the negation key for the guess and the bounds Want to know the math behind the button Calculators and computers solve functions with some form of Isaac Newton s method On a graph vrs Newton s method finds solutions or zeros x axis crossing points like this using your initial guess and the slope of the graph at the point of your guess it calculates where that slope a straight line of course crosses the x axis Then it takes the crossing point as a second guess it goes to the graph point directly above or below where the first slope crossed the x axis It calculates the new slope of the equation at that point goes to where that new slope crosses the x axis and repeats the process If a guess is close to a Zero or solution to a point where the graph crosses the x axis you can see that the graph s slope from that point will be almost parallel to the graph itself the slope s crossing point on the x axis will be close to the crossing point of the graph Only a few repeats will be needed before it homes in on the actual crossing point In reality graphing calculators perform so
31. ts are not included is called an open interval and shown as An interval where only one end point is included is called a half open interval and shown as either or We can form the union of two intervals in the same way that we form the union of two sets Remember that u is the symbol for union K x lt 5 xe Rh 0 lt x lt 4 xe R can be written as 0 5 U 0 4 Most rational functions have restrictions because the denominator of a function cannot be zero The domain and range both have restrictions Interval notation is a convenient way to express a restricted range or domain Module 1 18 Section 1 Lesson 2 Principles of Mathematics 12 Example 4 1 The rational function x _4 has asymptotes at x 2 and y 0 It has a y intercept at y 1 but no x intercepts Domain x 2 2 In interval notation this would be 090 2 U 2 2 U 2 0 which is very awkward Range 4 y lt 1 0r y gt 0 In interval notation this would be 1 4 L 0 9 y Note Remember an asymptote is a line that a curve approaches to infinity Combination of Functions Two functions can be combined arithmetically by xor The normal rules about addition subtraction etc apply 1 For example if 57 and g x v3x 2 then we can create new combined functions by simple arithmetic like this A a E t Neram Note There are two ways to show the addition of functions f x g x or f
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