Chapter 2 Analysis of Graphs of Functions

Transcription

1 Chapter Analysis o Graphs o Functions Chapter Analysis o Graphs o Functions Covered in this Chapter:.1 Graphs o Basic Functions and their Domain and Range. Odd, Even Functions, and their Symmetry.. Translations o Graphs: Vertical and Horizontal Shit, Stretching and Compressing, and Relections.. Graphs o Absolute Value Functions. Equations, Inequalities, and Applications..4 Piecewise Functions, Greatest Integer Function, and their Graphs..5 Algebra o Functions and their Composition. Dierence Quotient. Answers to Eercises. -48-

2 Chapter Analysis o Graphs o Functions Deinitions: Continuous Functions:.1 Graphs o Functions A unction is called continuous on its domain i it has no tears, gaps, or holes. That is, i it can be drawn without liting your pencil when you draw it on paper. Increasing and Decreasing Functions: A unction is called increasing on an interval I i whenever in I 1 1 A unction is called decreasing on an interval I i whenever in I 1 1 y B D y C A 1 0 a 1 b c d Increasing Increasing Decreasing Constant Functions: A unction is called constant on an interval I i whenever in I (or in general, whenever ) y y 5 Symmetry o Functions: Even Functions y y A unction is called an EVEN unction i it satisies the condition or all in its domain. i.e. I the point, is on the graph o, then the point, must be also on its graph. An EVEN unction is symmetric about the y-ais

3 Section.1 Graphs o Functions Eamples o Even Functions y 0 Odd Functions y y A unction is called an ODD unction i it satisies the condition or all in its domain. i.e. I the point, is on the graph o, then the point, must be also on its graph. An ODD unction is symmetric about the origin. Eamples o Odd Functions y 0 Tests or Symmetry: Symmetry about the y-ais (even unctions): Replace with in ( ). I ( ) ( ), then the graph is symmetric about the y-ais. Symmetry about the origin (odd unctions): Replace with in ( ). I ( ) ( ), then the graph is symmetric about the origin. Symmetry about the -ais: Replace y with y in the equation. I you get the same equation, then the graph is symmetric about the -ais. Remember that graphs symmetric about the -ais are not unctions. See the eample below. -50-

4 Chapter Analysis o Graphs o Functions Eample 1: Decide whether the unction is odd, even or neither. 4 Solution: First replace with in the given unction ( ) ( ) 5( ) 10 4 ( ) 5 10 ( ) ( ) ( ) The given unction is EVEN. (i.e. The unction is symmetric about the y-ais.) 5 10 Eample : Decide whether the unction is odd, even or neither. Solution: First replace with in the given unction. 5 4 ( ) 5( ) 4( ) ( ) 5 4 ( ) (5 4 ) ( ) ( ) nor ( ) The given unction is NEITHER odd nor even. 5 4 Eample : Decide whether the unction is odd, even or neither. Solution: First replace with in the given unction ( ) 5( ) 4( ) ( ) ( ) The given unction is ODD. (i.e. The unction is symmetric about the origin.)

5 Basic Functions and their Graphs Section.1 Graphs o Functions Identity Function Absolute Value Function y Domain: or (, ) Range: or (, ) Odd Function Symmetric about the origin y Domain: or (, ) Range: y 0 or [0, ) Even Function Symmetric about the y-ais Quadratic Function Square Root Function y Domain: or (, ) Range: y 0 or [0, ) Even Function Symmetric about the y-ais y Domain: 0 or [0, ) Range: y 0 or [0, ) Function is neither odd nor even Cubic Function y Domain: or (, ) Range: or (, ) Odd Function Symmetric about the origin -5-

6 Chapter Analysis o Graphs o Functions Cube Root Function y Domain: or (, ) Range: or (, ) Odd unction Symmetric about the origin ******************************************************** Eercises.1 In eercises 1 1, decide whether the unction is odd, even, or neither and state whether its graph is symmetric about the origin, y-ais, or neither. 4 1) ) 4 ) 5 4) ) 6 6) ) 1 8) 4 9) 1 10) ) )

7 Section.1 Graphs o Functions In eercises 1 4, decide whether the graph is symmetric about the -ais, y-ais, or the origin. 1) 14) 15) 16) 17) 18) 19) 0) 1) ) ) 4) -54-

8 Chapter Analysis o Graphs o Functions. Translations o Graphs In this section, translations o graphs will be discussed. We will be using a screen produced by a TI 84+ showing an original unction drawn in a thick line and some other unctions that were produced rom that original unction. We should be able to come up with a conclusion rom all o these screens. Vertical Shits The above two screens show that i ( ) is a unction and c is a positive number, then the graph o ( ) c is eactly the same as the graph o ( ) but shited down c units. The above two screens show that i ( ) is a unction and c is a positive number, then the graph o ( ) c is eactly the same as the graph o ( ) but shited up c units. Horizontal Shits The above two screens show that i ( ) is a unction and c is a positive number, then the graph o ( c) is eactly the same as the graph o ( ) but shited to the right c units. -55-

9 Section. Translations o Graphs The above two screens show that i ( ) is a unction and c is a positive number, then the graph o ( c) is eactly the same as the graph o ( ) but shited to the let c units. Vertical Stretching The above screens show that i ( ) is a unction and c is a positive number greater than 1 ( c 1), then the graph o [ c ( )] is eactly the same as the graph o ( ) but vertically stretched by a actor o c Vertical Shrinking The above screens show that i ( ) is a unction and c is a positive number such that 0 c 1, then the graph o [ c ( )] is eactly the same as the graph o ( ) but vertically shrunk by a actor o c. -56-

10 Chapter Analysis o Graphs o Functions Horizontal Shrinking The above screens show that i ( ) is a unction and c is a positive number greater than 1 ( c 1), then the graph o ( c) is eactly the same as the graph o ( ) but horizontally shrunk by a actor o c. Horizontal Stretching The above screens show that i ( ) is a unction and c is a positive number such that 0 c 1, then the graph o ( c) is eactly the same as the graph o ( ) but horizontally stretched by a actor o c. Relection About the -ais The above screens show that i ( ) is a unction, then the graph o [ ( )] is eactly the same as the graph o ( ) but relected about the -ais. -57-

11 Section. Translations o Graphs Relection About the y-ais The above screens show that i ( ) is a unction, then the graph o ( ) is eactly the same as the graph o ( ) but relected about the y-ais. Now, that we know all operations that can be perormed on a given graph, we should be able to answer the ollowing eamples. Eample 1: The graph o the unction ( ) is shited to the right units, then vertically stretched by a actor o 4, then relected about the y-ais, and inally shited down 5 units. Write the inal translated unction h() and give its graph. Solution: ( ) Shited to the right units Vertically stretched by a actor o 4 4 Relected about y the -ais 4 Shited down 5 units 4 5 h( ) -58-

12 Chapter Analysis o Graphs o Functions Eample : The graph o the unction ( ) is shited to the let 4 units, then vertically shrunk by a actor o 1/, then relected about the -ais, and inally shited up units. Write the inal translated unction h() and give its graph. Solution: ( ) Shited to the let 4 units 4 Vertically shrunk by 1 a actor o 1/ 4 Relected about 1 the -ais 4 Shited up units 1 4 ( ) h Eample : The graph o the unction on the graph o the unction is obtained by perorming the ollowing transormations g in the given order: shited horizontally 5 units to the let, then vertically stretched by a actor o 4, relected about the y-ais, and inally shited downward units. Find the rule o the unction g. Solution: We will show two methods to solve these types o problems. Method 1: Following the eact steps as given and equating the inal answer to shited to the let vertically stretched relected about g g g g 5 units by a actor o 4 the y-ais shited down 4g 5 4g 5 ; solve or g units 4g 5 ; add to both sides o the equation 4g 5 ; multiply both sides o the equation by 1/4

13 1 g 5 ; relect about the y-ais g g ; shit to the right 5 units to obtain g Section. Translations o Graphs Note: The absolute value o any negative number is equal to the absolute value o the number itsel. Method : The unction g is shited to the let 5 units, vertically stretched by a actor o 4, relected about the y-ais, and inally shited down units to obtain the unction Perorm the opposite o each transormation in reverse order on g. The unction. to obtain the rule o the unction is shited up units, relected about the y-ais, vertically shrunk by a actor o 1/4, and inally shited to the right 5 units. Shit Relect up units: about the y-ais: 1 Vertically shrink Shit g by a actor o 1 4 : to the right 5 units: 5 5 g

14 Chapter Analysis o Graphs o Functions Eercises. In eercises 1 8, write the unction g whose graph can be obtained rom the graph o the unction by perorming the transormations in the given order. Graph the unction g. 1) ; shit the graph horizontally units to the right and then vertically downward 5 units. ) 1; shit the graph horizontally units to the let and then vertically upward 4 units. ) 1; relect the graph about the y-ais, then shit it vertically downward units. 4) 4 4; relect the graph about the y-ais, then shit it vertically downward 1 unit. 5) 1; relect the graph about the -ais, then shit it vertically upward units. 6) 4 4; relect the graph about the -ais, then shit it vertically upward units. 7) ; shit the graph horizontally to the right 4 units, stretch it vertically by a actor o, relect the graph about the y-ais, then shit it vertically upward 5 units. 8) ; shit the graph horizontally to the let units, stretch it vertically by a actor o, relect the graph about the y-ais, then shit it vertically downward 1 unit. In eercises 9 14, use the graph o the unction g in the given igure to sketch the graph o the unction. 9) g 10) g 11) g 1) 0.5g 1) g 14) g -61-

15 Section. Translations o Graphs In eercises 15 0, use the graph o the unction g in the given igure to sketch the graph o the unction. 15) g 1 16) g 17) g 18) 0.5g 19) g 0) g 1 1) The graph o the unction is obtained by perorming the ollowing transormations on the graph o the unction g in the given order: shited horizontally units to the right, then vertically shrunk by a actor o 0.5, relected about the -ais, and inally shited downward units. Find the rule o the unction g. ) The graph o the unction is obtained by perorming the ollowing transormations on the graph o the unction g in the given order: shited horizontally units to the let, then vertically stretched by a actor o, relected about the y-ais, and inally shited upward 5 units. Find the rule o the unction g. ) The graph o the unction is obtained by perorming the ollowing transormations on the graph o the unction g in the given order: shited horizontally 1 unit to the right, then vertically shrunk by a actor o 0.5, relected about the y-ais, and inally shited downward units. Find the rule o the unction g. 4) The graph o the unction is obtained by perorming the ollowing transormations on the graph o the unction g in the given order: shited horizontally 4 units to the let, then vertically shrunk by a actor o 0.5, relected about the -ais, and inally shited downward units. Find the rule o the unction g. -6-

16 Chapter Analysis o Graphs o Functions Absolute Value o a Function:. Absolute Equalities and Inequalities i 0 i 0 The above notation is simply saying that when the graph is zero or positive, leave it as it is. I the graph is negative then relect it about the -ais to make it positive. 4 Eample 1: Given that 0.1, use your graphing calculator to graph y. Solution: The irst screen below shows the graph o the unction (). The second screen shows both the graph o the unction in thick lines and the graph o its absolute together. The last screen shows only the graph o the absolute o the unction. Absolute Equations: i 0 i 0 Absolute value o measures the distance rom to zero. Remember that distance must be non-negative. Thereore 7 7, 1 1, 0 0, 5 5, 10 10, and so on since the distance rom any number to zero is the positive case o that number. Eample : Solve the absolute equation. Solution: The problem is simply saying that i the distance rom to zero is units, what is? We know that the distance must be non-negative. I we want to measure units rom zero, we have two ways to do so. Either we count units to the right (this case we get +) or we can count units to the let (this case we get -). This means we have two answers or, + and -. The igure below shows how we got these two answers. units units

17 Section. Absolute Equations and Inequalities To generalize the method o solving absolute equations, we have the ollowing rules: I a b c where c is a positive number ( c 0) then, a b c or a b c I a b c where c is a negative number ( c 0) then the equation has no solution I a b 0 then, a b 0 Eample : Solve the absolute equation 5. Solution: First we need to apply the rule above and remove the absolute symbol by writing two equations. 5 5 or 5 5 ;, 8 8 Eample 4: Solve the absolute equation Solution: We have to isolate the absolute epression beore removing the absolute symbol. Thereore the equivalent equation is Now we should be able to write two equations and solve the problem or ;,9 9 Eample 5: Solve the absolute equation 7 8. Solution: We have to isolate the absolute epression beore removing the absolute symbol. Thereore the equivalent equation is 7 5. But, we know that absolute value must be non-negative all the time. Thereore, the given absolute equation has no solution. Eample 6: Solve the absolute equation Solution: In this type o problems when you see zero on one side o the equation, simply remove the absolute symbol and solve the equation as usual

18 Chapter Analysis o Graphs o Functions Solving Absolute Equations o the Type a b c d I a b c d, then a b c d or a b c d Eample 7: Solve the absolute equation 7 5. Solution: Follow the above rule and write two equations to solve the problem , 5 1, 5 Eample 8: Solve the absolute equation 5. Solution: Write two equations and solve the problem No Solution Note: In the last eample, don t write the answer is no solution or -1/. Just write -1/ Solving Inequalities o the Form a b c Solving Absolute Inequalities I a b c where c is a positive number c 0, then a b c and a b c I we write the above as continued inequalities, we can write the inequality as ollows: c a b c I a b c where c is a negative number c 0, then the inequality has no solution I a b 0, then a b 0 Note: In the above rule, due to space limitation, we used the symbol o inequality less than or equal. The less than symbol also applies to this rule. -65-

19 Section. Absolute Equations and Inequalities The igure below shows why we call the above inequality the AND case. Notice that the interval o the solution is one continuous interval. I is in the interval (-, ), we can see that is less than AND is greater than -. less than units less than units Eample 9: Solve the absolute inequality. 5 Solution: First we need to apply the rule above and remove the absolute symbol by writing continued inequalities as ollows: Set Notation: { 8 } Interval Notation: [-8, ] Eample 11: Solve the absolute inequality. 7 8 Solution: We have to isolate the absolute epression beore removing the absolute symbol. Thereore the equivalent inequality is 7 5. But we know that absolute value must be nonnegative all the time. Thereore, the given absolute inequality has no solution. Eample 10: Solve the absolute inequality Solution: We have to isolate the absolute epression beore removing the absolute symbol. Thereore the equivalent inequality is Now we should be able to write the inequality as continued inequalities and solve the problem Set Notation: { 9} Interval Notation: (,9) Eample 1: Solve the absolute inequality Solution: In these types o problems when you see that the inequality epression is less than or equal to zero, remove the absolute symbol, replace the inequality symbol with an equal sign and solve the equation

20 Chapter Analysis o Graphs o Functions Solving Inequalities o the Form a b c I I a b c where c is a positive number c 0, then a b c or a b c I a b c where c is a non-positive number c 0, then the solution is the set o all real numbers b a b 0, then the solution is all real numbers ecept a b b that is:,, a a Note: In the above rule, due to space limitation, we used the symbol o inequality greater than or equal also applies to this rule.. The greater than symbol The igure below shows why we call the above inequality the OR case. Notice that the interval o the solution is two separate intervals. I is in the interval, then is less than -. I is in the interval, then is greater than. Since cannot be in two places at the same time, we have to say that is less than - OR is greater than. greater than units greater than units Eample 1: Solve the absolute inequality 5 Solution: First we need to apply the rule above and remove the absolute symbol by writing the absolute inequality as two separate inequalities or 5 5 Set Notation: { 8 or } Interval Notation: (-,-8] [, ) Eample 14: Solve the absolute inequality 7 8 Solution: We have to isolate the absolute epression beore removing the absolute symbol. Thereore, the equivalent inequality is 7 5. But, we know that absolute value must be non-negative all the time. Thereore, the solution to the given absolute inequality is the set o all real numbers. -67-

21 Section. Absolute Equations and Inequalities Eample 15: Solve the absolute inequality Solution: We have to isolate the absolute epression beore removing the absolute symbol. Thereore, the equivalent inequality is Now we should be able to write the inequality as two separate inequalities and solve the problem or Set Notation: { or 9} Interval Notation: (, ) (9, ) Eample 16: Solve the absolute inequality 10 0 Solution: In these types o problems when you see that the inequality epression is greater than zero, remove the absolute symbol, replace the inequality symbol with an equal sign and solve the equation The solution is all real numbers ecept 5 or ecept 5. Interval notation: (-,5) (5, ). Now that we have discussed all types o absolute equalities and inequalities, we should be able to solve an eample with absolute equality and its related inequalities. Eample 17: Solve the absolute equality and its given related inequalities. Solution: a) 7 b) 7 c) 7 a) 7 7 or 7 7 or 7 4 or 10 or 5 {,5} b) Interval Notation: [,5] Set Notation: { 5} -68- c) 7 7 or 7 7 or 7 4 or 10 or 5 Interval Notation: (, ] [5, ) Set Notation: { or 5}

22 Chapter Analysis o Graphs o Functions Eample 18: Use your graphing calculator to solve the absolute equality and its related inequalities. a) b) c) Solution: First enter the let side o the equality as Y 1 and the right hand side as Y. Move cursor close to the other intersection From the inal screen we can see that: a).5,7.5 b) [-.5,7.5] c) (,.5] [7.5, ) -69-

23 In eercises 1 16, solve the absolute equations. Eercises. Section. Absolute Equations and Inequalities 1) 5 4 ) ) 5 4) 1 7 5) 7 8 6) 4 1 7) ) ) 5 10) ) ) ) 5 14) ) 1 16) In eercises 17 40, solve the absolute inequalities. 17) 4 18) ) 5 0) 4 1 1) 4 5 ) 4 6 ) ) 6 5) 5 6 6) ) ) ) 5 0) 4 1) 5 ) ) 4 4) 5 8 5) ) ) 5 6 8) ) ) In eercises 41 46, solve the absolute equation and its related inequalities (see eample #17). 41. a) 1 4. a) 6 4. a) 1 9 b) 1 b) 6 b) 1 9 c) 1 c) 6 c) a) a) a) 4 5 b) 4 5 b) 5 11 b) 4 5 c) 4 5 c) 5 11 c)

24 Chapter Analysis o Graphs o Functions.4 Piecewise Functions Deinition: A piecewise unction is a deined unction that consists o several unctions (i.e. The graph o a piecewise unction consists o several graphs. Each one o these graphs is also a unction). Graphs o parts o a piecewise unction usually have one or more o the ollowing: holes, sharp corners, and jumps. Eample 1: Graph the ollowing piecewise unction and evaluate it at the given values., i 0 ;,, 0 5, i 0 Solution: To graph the given piecewise unction, we need to graph each part separately. Be sure to graph each part within its deined domain. You can see that the irst part which is a parabola is deined only over the interval (,0] and the point zero is included (closed circle). The second part represents a line. The line is deined only on the interval (0, ) where point zero is ecluded (open circle). Now, press TRACE and enter the values or at which you want to evaluate the unction. 5 1, use the second unction since is in its domain only. 4, use the irst unction since - is in its domain only 0 0 0, use the irst unction since 0 is in its domain only Computer generated graph : -71-

25 Section.4 Piecewise Functions Note: There are other ways to enter the piecewise unction o eample #1 in our TI. See the ollowing screens to see how this can be done. In our opinion, the irst method is easier than these other methods. Method Method Greatest Integer Function (Step Function): Notation: ; which means the greatest integer that is less than or equal to. Equality will occur only when is an integer. I not, always round down. Eample : Evaluate the ollowing:.4, 5, 1., 0, 1.45, 7 Solution: Since the value o the unction is equal to i is an integer, we need to round down i is not, 5, 1,0,, 7 an integer, Graphing Calculator Solution o Eample : Press MATH NUM 5:int( Graph o the Greatest Integer Function : Computer G enerated G raph : Domain: The set o all real numbers or, Range: The set o all integers -7-

26 Chapter Analysis o Graphs o Functions Using a graphing calculator to graph : To ind "int", press MATH NUM 5:int( Don t be deceived by the given TI graph. This is a lack o technology. All vertical segments are not part o the graph. I you want to see a better graph, you need to change the MODE in your TI rom Connected to Dot. To do this, press MODE, move the cursor down to Dot and press ENTER to highlight Dot. I you do this you ll get the screen below. Now you can see why we call this unction Step Function. Note: The TI graphing calculator will not show the open or closed end o each segment. Eample : Graph the unction 1, evaluate 1.,,., 4,and 7.1 Solution: Since the value o the unction is equal to i is an integer, round down i is not an integer

27 Section.4 Piecewise Functions Computer G enerated G raph : Using the Table Feature: Press nd TBLSET, highlight Ask to the right o Indpnt:, and highlight Auto to the right o Depend:. Press nd TABLE, enter the value or at which you want to evaluate the unction, and then press ENTER. From the last screen, 1., 4,. 1, 4, and

28 Chapter Analysis o Graphs o Functions Eercises.4 In eercises 1 10, graph the unction and evaluate it at the given values. i 1), evaluate 4, 5,and 5 i i 1 ), evaluate 4, 5,and i 1 1 i 1 ), evaluate 0, 1, 1,and i 1 i 4), evaluate 0,,,and i i 0 5), evaluate 0, 1,and 1 4 i 0 i 0 6), evaluate 0, 1,and 1 i 0 i 1 7), evaluate, 1,and 5 i 1 i 8), evaluate 0,,and i 1 i 9) i 1, evaluate 0,, 1,,and 4 1 i 1 5 i 1 10) i 1, evaluate 0, 1,,,and 4 9 i -75-

29 Section.4 Piecewise Functions In eercises 11 18, graph the unction and evaluate it at the given values. 11), evaluate., 5, 4.1,,and 5.1 1) 1, evaluate 1.,,., 4,and 7.1 1) 1, evaluate 1., 5., 0,.5,and 7 14), evaluate., 4, 0.1, 6,and ) 1, evaluate 1.,,.1,,and ) 1, evaluate 4., 1, 0, 5,and.1 17), evaluate., 1, 0,,and 7. 18), evaluate 5.,, 0,.,and 5-76-

30 Chapter Analysis o Graphs o Functions.5 Algebra o Functions and their Composites Two unctions and g can be combined to orm a new unction h eactly the same way we add, subtract, multiply, and divide real numbers. Let and g be two unctions with domains A and B respectively then: 1 g g h ( ) g g domain { A B} h ( ) g g Algebra o Functions domain { A B} h ( ) g g domain { A B} h4 ( ) domain { A B and g 0} The graph o the unction h( ) ( g)( ) can be obtained rom the graph o and g by adding the corresponding y-coordinates as shown on the igure below. Similarly, we can ind the dierence, product, and division o and g. In case o division, remember that g ( ) 0. So any value o that makes the denominator equal to zero must be ecluded. Eample 1: I ( ) 5 and g( ), ind ( g)( ), ( g)( ), ( g)( ), the ( )( ). g Also evaluate ( g)(1) and ( )(). g Solution: g ( ) ( ) g( ) 5 g ( ) ( ) g( ) g ( ) ( ) g( ) 5 15 ( ) 5 ( ), g g( ) To evaluate g(1), and () : g g (1) (1) () 5 1 () g 5

31 Composition o Functions Notation: g g Section.5 Algebra o Functions and their Composites The above can be read in many ways. It can be read as composed with g at, composed with g, the composition o and g, o g o, or circle g. The Composite Function gis deined by g g Its domain is all numbers in the domain o g such that g is in the domain o g g g g Domain o g Domain o Range o g Range o Eample : I g g g Solution: 5 and, ind and. g g g g g Note: In general g g Eample : I 7 and g 5 1, ind g Solution: We can use two dierent methods to solve the problem. First Method: Find g g and then substitute or g g g () 5 5 Second Method: g g g, evaluate g and substitute the result into g g

32 Chapter Analysis o Graphs o Functions The Dierence Quotient The dierence quotient epression is very important in the study o calculus. Looking at the graph below, the dierence quotient is the slope o the line PQ which is called the secant line. y Q h, h P, h h Secant line h From the graph, we see that the slope o secant line PQ is m PQ change in y Rise h. change in Run h Dierence Quotient h m h Eample 4: Let 5, ind the dierence quotient and simpliy your answer Solution: For clarity, we will solve this eample step by step. ( ) h h h 5 h h h 5 h h h h h h h h 5 h h h h h h ( ) h h( h ) m h h h 5 Note 1: The simpliied orm o the numerator h should have h as a common actor. Note : Review your work i all o the terms o the original unctions () don t cancel in the inal epression o the dierence quotient. -79-

33 Let and g 4 5 Section.5 Algebra o Functions and their Composites Eercises.5. Find the ollowing g g g g g 6. g 7. g 8. g g g 1 Let 4 and g. Find the ollowing g g g g g g g g g g Let 4 1 and g. Find the ollowing. 1. g. g. 4. g g g g 8. g g Let and g 1. Find the ollowing. 9. g 0. g 1.. g g g g 6. g g In eercises 7 44, ind the dierence quotient o each given unction and simpliy the answer completely

34 Chapter Analysis o Graphs o Functions Answers - Eercises.1 1. Odd. Even. Even 4. Neither 5. Odd 6. Neither 7. Even 8. Odd 9. Neither 10. Even 11. Odd 1. Odd 1. y-ais 14. Origin 15. Origin 16. Origin 17. y-ais 18. All 19. All 0. All 1. All. All. -ais 4. y-ais Answers - Eercises. 1. g. g. g 4. g 4 5. g 1 6. g g g g 6. g g g 4 4 1

35 Answers to Chapter Problems Answers - Eercises. 1. 1,9. 1,5. 4,1 6., , 5 5 8, 17. 1, , , , , , , ,. 10 8, , 7, 0., 71, 1., 1,. 9,, 4. 5, 5, 7 5., 1, 5.,5 5, 6.,, a),1, b), 1,, c),1 4. a) 4,, b), 4,, c) 4, 4. a) 5.5,.5, b), 5.5.5,, c) 5.5, a) 6.5, 4.5, b), ,, c) 6.5, a), b), c) 46. a), b), c) Answers - Eercises , -8,. -15, -7, 1. 1,, 5, 8 4., -4, 1, 5., 1, , -5, , 1, , -7, -9-8-

36 Chapter Analysis o Graphs o Functions 9. 0, -,, -5, , -6, 7, -8, , -7,, 1, 1. -, -4, 1,, , -5, 1,, , -, 1, 8, ,, 1, 7, , -5, -4, 1, , 1,, 4, , -5, -, -1, 1) 6 ) 8 ) 8 15 Answers - Eercises.5 4), ) 8 6) 1 7) 4 8) 8 9) 17 10) 1 11) 1 1) 7 1) 5 14) 4 ; 1 15) 7 16) 8 17) 4 18) 7 19) 0) 1 1 1) 8 11 ) 8 1 ) ) 4 9 5) 19 6) 15 7) 1 8) 9 9) 9 6 0) 10 1) 6 6 ) 9 4 ) 4) 10 5) 6) 5 7) 4 8) 9) 5 40) 41) h 4) h 4) h 44) 4 h -8-