FUNCTIONS AND RELATIONS CHAPTER OBJECTIVES:. Concept of function f : f ( ) : domain, range; image (value). Odd and even functions Composite functions f g; Identit function. One-to-one and man-to-one functions. Inverse function f including domain restriction. Self-inverse functions.. The graph of a function; its equation f ( ). Investigation of ke features of graphs such as maimum and minimum values, intercepts, horizontal and vertical asmptotes, smmetr, and consideration of domain and range. The graphs of the functions, f ( ) and f ( ). The graph of f given the graph of f ( ).. Transformations of graphs: translations; stretches; reflections in the aes. The graph of the inverse function as a reflection in.. a b The rational function, and its graph. c d The reciprocal function is a particular case. Graphs should include both asmptotes and an intercepts with aes. BEFORE YOU START You should know how to:. Sketch linear relationships, find the gradient and aes intercepts for linear graphs. Find the equation of a line perpendicular to the given line.
m c is an equation of a straight line with gradient m and - intercept c. To find the -intercept, make 0. To straight lines are perpendicular if the product of their gradients equals -. Rule:. Appl three equivalent representations: tabulate values, determine the rule, and plot the graph. Table: - 0-5 7 Graph: = + - - - -. Sketch graphs of quadratic functions with aes intercepts and a turning point. a( h) k is an equation of a quadratic function in a turning point form, where a is called a dilation factor. a b c is the general equation of a quadratic function. a( )( ) is a factorised form of a quadratic function.
. Use set and interval notation. [, ) 5. Evaluate epressions. 5 5 0 0 CHECK IN EXERCISE. The linear graph is given b the following rule 6. a. Find the gradient and aes intercepts. b. Sketch the graph. c. Find the equation of the line perpendicular to 6and passing through the point (,).. Annika works as a demonstrator for Tupperware. The compan offers her $95 per one part hosted, plus $8 for each set of baking products sold. a. Create a table of values showing Annika s income per part in terms of the number of baking products sold. b. Form a rule linking her income to the number of baking products sold. c. Plot our rule on the coordinate aes.. A parabola is defined b the equation 6. a. Epress the equation in a turning point form. b. Epress the equation in the factorized form. c. Sketch the parabola, showing the coordinates of the turning point (verte) and aes intercepts.. Solve the following inequalit interval notation. 5. Evaluate. Give our answer in set and a) b) 0 c) d) 0.
CHAPTER FUNCTIONS & RELATIONS: Eploring the power of smbolic language. In our studies of mathematics ou have learnt to use mathematical language and mathematical notation. It consists of definitions, smbols and special names given to mathematical objects and operations. When scientists tr to communicate with other intelligent beings in space, the code the information using mathematical smbols. It is generall accepted that mathematical language is universal and thus should be understood b other civilizations. There is evidence that the number eists in nature and has not merel been invented b mathematicians. Mathematical language is ver powerful; it can communicate mathematical knowledge much better than words. In this chapter ou will learn how to use function notation and how to use smbolic language to describe operations linked to functions. Did ou know that the notion of function was not clearl defined until about 67? It was Leibniz (66-76), who introduced the term function into mathematics. Before Leibniz, Descartes (596-650) clearl stated that an equation in two variables, geometricall represented b a curve, indicates dependence between variable quantities. Mentions and eamples of functions ma be found in ancient times; for eample, counting, where a particular relationship eists between a set of given objects and a sequence of counting numbers. The concept of function did not happen in mathematics b chance. It appeared as the necessar mathematical tool to eplain natural phenomena, begun b Galileo (56-6) and Kepler (57-60) in their stud of planets, followed b Newton to model his laws of motion and describe the relationship between force, mass and acceleration. It is difficult to imagine contemporar mathematics without one of the most useful tools such as functions.
. Functions and relations. Function Machine Investigation A function machine feeds on special fractions. When we input a fraction, the function machine will produce an output fraction output. fraction a. We will call output and input. Therefore. b. Find the output for inputs,, and. What do ou notice? 5 c. Feed back our answers from b into the function. What are the outputs? d. Repeat steps b and c with a few more fractions where the denominator is one more than the numerator. What do ou notice? e. For an original input fraction a, what would the result be after one a million processes? Tr to prove our result algebraicall. Hint: substitute for into function. a a Testing for a function. As observed in the investigation, when we define a function we need to define an input (often called an independent variable), in this case, and an output given b a rule, usuall called, the dependent variable. Not all relations between two variables are functions. There are specific conditions which need to be fulfilled for a relation to be a function. A relation is a relationship between two sets, sa A and B, such that each element in set A will have a corresponding element in set B. We often sa that the elements of set A are mapped into elements of set B. This mapping ma be assigned in different was. Let sa that we consider set A as all students in our class and set B as all pets which belong to the students. Then we create a special relationship or mapping students pets. Of course it is possible that a particular student will have more than one pet, some students will have no pets and some will have 5
.5 0.5 0.5.5.5 0.5 0.5.5 just one. This relation is not considered to be a function due to the fact that some students from set A ma be assigned more than one element in set B. Thus a relation is a set of ordered pairs, usuall defined b a rule. It can be plotted on the coordinate grid. Here are graphs of si relations between variables and. Relations a, c and d are functions. Relations b, e and f are not functions. What differences do ou notice between the graphs of the functions and the other graphs? a) b) c) 5-5 - - - - - - - - 5 6 7 d) e) f) 6-0 -5 5 0 - - -6 5 - - - 5 - - - -5 There is a simple test called the vertical line test to test for a function. If a vertical line cuts the graph once onl anwhere, it is a function. If it cuts more than once at some points, the relation is not a function. Appling the vertical line test to the graphs above results in the following: a) b) c) 5 - - - - - 5 - - - - 5 6 7 d) e) f) 6-0 -5 5 0 - - -6 5 - - - 5 - - - -5 6
Clearl, the vertical line cuts the graph more than once in graphs b) e) and f). To define a function in a more formal wa, we need to sa: For a relation to be a function, the following condition must be true: for an value of there is one and onl one value of. It follows that all functions are relations but not all relations are functions. A function is a mapping of elements in one set to elements in another set in such a wa that an element in the first set has one and onl one element corresponding to it in the second set. You can represent a rule in a mapping diagram. f ( ) f(-)= f(-)= f(0)=0 f(0.5)=0.5 f()= f()=8 and so on. - - 0 0.5 8 0 0.5 We can see that one value of f(). f ( ) is a function because each value of maps to onl The relation is not a function. It can be re-arranged to. The mapping diagrams show that there are two values of for an value of ecept for. 7
(0) () 0 ( ) and so on. 0-0 - - Eample Is the relation {(,),(,),(,0),(,),(0,),(,),(,),(,5)} a function? Enter the values in the spreadsheet on our GDC and plot XY Line Plot. Now appl the vertical line test to conclude that the relation is a function. 8
Eample Is the following relation a function? The relation is a circle and vertical line test cuts more than once, therefore it is not a function. Eample Is the following relation a function? This time we onl have the upper semicircle which is a function. One-to-one and man-to-one functions There are two tpes of functions: one to one and man to one functions. Look again at these functions. Graphs a) c) and d). Graphs a) and d) show a oneto-one functions while graph c) shows a man-to-one function. For a one-toone function, each -value maps to one onl -value. For a man-to-one function, several -values map to the same -value. 9
To test for one-to-one function, we appl the horizontal line test. If the horizontal line cuts the graph of a function once onl, anwhere, it is a one-toone function. If it cuts more than once, at an point, it is man-to-one function. Appling the horizontal line test to graphs a), c) and d) results in the following: a) c) d) 5 - - - - - - 5 6 7 6-0 -5 5 0 - - -6 Therefore, graph c) shows a man-to-one function and graphs a) and d) show one-to-one functions. Eample Consider the two functions f ( ) and f ( ). Which of these is a one-to-one function? An straight line is a one-to-one function and an parabola is a man-to-one function which is also illustrated b graphing both functions and appling horizontal line test. 5-5 - - - - - 5 - - - -5 - - - - - - 0
Eercise A. Appl vertical line test to determine which of the relations are functions. a) b) c) 0 5 - - - - - - - -0-5 5 0-5 - - -0 d) e) f) -0-8 -6 - - - -6 - - 6 - - - 0 5 - - -5-0. Which of the functions in Question are one-to-one?. Which of the following relations is a function? a) 5 b) c) {(,),(,),(,5),(,),(,)} d) 0 e) f). Use a GDC to sketch the following graphs and in each case appl the horizontal line test to determine if a given function is one-to-one or man-toone. a.. b. c. d. e. f. ( ) g. f. h. 5. Draw graphs of: a) three relations which are not functions b) three relations which are functions. 6. Draw two graphs of: a) one-to-one functions b) man-to-one functions.
. Function notation. Domain and range. Independent and dependent variable. There are man was we can define a function. We use small letters such as f, g, h to name a particular function. Ever time we define the function we need to specif the argument and the domain. The domain is the set of all the first elements (independent variable) of the ordered pairs (the -values on the Cartesian plane). The range is the set of all the second elements (dependent variable) of the ordered pairs (the -values on the Cartesian plane). We often talk about variables. When we consider an equation, is an independent variable and is a dependent variable. With respect to functions, the independent variable is often called an argument of the function. The dependent variable is called the value of the function. In real life situations we come across other variables. For eample we ma have the relationship between volume of a tank being filled with water as a function of time. In this situation time is considered to be independent variable and volume is a dependent variable. If the tank is being filled with water in 0 minutes then the domain of the function will be all times from 0 minutes to 0 minutes. Assuming that the tank can hold 00 litres of water, the range will be all values of volume between 0 litres and 00 litres. It can be written using interval notation as: t [0,0] V [0,00]... Function notation Eample 5 The equation R R, can be re-written in function notation as follows: What it means is that the set of real numbers R (domain) is mapped onto set of real numbers R (co-domain) following the mapping such that each value is mapped into value. To determine the range of the function we ma consider its graph: = - - - - Intercept Local Minimum ( 0, - )
It can be seen from the graph that can onl takes values greater or equal to -. Therefore the range of the function R R, is or [, [ Note the difference between co-domain and range. In general co-domain is usuall R whilst range will var depending on the rule. Eample 6 Determine the domain and range of the function R R, 5 Use GDC to sketch the graph and find the coordinates of the verte. Domain: R Range: ],6.] Eample 7 Given that function f() is defined as f :[0, ) R,, a) a) find the domain and range of f(). b) solve the equation f( ).
The square root is onl determined if the argument is greater or equal to zero. It is indicated in the definition of function f(). The range can be read off the graph on GDC above. Domain: [0, ) Range: [0, ). b) f( ) Square both sides: 6, 8 Eample 8 Given that function g() is defined as g : R \{0} R,, find the domain and range of g(). GDC gives the following graph: An fraction is undefined if the denominator is equal to zero. Therefore 0 has to be ecluded from the domain. Also, there is no value of for which would equal 0. R \{0}; R \{0} determines the domain and range of g().
Evaluating functions. One we have defined a function, we ma then evaluate it for different arguments. Eample 9 Given two functions: f : R R, g : (0, ) R, evaluate the following: f f (0) 0 0 ( ) ( ) 8 f 7 f ( a) a g() g() gb ( ) b g(0) undefined g( ) not in domain Using a GDC to evaluate functions: 5
Restricting the domain. Consider the following functions: f : R R, g :[,] R, h : [0,) R, = 0 5 - - -5-0 = ; 0 8 6 - -.5 = ; 0 < 0.5 - -0.5 - We used different letters, f, g, and h to name three functions to emphasize the fact that even if the rule is the same but domain is different, it then becomes a new function. We call g() and h() restrictions because their domains are subsets of the domain of f(). Restricting domain on GDC To restrict domain on a GDC, we use the ctrl = as shown in the screen shots below: sign, which can be found pressing The GDC snta is: GDC onl displas the part of the f ( ) parabola with the restricted domain. 6
Eample 0 Sketch the graph of the following function: f :[,6) R, and determine its range. f( ), f(6) 5 8 6 - - 6 8 - - -6 Range: ( 5,] or 5. Eample Use function notation to define the function in the diagram below: 0 ( 5, ) 8 6 (, 8 ) gradient = 8 5 equation: 6 8 0 7
8 ( ) 6 Domain: 5 Range: 8 6 f ( ) : (,5] R, Implied domain. It is often necessar to determine the maimal domain of the function, also called the implied domain. At this stage we define functions on the set of real numbers R. Clearl, there are some values which are undefined. For eample the square root of a negative number or a fraction with denominator equal to zero. Eample Find the maimal domain for the following equation The equation is undefined for. It implies that can take an value but. Implied domain: R\{} Eample Consider the following equation. Find the implied domain. The number inside square root has to be greater or equal to 0. 0 Implied domain: Eample Find the implied domain for the following equation 5 This time we have a combination: fraction and square root. Since the square root appears in the denominator, the argument had to be positive. 5 0 5 Implied domain: ( 5, ) 8
Eample 5 Use GDC to find the implied domain of Sketch the graph over the appropriate window: Look up the table: ctrl T Alternativel tr a few values in the calculator screen: Implied domain: Real-life modelling problem. A clinder is inscribed in a sphere as shown in the diagram. The radius of a sphere is cm. cm a. Define function V, which gives the volume of a clinder in terms of its radius r. b. Sketch the volume function on GDC. c. State domain and range for the volume function. d. Find the maimum volume of the clinder. r h Using Pthagoras we can form the equation 9
h r The volume of a clinder is given b the formula V r h a) Epressing h in terms of r gives h 6 r V r 6 r V :[0,] R, r r 6 r b) V :[0,] R, r r 6 r c) Domain 0r Range 0 V 65. d) The maimum volume of the clinder is 65. cm. Eercise B. Determine domain and range of the following: a) b) 0 d) ( ) 5 e) ( )( ) f) c). Use function notation to define functions pictured below: a) b) Intercept ( -0.5, 0 ) 6 5 - - - - - - - - - 5-6 Intercept ( 0, ) Intercept ( -, 0 ) 6 - - - - Intercept Local Maimum ( 0, ) Intercept (, 0 ) 0
. Find the implied domain of the following functions: a) b) 5 6 c) d) e) f) 5. Sketch the following functions: a) f : (,] R, b) g : (,0] R, c) d) k :[, ) R, 7 h :[ 5,6) R, ( ) 5. Use a GDC to find domain and range of the following functions: a) b) c) d) 9 e) f) 6. Use a GDC to find the implied domain for the following: a) b) c) 9 d) 5 e) f) 7. Given the function f : R R,, a) find i) f () ii) f iii) f( a )
b) solve f( ) 0. 8. The right circular clinder is inscribed in a right circular cone with radius cm and height 9 cm as shown in the diagram. a) Define function V, which gives the volume of the clinder in terms of its radius r. State the domain and range. b) Define function V, which gives the volume of the clinder in terms of its height h. State the domain and range. c) Sketch both functions on GDC and find the maimum volume.. Eploring ke features of graphs with GDC. In this section we will investigate properties of some chosen graphs such as hperbolas, absolute value function, square root graphs and simple polnomial graphs. Galler of graphs:,,,, Basic shapes / features without transformations. Domain and range.. Transformations of graphs. Graphs can be subjected to transformations such as reflection in either or aes, translation verticall or horizontall and dilations (stretches). Appling transformations to the galler of basic graphs in. For each of the graphs below, define each function, sketch its graph and then perform the required transformation. Make a conjecture after each part. Functional notation to describe transformations: f ( ) f ( ) f ( ) f ( ) f ( ) k f ( ) f ( ) f ( k ) f ( ) f ( ) b f ( ) f ( a) And the combination of the above.
Real-life modelling: Sdne Harbour Bridge The bottom arch of the Sdne Harbour Bridge in Sdne (as in the diagram below) can be modelled b the quadratic function h :[0,50] R, 0.008655.986978, where is the distance (in metres) from the left plons and h is the height (in metres) of the arch above the water. Round all answers to the questions below to the nearest metre. a) With the help of GDC sketch the graph of the function modelling the shape of the lower arch of the bridge on the set of aes below. Mark all important features on our graph (verte and aes intercepts). b) Determine the coordinates of the highest point of the bottom arch. c) i) Factorise the epression h( ) 0.008655.986978 ii) Hence find the horizontal span of the lower arch of the bridge at the water level. Eplain how the horizontal span can be found from the factorised form. d) Find the height of the bottom arch of the bridge above water level at a distance of 00 metres from the left plons. e) Find the length of the road enclosed b the bottom arch. Assume the road is 60 metres above the water level. f) Assuming that the top arch can be formed b translating the bottom arch 0 metres up, suggest the equation for the top arch. Justif our answer. g) In real life the top arch passes through the following points (0,59), (5.5,6), (50,59)whilst the lower arch passes through these points (0,0), (5.8), (50,0). Find the equation of the top arch. In the light of the added information comment on our answer to part f.
.5 Piecewise functions and continuit. Piecewise functions are functions which consist of two or more parts (pieces). What it means is that a piecewise function ma have two different rules over its domain or three different rules over R. You are alread familiar with one eample of a piecewise function such as. According to definition of the absolute value function we have two rules defined as follows:, 0, 0 We can create more piecewise functions in a similar manner. Eample Use our GDC to sketch the graph of, f( ) 0., Hence evaluate a) f ( ) b) f ( ) c) f (0) d) 7 f We can use the special template to define a piecewise function on GDC: a) f ( ) 0.( ).6 b) f ( ) ( ) c) f (0) (0) d) f 7 7 0..9 It is important to use the appropriate rule when evaluating piecewise functions.
Eample Find the rule for the piecewise function shown in the diagram below: Intercept ( -, 0 ) Intercept (, 0 ) - 5 - - - - - 5 - - - - 5-6 The piecewise function above has two linear parts. When 0, straight line passing through two points (,0) and (0,) will have the equation. When 0, another straight line passing through the points (0,) and (,0) will have the equation. Therefore the piecewise function is defined as:, 0 f( ), 0 Piecewise functions considered in Eamples.6. and.6. are not joined together. Can we create a piecewise function in such a wa that two pieces join together? Eample Let h be the function defined b, h ( ) m ( ), where m is a constant, m R. (m can also be called a parameter). Find the value of m that makes two pieces to join together. For two pieces to join together, the values must be equal at. 5
m( ) m m We can easil check if our answer is correct b sketching the graph of h() with m. When two pieces of a piecewise function join together, such a function is continuous. When two pieces of a piecewise function do not join together, such a function is discontinuous. There are other eamples of discontinuous functions. Whenever the graph of a function has a break, hole, vertical asmptote or an open dot, such a function is considered to be discontinuous. We can take an informal intuitive approach at this stage to test for a continuous function: If ou can trace the graph of a function without taking the pencil from the paper over the entire domain, the function is continuous. Eample Consider the following piecewise function: 7, g ( ) 5 Sketch its graph and determine if g() is continuous over R. It appears that all three pieces join together, so g() is continuous over R. Eample Use piecewise function notation to write an equivalent epression for each of the following: a) f ( ) 5 b) f ( ) c) a) f ( ) d) 5, 5 f ( ) 5 5, 5 f( ) 6
b), f ( ), 5,, f( ), c) d) f ( ) f( ) Eercise, (, ] [, ), (,),,. Draw the following piecewise functions without GDC.. Use GDC to graph the following piecewise functions.. Which of the functions in questions and are continuous?. Find the values of parameters for the following piecewise functions to be continuous: a. b. c. d. 5. Find the rule for the piecewise functions shown in the diagrams below: a. b. 6. Define the following absolute value functions using the piecewise function notation. 7. Draw a graph of a piecewise function. Swap the graphs (without equations) with our classmate and determine the rules of the graphs. Discuss an discrepancies. 8. For the piecewise function defined as evaluate a. b. c. d. 9. Write down the following functions using piecewise function notation: a) f ( ) b) f ( ) c) f ( ) d) Real-life modelling problem. f( ) An athlete walks at km/h for 5 minutes and then runs at 6 km/h for 5 minutes. Let d be the distance the athlete has covered after t minutes. 7
The distance d can be described b the piecewise function: at, 0 t 5 dt () bt c, 5 t 0 a) Find the values of a, b and c. b) Sketch the graph of dt ( ). c) State the domain and range of dt ( ). d) What distance has the athlete covered in 5 minutes? a) km / h km / min 60 5 a 5 6 6 km / h 0. km / min 60 0 b 0. To find c substitute point (5,) into d 0.t c 0.(5) c c 0.5 Therefore the piecewise function is: t, 0 t 5 dt () 5 0.t 0.5, 5 t 0 b) d c) Domain 0 t 0 Range 0 d.5-0 0 0 0 t d) d(5) 0.(5) 0.5 km 8
.6 Inverse functions GDC Investigation on Inverse Functions..7 Composite functions. We can appl different operations to functions. We can multipl two functions thus forming their product. We can add or subtract two functions. We deal with composite functions when we have one function inside another function. Notation and terminolog ( f g)( ) stands for a composite function f of g in. ( f g)( ) can also be written as f ( g( )) In the composition above gis ( ) an inside function and f( ) is an outside function. We can illustrate the composition of two functions b means of a diagram using two boes, each representing a function with respective inputs and outputs to each bo. Assume that f ( ) and g( ). Recall the function machine s picture at the start of the chapter. This time we will need to have two function machines g and f respectivel represented b boes: input g output g() f output f(g()) so f ( g( )) ( g( )) ( ). In other words the output from g gets squared in the second bo. 9
Eample Given f ( ) and g( ) form: a) f g b) g f c) hence evaluate ( f g )() and g( f( )) To form f gwe replace in f() with g() as follows f g ( ) 6 5 6 Similarl to form the composition g replace in g() with f(): g f ( ) (9 6 ) 8 8 ( f g)() 56() 56 f we start with g() and g( f( )) 8( ) ( ) 8() 6 9 As can be seen f gis different to g f Thus, the order in which functions are composed will make a difference in the end result. Can ou think of two such functions that f g= g f? What can ou sa about the domain and range of f(), g() and both compositions? The two functions in eample.8. both had the set of real numbers as their domain. We need to consider another eample where we compose two functions with other implied domains in order to answer the question: can we form an composition of functions? Eample Consider two functions f ( ) compositions f gand g f? and g( ). Is it possible to form both Let use a graphical approach first. On GDC in calculator screen define f ( ) graph screen sketch a) f( f( )) b) f( f( )) Two graphs are shown below: and f ( ). Then in 0
f ( g( )) g( f ( )) Evaluating some values of compositions: Clearl not all values are defined. To help us decide which composition is possible we need to consider the domain and range of each of the component functions. It is practical to use the following table: domain range f [0, ) [0, ) g R (,] For the composite function f(g()) to eist, range of g has to be the subset of domain of f. ran g dom f For the composite function g(f()) to eist, range of f has to be the subset of domain of g. ran f dom g
For two functions under consideration we can see from the table that the following holds: [0, ) R (,] [0, ) Therefore g f eists, but f gdoes not (unless we restrict the domain of g). Thus we can conclude that g f, 0 Note that the domain of g f is equal to the domain of f. In general: For f gto eist, ran g dom f. Domain of f gis equal to domain of g. Eample a) Given the functions f ( ) 5 and g ( ), find the largest possible domain for which the composite function f defined. b) Sketch the graph of g( ) on this restricted domain. c) Find the rule for f ( g( )) and state its domain and range. gis a) domain range g R \{ } R \{0} f (,5] [0, ) f gis defined when range of g is the subset of g domain of f. For range of g to be a subset of f, we need to restrict the domain of g. 5 5 0 7 5 So the domain of g: Alternativel: 7 (, ) [ ; ) 5 7 R\[, ) 5
b) ( -., 5 ) 6 = - g() restricted - - - - c) f ( g( )) 5 domain of f ( g( )) = domain of g() Range of the composite function can be found on the calculator b sketching the graph. [0, 5) ( 5, ) Eample Given the composition f ( g( )), list f( ) and g ( ). a) b) c) d) 8 5 6 ( ) 9 e) 0 f) 5 ( ) Is there onl one da of decomposing function into two component functions? The following gives one wa of decomposing each function into f() and g();
a) f g ( ), ( ) 6 b) f ( ), g( ) 6 c) d) 5 f ( ), g( ) f g ( ) 7, ( ) e) f ( ) 0, g( ) f) f g 5 ( ), ( ) Composition of a function and its inverse investigation. The aim of the investigation is to conjecture what will result in the composition of a function f( ) and its inverse f ( ).. Consider the following functions a) f ( ) b) c) f( ) f d) f ( ) ( ). For each of the functions above find the rule for f ( ).. For each of the functions under consideration, form two compositions: f f and f f.. Make a conjecture stating what happens when we compose function f with its inverse Does the order matter here? f. 5. Eplain wh our conjecture holds for the composition of a function and its inverse? 6. Will our conjecture be helpful in checking whether our rule for the inverse function is correct? Eplain how.
.8 Odd and even functions and their properties. ODD OR EVEN? - GDC INVESTIGATION There are some functions which are odd and other functions which are even. Not ever function will be either odd or even. For a function to be odd, the following applies: f ( ) f ( ). The graph of an odd function is smmetrical about the origin. For a function to be even, the following is true: f ( ) f ( ). The graph of an even function is smmetrical about the -ais. Eample Show algebraicall and graphicall that odd function. Algebraicall we need to find f ( ) ( ) f ( ) f ( ) f ( ) an odd function. f ( ) is an Graphicall, b inspection the graph of f ( ) is smmetrical about the origin, so it is an odd function. 6 = - - - - -6 Eample Show algebraicall and graphicall that g( ) is an even function. g g ( ) ( ) ( ) g( ) g( ) even function And the graph of g -ais. Eample ( ) is smmetrical about the Verif analticall and graphicall whether each of the functions below is even, odd or neither. a) f ( ) b) g( ) c) d) f ( ) h ( ) -6 - - 6 - - -6 = 5
Eercise. Use our GDC to sketch some of the common graphs. In each case determine whether a given function is even, odd or neither.. Show algebraicall and graphicall that the following functions are odd.. Show algebraicall and graphicall that the following functions are even.. Verif analticall and graphicall whether each of the functions below is even, odd or neither. 5. Consider two even functions. Show that their sum and product will be even. 6. Consider two odd functions. Will their sum and product be odd or even? 7. Transformations with odd/even functions. What will the result be? 8. Absolute value function. Even/odd after the absolute value sign applied..9 Reciprocal graphs. Reciprocal graph of f( ) is denoted as. f( ) We will start with a short GDC investigation to infer the relationship between the graph of f( ) and. f( ) Reciprocal graphs investigation. We can infer the following relationships between the graph of a function and its reciprocal graph. a b.0 Rational functions c d Changing to another form, where the asmptotes can be seen b algebraic manipulation onl; no long division at this stage. Aes intercepts. Finding the equation, given the graph. 6
. Modulus graphs. ABSOLUTE VALUE INVESTIGATION Use GDC to sketch the pairs of graphs as indicated below. Cop each pair of graphs to our workbook and label them clearl.. Sketch the graphs of and on the same set of aes. Comment on the relationship between two graphs.. Repeat for the following pairs of graphs: a. and b. f ( ) and f ( ) c. g( ) ( ) and g ( ) d. h ( ) and h ( ) e. k ( ) and k ( ) ( ). State the general rule which will eplain what happens to the graph of f( ) when we want to sketch the graph of f( ).. Sketch the following pairs of graphs on the separate set of aes for easier comparison: a. f ( ) and b. g( ) 5 and g( ) 5 c. h ( ) and h d. k ( ) ( ) f ( ) and k( ) 5. State the general relationship between the graphs of f( ) and f( ). 6. a. For the graph of h( ), sketch without a calculator h( ). b. Now check on our GDC. 7. What new have ou learnt from this investigation? 7
GRAPHS INVOLVING f( ) and f( ) CONCLUSIONS. Please answer the questions below without using our GDC.. a. Write in words what needs to be done when ou are given a graph of f( ) and ou are asked to sketch the graph of f( ). b. Illustrate with an eample using a linear or quadratic function of our choice.. a. Write in words what needs to be done when ou are given a graph of f( ) and ou are asked to sketch the graph of f( ). b. Illustrate with an eample using a function of our choice.. a. Write in words what needs to be done when ou are given the graph of f( ) and ou are asked to sketch the graph of f( ). b. Appl our rule from.a. to sketch the graph of g( ) for the graph of gsketched ( ) below: Intercept (, 0 ) - 6 - Intercept ( 0, - ) -. Given that h ( ) is an even function, conclude what happens when we want to sketch the graph of h( ). 8
TOK section Quotes b Albert Einstein:. As far as the laws of mathematics refer to realit, the are not certain; and as far as the are certain, the do not refer to realit.. How can it be that mathematics, being after all a product of human thought independent of eperience, is so admirabl adapted to the objects of realit?.. Mathematics deals eclusivel with the relations of concepts to each other without consideration of their relation to eperience Topics for discussion: How is it that mathematics fits the real world? Can everthing be epressed in mathematical language? Is mathematics about truth or validit? Mathematics ma have some limitations, but in our human eperience we seldom come across those limitations. Mathematics remains a miraculous device for seeing the world more clearl. Is mathematical knowledge universal? 9