Module B4 Relations and functions. ModuleB4. Relations and functions

Transcription

1 Module B Relations and unctions ModuleB Relations and unctions B

2 Table o Contents Introduction..... What are relations and unctions? Domain and range o relations and unctions Function notation..... The linear unction Rate o change o a linear unction The inverse undoing a unction When two linear unctions meet The quadratic unction Sketching parabolas Rate o change o a quadratic unction..... Other unctions..... A taste o things to come Post-test Solutions....9

3 Module B Relations and unctions. Introduction The title o this module might conjure up images o aunts and grandathers, weddings and unerals. You might think that this does not have much to do with mathematics. But mathematics and its applications in business, science and engineering etc. are primaril about relationships. Just as we might have a range o dierent tpes o relationships with our relatives, or riends or that matter, so dierent variables can be related in dierent was. I we are luck, or unluck depending on the situation, we can usuall visualize our relatives and riends (or get a photograph i necessar. This module is all about using visual images o relationships (graphs to help us understand the nature o relationships between variables. More ormall when ou have successull completed this module ou should be able to: demonstrate an understanding o the concept o a unction use unction notation demonstrate an understanding o the relationship between an algebraic epression (linear involving two variables and its graph on the Cartesian plane recognize, sketch and use linear unctions demonstrate an understanding o the nature o change (gradient in a linear unction demonstrate an understanding o and ind the inverse o a linear unction recognize, sketch and use quadratic unctions demonstrate an understanding o the nature o change in a quadratic unction recognize and describe other unctions.

4 . TPP78 Mathematics tertiar preparation level B. What are relations and unctions? We know rom module 3 that i a quantit is not ied then we call it a variable and smbolize it with a letter or pronumeral. Oten one variable is related to one or more other variables, so we sa a relationship eists between the variables. The most simple case eists between two variables represented b a set o ordered pairs. For eample, consider the situation o return airares around the world depicted below. Each destination rom Brisbane has three dierent return ares associated with it (the charge dierent amounts in dierent seasons. I we recall some o our previous knowledge we could write this as a set o ordered pairs; (London, 99 (London, 699 (London, 799 (Frankurt, 99 (Frankurt, 99 (Frankurt, 699 (Paris, (Paris, (Paris, 6 and represent it as a graph. Destination Return are ($ London Frankurt Paris 6 Return Fare ($ rom Brisbane to London, Frankurt and Paris ReturnFare($ 8 6 London Frankurt Paris Destination

5 Module B Relations and unctions.3 In relationships o this tpe between two variables, we sa one o the variables is dependent on the other. In the case above, we do not know what the return are could be unless we irst know the destination. The Return Fare is dependent on Destination. Because o this we call Return Fare the dependent variable and Destination the independent variable. I ou are conused b these terms another wa to think o them is as input and output variables. The independent is alwas the input variable, while the dependent is the output variable. Conventionall in most disciplines, the independent variable (input is the irst member o the ordered pair and is placed on the horizontal ais on a graph (oten called the -ais. The dependent variable (output is the second member o the pair and is placed on the vertical ais (oten called the - ais. Eample In the ollowing epression which variable is the independent variable and which is dependent? In a small business the selling price o a commodit is related to the number ordered, so that i ou bu one tee shirt it costs $ each while i ou bu the will cost $ each. The selling price is determined b the number o shirts ordered so we sa Number ordered is the independent variable and Selling Price is the dependent variable. Eample Pressure and depth underwater were both recorded regularl throughout an eperiment on the eectiveness o a new wet suit. I ou had to graph this relationship which variable is the independent variable and should be placed on the horizontal ais. In this instance pressure is dependent on the depth o the diver, so we sa that depth is the independent variable and should be placed on the horizontal ais. Pressure will be the dependent variable and should be placed on the vertical ais. Activit.. In the ollowing epressions, which variable is the independent variable and which is dependent? (a In Australia, the amount o income ta paid is related to the amount o income earned. For eample, or an income earned over $, 8 cents in the dollar is paid as ta. (b Reading the label on a medicine bottle, a parent inds that the number o millilitres to be given to a sick child is related to the age o the child. I a child is 3 ears old, then a ml dosage is suicient, but a ear old child requires ml.. I ou had to graph the ollowing relationships decide which variable should be placed on the horizontal ais (independent variable and which should be placed on the vertical ais (dependent variable. (a The dail demand or the bulk purchasing o bottled water is related to the selling price o the water. When the selling price is $6. per unit, then units are sold, but when the selling price is $6. per unit then onl units are sold.

6 . TPP78 Mathematics tertiar preparation level B (b Scientists recorded data on the area covered b a plant species resistant to cold temperature etremes and the number o rost-ree das throughout winter. (c Saet tests perormed on a new model car required data to be collected on stopping distances o the car or a range o velocities. I we return to the airares problem we saw that this relationship was a set o ordered pairs we call this a relation. A relation is a set o ordered pairs. Relations or sets o ordered pairs can be represented in three was, oten depending on the tpe o relation. The can be represented as: groups o points as in the airares eample above or as in {(,, (,3, (,, 7,6}; ormulas or equations like, or z p (recall that given the value o either or p in these equations we can calculate the value o or z and so orm sets o ordered pairs; or graphs o the relationships (see below All o the above are called relations. But let s go back again to the airares problem. This relation does tell us something about the cost o ling to speciic places but unless we had some more inormation then we could not reall use the relation to calculate one price there are three dierent ares or each destination. It might be more useul to have a relationship that looked like this, with destination and time together as one variable.

7 Module B Relations and unctions. It would be graphed like this: Destination/time Return are ($ London in Februar 99 London in March 699 London in April 799 Frankurt in Februar 99 Frankurt in March 99 Frankurt in April 699 Paris in Februar Paris in March Paris in April 6 Return ares ($ rom Brisbane to London, Frankurt and Paris at dierent times London Februar London March London April Returnares($ Frankurt Februar Frankurt March Frankurt April Paris Februar Paris March Paris April Destination/time In this relation we see that each destination/time has onl one cost. It doesn t matter i the costs are the same or dierent destinations/time, as is the case or London in Februar and Frankurt in March, because we are tring to predict cost rom destination/time not vice versa. In man mathematical situations it is oten useul to have a relationship rom which we can calculate just one value o the dependent variable rom one value o the independent variable...this tpe o relation is called a unction. A unction is a relation in which there is onl one value o the dependent variable or each value o the independent variable.

8 .6 TPP78 Mathematics tertiar preparation level B Something to talk about You will see the deinition o a unction written in man dierent was in dierent books. Write a deinition o a unction in our own words and share it with other students either through personal contact or through the Discussion Group. Think about which deinition best captures the essence o the nature o a unction and send some eamples o this in practice to the group. Eample The ollowing are relations. Indicate which relations are also unctions. Give reasons or our choices. Relation Function? Reason Rainall in a small town is measured once per da in a mm rain gauge and is graphed against da o the month. The graph is shown at a town meeting. Yes In this relation, the da o the month is the independent variable and the amount o rain (mm is the dependent. As the rainall is measured onl once per da there must onl be one value o volume or each da. The relation must be a unction. {(,, (,, (,, (,} No In ordered pairs the irst coordinate is traditionall the independent variable and the second the dependent variable. For = there are two alternatives o = or =. The relation has two values o the dependent variable ( or the same value o the independent variable (. The relation is thus not a unction. {(7,, (3,, (,6, (,} Yes In ordered pairs the irst coordinate is traditionall the independent variable and the second the dependent variable. For each o the values shown in the set o ordered pairs there is onl one value. It does not matter that there are repeated values o ( = or dierent values. The relation is a unction.

9 Module B Relations and unctions No In graphs the horizontal ais traditionall represents the independent variable and the vertical ais the dependent variable. At one point on the graph or one value o (the independent variable there are two values o (the dependent variable. The relation is thus not a unction. k q No For ever value o q (the independent variable we can calculate a positive and a negative value o k (the dependent variable. That is, we have two values o the dependent variable or one value o the independent variable e.g. i q then k or. The relation is not a unction. Yes As ou move along the -ais (the independent variable there is onl ever one value on the -ais (the dependent variable that corresponds to it. The relation must be a unction. 6 6 Activit.. Complete the ollowing table indicating which o the ollowing relations are also unctions. Give reasons or our choices. Relation Function? Reason (a

10 .8 TPP78 Mathematics tertiar preparation level B (b (c (d (,, (3,,(,,(6, (e The amount o potassium absorbed b the lea tissue o a plant when placed in darkness is graphed against time. ( ( 3,7,( 9,, (,7,(3,.. Domain and range o relations and unctions Let s return to the relation we used in the airare problem, this time represented as a set o ordered pairs. (London, 99 (London, 699 (London, 799 (Frankurt, 99 (Frankurt, 99 (Frankurt, 699 (Paris, (Paris, (Paris, 6

11 Module B Relations and unctions.9 You will notice that there are restrictions on the values o the independent variable. We could not use the relation to determine the cost o ling to San Francisco because it does not eist in the relation. The conditions under which a relation will work are called the domain a range. The domain is the set o values or which the independent variable is deined. All relations, including unctions, will have domains, sometimes the domain will be ver obvious as in the ollowing: Eample Function Values o independent variable Domain Fish Species Gupp Goldish Rainbow Cichlid Cost c c $. $. Gupp, Goldish, Rainbow, Cichlid {Gupp, Goldish, Rainbow, Cichlid} (,, (,3, (,,, {,, }...this would include onl three values p a, where a is measured < a < < a < this between and. would include an ininite number o values between and Graph is onl drawn between the values o.3 and..3. Notice that the domain can be either a set o discrete values (e.g.,,, or an ininite number o points as in a continuous epression (e.g. < a < Sometimes, however, the domain o a unction is not so obvious. Consider these eamples.

12 . TPP78 Mathematics tertiar preparation level B Eample Final speed o a car (v is equal to the starting speed o m/s plus the product o acceleration ( m/s/s and time o travelling (t. What is the domain o this unction? I we translated this rom words to an algebraic ormula we would get v t vt In this instance the independent variable is time (t. Are there an restrictions on the values o time? I ou said es ou would be correct. In normal circumstances we would not be able to travel back in time and so time would alwas have to be a positive number or zero. We would sa that the domain o this unction would be t and theoreticall would etend to ininit. Eample Consider the unction, what is its domain? B convention the independent variable in this unction would be. Are there are restrictions on the values o or this unction? I ou said no ou would be correct. The domain or this unction contains an ininite number o values in act it will contain all real numbers. We would sa that the domain would be all real values o. Eample Consider the unction p, what is the domain o this unction? a In this unction, unless otherwise indicated, we would assume the independent variable to be a. Are there an restrictions on the values o a or this unction? I ou said es ou would be correct. Tr and calculate the value o p or a = it is not possible. is not a number so the unction is not deined or a =. We would sa that the domain o the unction is all real values o a ecept a =

13 Module B Relations and unctions. Eample Consider the unction, what is the domain o this unction? In this unction we would assume that the independent variable would be. Are there an restrictions on the values o? I ou said es ou would be correct. Let s think about the square root epression. Have ou ever tried to ind the square root o a negative number on our calculator? In the real number sstem it is not possible to calculate the square root o a negative number. So this means that an epression under the square root sign must not be negative. In our eample this means that. I we solve this inequation or we will get. This means that the domain o the unction is. In the above eamples the domain is sometimes called the natural domain. Activit.3. State the natural domain o the ollowing unctions. (a (b (c (d (e p v ( 3( m 8n Just as we have a word associated with the etent o the independent variable we also have a word associated with the etent o the dependent variable. The range is the set o values or which the dependent variable is deined. In most instances (unless we have a graph rom which we can read o the values we would irst determine the domain and then use it to determine the range. Look at our previous eamples.

14 . TPP78 Mathematics tertiar preparation level B Eample Fish species Gupp Goldish Rainbow Cichlid Function Domain Etent o the dependent variable Range Cost c c $. $. {Gupp, Goldish, Rainbow, Cichlid} All costs {.,.,.,.} (converted all cents to dollars (,, (,3, (, {,, }... All coordinates {, 3, } p a, where a is measured between and < a < All values o p when < a <. Limiting value o p is when a =, maimum value is when a =. < p <.3. All values o when.3.. From the graph when.3,.9, when.,. 9.9 < <.9 Determining the range or man unctions is oten not as eas as in the unctions we have described above. It involves knowledge o the domain o the unction and an understanding o the shape o the unction. We will investigate domains and ranges o speciic unctions later in this module.

15 Module B Relations and unctions.3 Activit.. Find the domains and ranges o the unctions with these graphs. (a 6 6 (b. State the range o the ollowing unctions. (a a b 3, where the domain is all real values o b (b I r and q are continuous variables in the eample detailed below. r 3 8 q 3 (c where the domain is

16 . TPP78 Mathematics tertiar preparation level B.. Function notation We know now what a unction is and in some cases how to determine its domain and range, but because unctions are so important and used so requentl a special notation has been developed to simpli their description. Let s look at the ollowing unction rom economics. In a monopol, the selling price o goods can be set so that the total revenue is a unction o the output quantit. I we call Total Revenue (R and quantit (q then more briel we would sa: Total revenue is a unction o quantit R is a unction o q R = (q In the monopol eample, a speciic total revenue unction is R.6( q 7 wanted to calculate values o R or q we would do the ollowing:. I we I q, then R.6( 7 which we could now write more simpl as: ( in words we would sa the value o the unction at is. The letter was chosen to represent unction or obvious reasons, but i we have dierent unctions using the same independent variable we can choose an letter or smbol we like. For eample in the same monopol, Price (P is also a unction o output quantit (q, P q 3 so we could write this as g( q q 3. I we wanted to ind the Price when output quantit was then we would calculate: g( 3 in words we would sa the value o the unction g at is. In these two eamples above both (q and g(q can still represent the dependent variable and when we graph the unctions, these values would conventionall be graphed on the vertical ais. We will return to this later. Let s look at some eamples o the use o this notation.

17 Module B Relations and unctions. Eample I ( ind the values o the unction when and ( ( ( ( 9 The value o the unction at is 9. ( ( ( ( The value o the unction at is. Eample I q( t 3( t, evaluate q(a and q( + a i a is a constant. qt = 3t + qa = 3a + qa = 3a 8a qa = 3a a qa = 3a a + everwhere t occurs replace b a epand squared term qt = 3t + q + a = 3 + a q + a = 3a 3 + q + a = 3a + 6a q + a = 3a 8a q + a = 3a 8a + 9 everwhere t occurs replace b + a epand squared term

18 .6 TPP78 Mathematics tertiar preparation level B Activit.. Write the ollowing using unction notation. (a The maimum magniication, M, obtained when using a converging lens is a unction o the ocal length,, o the converging lens, so that M. (b The dail cost, C, o producing a product is a linear unction o the number, q, o the product made, so that C q. (c The dail sales, S, o a product is a unction o the time the product has been on the market, so that S.. 3t e (d The surace area o a sphere (S is a unction o the radius (r o the sphere so that S r.. Find the value o the ollowing unctions when = 3, =., =, = (+t, = a (a (b (c (d ( ( 3 ( 9 ( 3. Given the unctional relationship, g( t 3t t ind (a (b g( g( a b. The weight, w, in kilograms o an object is a unction o the distance, d, in kilometres rom the surace o the earth. For a person weighing 6 kg, the unction can be written, 6 6 w( d ( d 6 Find the weight o a person standing on a mountain 3 m above sea level.

19 Module B Relations and unctions.7. The linear unction Linear unctions are unctions whose graphs represent straight lines. You should have come across unctions like this beore. The are called linear because in each case the variables are o power one and no two variables are multiplied together (we will add to this deinition later in the module. The are called unctions because each value o the input variable has onl one value or the output variable. Eamples we have seen beore include: I P, where P was the price o shirts and I was the income 9 F C 3, where F was the temperature in Fahrenheit and C the temperature in Centigrade. Other eamples would be: B.S, where B is the temperature in Brisbane and S the temperature in Sdne in C. H 3V 69, where H is an animal s heartbeat (per min and V is the volume o a drug (ml A business consultant s ee is a retainer o $7 plus $ per da or part o a da. All o the above can be graphed on sets o coordinate aes. Remember we choose the aes so that the independent variable is on the horizontal ais and the dependent is on the vertical. You can graph linear unctions in a range o was. In the past ou might have drawn up a table o values and then plotted the points. For straight line graphs ou onl need to plot two points with possibl a third as a check. Let s have a look at another wa to graph straight lines. Consider the unction B.S, where B is the temperature in Brisbane and S the temperature in Sdne in C. We would irst draw the aes with the independent variable (S on the horizontal ais and the dependent variable, B, on the vertical. Recall rom our previous work on unctions that we could have just as easil described this unction as B ( S.S. The net step is to ind and plot two points. We could choose an two points and man people would choose S and S because the are eas to calculate. But it is oten useul to determine the intercepts on the vertical and horizontal aes. So we will ollow that method. Note that in general where the vertical ais is the -ais and the horizontal ais is the -ais we would have the ollowing.

20 .8 TPP78 Mathematics tertiar preparation level B Along the ais all values o are = Along the -ais all values o are = So in our case where the vertical ais is the B-ais and the horizontal ais is the S-ais and B.S to see where the line cuts the B-ais we would put S. I S, then B.. So the vertical intercept is B. This is the point (,. To see where the line cuts the S-ais we would put B. I B, then.s.s S. S.7 So the horizontal intercept is S.7. This is the point (.7,. We can graph these two points on the aes and then draw a line through and beond these two points.

21 Module B Relations and unctions.9 B 3 S 3 I ou have access to a graphing package then check the accurac o this method b drawing the graph with the package and reading o the intercepts on the horizontal and vertical aes. Activit.6. For each o the ollowing linear unctions ind the vertical and horizontal intercepts. (a 3 ; where is the independent variable. (b ba ; where a is the independent variable. (c s 8 t ; where s is the independent variable. (d 9 m n ; where m is the independent variable. (e ( ( 3 ; where is the independent variable. q 3 ( p ; where p is the independent variable.. Graph the linear unctions in question using the - and - intercepts.

22 . TPP78 Mathematics tertiar preparation level B 3. Graph the ollowing unctions or the domains indicated. (Hint: ou might want to rearrange the equation irst. (a 3 (b (c (d (e This method is all ver well when we are given the equation o the linear unction but what is the procedure in a real world situation. In man instances we would just be presented with something like this. You are the inance manager or a small compan and are asked to develop a ear plan or a business consultant s services over this time. The eas wa to do this is to develop a unction ou can put into a spreadsheet rom which ou can predict consultant s ees rom number o das emploed. The onl inormation ou have is A business consultant s ee is a retainer o $7 plus $ per da or part o a da. (Note spreadsheets are tables o values. This terminolog is oten used when the table o values is generated b a speciall designed computer package. Step : Name the variables and identi which would be the input variable and which would be the output variable. In this case it is clear that: number o das (or parts o a da worked is the input (independent variable, call it n total ee is the output variable (dependent variable, call it T. Step : Generate the linear unction. In this case it would be: T 7 n in unction notation this would be ( n 7 n Step 3: Determine i there are an restrictions on the domain. In this case there are restrictions. n, the number o das (or parts o das worked cannot be negative so there is a lower limit that n must be greater than zero. Also because we were asked to onl develop a ear plan there is an upper limit o ears or 8 das. The domain would thus be n8. Once we have the above unction and know its domain we can then happil construct a table o values to determine costs. I we had wanted to we could have also drawn a graph (as beore o the unction rom which we could have made approimate predictions.

23 Module B Relations and unctions. Eample The temperature in a swimming pool cools down each night to C. When the owner puts the heater on in the morning the temperature increases at.c per minute until it reaches 7C, the temperature set on the thermostat o the pool heater. Draw a graph o the relationship between pool temperature and time in minutes, during this heating phase. Step : The variables o interest are pool temperature in degrees Celsius (call it P and the time in minutes (call it t. t is the independent variable and P is the dependent variable. Step : The linear unction in this case would be P.t in unction notation it would be ( t.t Step 3: There are restrictions on the domain o the unction. There is a lower value because the time can never be negative. There will be an upper value because the temperature was onl measured until it reached 7C (i.e. during the heating phase. From the equation above we know that the temperature reaches 7C ater 7 minutes. So the domain will be t 7. Step : To draw the graph it is conventional to place the independent variable on the horizontal ais and the dependent variable on the vertical ais. To calculate two points rom which to sketch the straight line, we can choose the intercept on the vertical ais and the intercept on the horizontal ais. The intercept on the vertical ais occurs when t =, substituting this in the equation P.t, we get P = giving us the point (,. The intercept on the horizontal ais occurs when P =, substituting this in the equation P.t we get.t.t t. t giving us the point (,

24 . TPP78 Mathematics tertiar preparation level B Using these two points we can sketch the graph Relationship between Pool Temperature (degrees Celsius and Heating Time (minutes Temperature (degrees Celsius 3 Time (minutes Because o the restrictions to the domain the inal graph would look like this (note we have modiied the scale so that the graph is more usable. Temperature (degrees Celsius 3 Relationship between Temperature (degrees Celsius and Heating Time (minutes Time (minutes Notice that the range o the unction is rom to 7C inclusive, which its with the inormation given.

25 Module B Relations and unctions.3 Activit.7. Establish a linear relationship or each o the ollowing and speci the domain and range. Remember to deine the variables. (a A candle 8 mm long burns so that ever hour it is mm shorter. (b A water tank has a capacit o litres. Water leaks rom two holes at the bottom o the tank so that ater hour it has lost litres and ater hours has lost litres. (c A screen printing irm is interested in its total costs or printing t-shirts or its customers. The ied costs or the irm are $8. or setting up the machine and the variable costs are $. to print ever t-shirts. (d A driver is interested in how ar he is rom his holida house at a particular time on his journe. He starts his journe at his home, km rom his destination and than travels 9 km ever hour.. Choose an appropriate scale to graph the linear unctions in question... Rate o change o a linear unction Previousl we have deined a linear unction to be a graph o a straight line or a unction in which the variables are onl raised to a single power. But what does that mean? In both o these deinitions what we are reall saing is that no matter where we are in the graph, one variable is changing at the same rate as the other variable. The quantit that we use to measure how one variable changes with respect to another is the gradient or slope o the straight line. It is sometimes also reerred to as the rate o change o the unction. Recall that the gradient puts a value on the steepness o a straight line b comparing the change in height with the change in horizontal distance. gradient change in height change in horizontal distance rise run Let s write this in unction notation. I we have the linear unction points on it called and. (, which has two

26 . TPP78 Mathematics tertiar preparation level B ( ( (,( ( change in horizontal distance (,( change in height ( ( So the gradient (called m in unction notation is written, change in height m change in horizontal distance ( ( This means that i we know an two points on a straight line graph we can use them to calculate the gradient o the unction.

27 Module B Relations and unctions. Eample A unction which relates two variables, p and q, where p is dependent on q is ( q p q 3. I ( and ( 3 6, determine the gradient o the unction. p= (q 6 6 q m ( q q ( q q 6 3 ( Gradient is. Activit.8. Find the gradient o a line which passes through the points with coordinates (a (,, (, (b (,, (,. A unction relates two variables, a and b, where b is dependent on a. I ( 6 and (, determine the gradient o the unction. 3. A unction relates two variables, s and t, where t is dependent on s. I g( 3 and g(, determine the gradient o the unction.. Find the gradient o a line i its - and -intercepts are given respectivel. (a 3 and (b -- and

28 .6 TPP78 Mathematics tertiar preparation level B You will have seen rom the above activit that gradients can have a range o values. In all instances however the gradient will be the same at all parts o the graph In all the graphs as increases the value o the unction (( increases. Line : gradient is. Line : gradient is Line 3: gradient is The rates o change o with respect to are all constants. So in line the values o the unction are increasing more slowl that the values o the unction in line In all the graphs as increases the value o the unction (( decreases. Line : gradient is. Line : gradient is Line 3: gradient is The rates o change o with respect to are all constants. So in line the values o the unction are decreasing more slowl that the values o the unction in line 3. In all the graphs as increases the value o the unction (( remains constant Line : gradient is Line : gradient is Line 3: gradient is So the rates o change o with respect to in all three unctions is zero. 3

29 Module B Relations and unctions.7 A unction is increasing i the values o ( increase as increases. The gradient will be positive. A unction is decreasing i the values o ( decrease as increases. The gradient will be negative. A unction is constant i the values o ( remain constant as increases or decreases. The gradient will be zero. Recall that we can read o the gradient rom its equation i it is written in the slope intercept orm. m b, where m is the slope and b is the intercept on the vertical ais. We can now revise our deinition o a linear unction to one in which the rate o increase or decrease is constant. Eample A amil regularl commutes between Brisbane and Toowoomba in their Toota Camr station wagon. p is the amount o petrol in the tank (in litres ater a number o trips and d represents the distance travelled in kilometres. I the equation representing this relationship is p 7. 7d, ind the gradient and the intercepts on the vertical and horizontal aes. Eplain what these quantities represent. Using the equation p.7d 7, which is in the slope intercept orm, we know that the slope is.7 and the vertical intercept is 7. To ind the horizontal intercept we have to put p to get.7d 7 7 d.7 The gradient is negative so this means that as the number o kilometres travelled increases the amount o petrol in the tank decreases. The value o the gradient indicates that the petrol is decreasing at a rate o.7 litres per kilometre or.7 litres per kilometres (standard wa in most motoring magazines. The vertical intercept represents the original amount o petrol in the tank, 7 litres. The horizontal intercept represents the total number o kilometres that the car could travel on one tank o petrol, km.

30 .8 TPP78 Mathematics tertiar preparation level B Activit.9. For the questions rom activit.7 question eplain the meaning o the gradient, the vertical intercept and the horizontal intercept.. A tai driver epects that the value o his car will decrease at a linear rate over time. When the car was irst purchased it cost $8. He epects that ater three ears, when he intends to sell his car, it will be worth $7. (a For the unction that represents the relationship between car value in dollars and time in ears, determine the slope using the two point ormula. (b Write the equation or this relationship and eplain what each o the quantities mean... The inverse undoing a unction Have ou ever wanted to reverse a unction or eample calculate a Fahrenheit temperature rom the Centigrade temperature instead o vice versa. We have done something like this when we changed the subject o a ormula in module 3. Finding the inverse unction is another perspective on this issue. Let s consider the relationship between Centigrade and Fahrenheit temperatures. We could represent some o the temperatures b a set o ordered pairs. Centigrade (C Fahrenheit (F I Fahrenheit is a unction ( o Centigrade, or the irst point we might sa ( 3 or or the second point (. I we wanted to go the other wa and write centigrade as a unction (g o Fahrenheit we would write the irst point as g( 3 and the second point as g(. Now because the second unction has reversed or undone the process involved in the irst unction mathematicians would not just call it g but give it a special name the inverse unction, call it.

31 Module B Relations and unctions.9 is a special notation and is not the same as the inde notation a introduced in module 3. a which was So our set o points would now look some thing like this Centigrade to Fahrenheit Fahrenheit to Centigrade ( 3 (3.... The Original Function The two unctions are dierent but have the same meaning looked at rom dierent perspectives. I we graphed them the would look like this. ( ( ( 68 (68 ( 3 86 (86 3 ( (.... The Inverse Function F C C F Graph when Centigrade is the independent variable and Fahrenheit the dependent variable. Graph when Fahrenheit is the independent variable and Centigrade is the dependent variable.

32 .3 TPP78 Mathematics tertiar preparation level B _ 9 = ( + 3 _ = 9( 3 When both graphs are put on the same aes we get a picture like this. We can continue this undoing process to ind the ormula or the Centigrade/Fahrenheit relationship. 9 F C 3, where F was the temperature in Fahrenheit and C the temperature in Centigrade. I we want to undo the process we have to subtract 3 then multipl b. 9 So i the original unction is 9 ( C C 3 = F then its inverse will be ( F ( F 3 = C 9 Let s have a closer look at inverses o linear unctions. Note that it is not alwas possible to ind the inverse unction o unctions which are not linear. Graphs o linear unctions and their inverses. We showed the graphs o the two Centigrade/Fahrenheit unctions previousl, now let s look at this process in more detail using a simpler unction. Consider the linear unction, ( to ind the inverse we would have to undo the multiplication b two b dividing b two. The inverse would be (. I we represented this as a set o points we would get the ollowing:

33 Module B Relations and unctions.3 ( ( I we were to graph the original unction and its inverse o the same set o aes we would get the ollowing. ( ( Graph o ( Graph o ( ( = ( = _ 6 6 = Graph o ( and ( on the same set o aes.

34 .3 TPP78 Mathematics tertiar preparation level B Notice that the original unction and its inverse appear to be relections about the line. I ou are not sure about this, a good eperiment is to draw all o the unctions on a piece o square graph paper. Then draw in the line and old the piece o paper along this line. You will notice that the two unctions will line up, because the are mirror images o each other in the line. The and coordinates are interchanged or reversed. Further, just as the coordinates are reversed to determine the inverse unction, so the domain and range will be swapped over. The domain o the original unction will become the range o the inverse, while the range o the original will become the domain o the inverse. The important things to remember about inverses are: I an inverse o the original unction ( eists we call it ( I an inverse unction eists then it is deined b the epression ( t d meaning ( d t An original unction and its inverse are relections in the line The domain o the original unction becomes the range o the inverse unction The range o the original unction becomes the domain o the inverse unction Eample The manuacturing costs o producing products is given b the unction ( 3. The government has placed quotas on production and will allow the compan to produce onl o each item. What is the ormula or the inverse unction and how would ou interpret it? I the original unction is ( 3, and valid onl or the domain o. The restriction on the domain will mean that the range will also be restricted. The range or value o the unction at is ( 3 3, and at = is = + 3 = range is 3 This unction will allow ou to calculate the cost o producing a given number o products. To get the inverse we must reverse the processes b subtracting and then dividing b three to get (. 3 The domain will also be aected with the domain o the inverse unction being now equivalent to the range o the original unction. Domain o the inverse unction is 3. The range or the value o the inverse unction at 3 is 3 (3 3 The range o the inverse unction is the domain o the original unction. The inverse unction allows ou to calculate the number o products produced at a given cost.

35 Module B Relations and unctions.33 Eample The price o some goods in dollars, P, is a unction o number o items sold, n, so that P = n. Eplain what is meant b (3 and ( in terms o prices and quantities sold. (3 means that i the number o items sold is 3 then the price o goods will be value o the unction at 3. (3 the ( means that i the price o the goods is $ then the number o items will be the value o the inverse unction at. ( Activit.. A unction is given as ordered pairs. Write the inverse o the unction and state its domain and range. State clearl i the inverse is not a unction. (a ( 7,, (,, (,, (,7 (b (,, (,, (3,, (,. Find the inverse relation o each o the ollowing unctions. State whether or not each inverse is a unction. I the inverse is a unction state the domain. (a (b (c (d g( h( 3 ( 3 g( ( 3..3 When two linear unctions meet In module 3 we looked at problems associated with equations with two unknown variables. Let s revisit that section now and see the graphical approach to the solution. The original problem in module 3 was A cit baker sold bread rolls on Sunda, with sales receipts o $ 6. Plain rolls sold or 9 cents each while gourmet rolls sold or $. each. How man o each tpe o roll were sold? Previousl we solved the problem b generating two equations and then solving them algebraicall. The equations were P G and.9p.g 6, the inal solution was number o gourmet rolls (G was and number o plain rolls (P was 3.

36 .3 TPP78 Mathematics tertiar preparation level B Each o the equations can be rewritten as a linear unction and drawn on a set o coordinate aes. Rearranging the equations we will get: P G.9P.G 6 will become will become P G 6.G P.9 Note we could have just as easil made G the subject o the ormula and graphed it as the dependent variable. Graphing the two equations we will get P G So the point o intersection o two straight lines is the same as the solution to two simultaneous equations with two unknowns. The graphical approach to solving simultaneous equations like the one above is oten the irst step in solving more comple problems in which the algebra is time consuming or impossible. Eample Economics uses the graphical and algebraic approach in a tpe o analsis called break-even analsis. The break-even point or a business is the point at which total revenue equals total costs. To determine the break-even point or a business a compan needs to consider two unctions. For a speciic compan this might be: The Cost unction c( The Revenue unction r( 9,

37 Module B Relations and unctions.3 Where is the quantit o goods. Note in break-even analsis we are interested in onl inding value o the -coordinate i.e. the quantit o goods that will result in the equalit o the total revenue and total costs. Break-Even Analsis Total revenue/total cost ($ Quantit The -coordinate o the point o intersection or the break-even value o the quantit rom the graph is units. To test this algebraicall substitute into both equations and it should give the same value or cost and revenue. c(, c = + = r( 9, r( 9 So the break-even value o the quantit is units. Activit.. Graph the pair o linear unctions or the given domain. Write as an ordered pair (,, the graphical solution o the simultaneous equations. Prove our solution is correct algebraicall. (a (b (c (d

38 .36 TPP78 Mathematics tertiar preparation level B. (a A local rent-a-car irm charges a ied dail rate plus a ied amount per kilometre. A total charge or the irst week was $83 with km on the clock. The nd week s bill was or $7 with another 8 on the clock. Determine graphicall the rental charge and the cost per kilometre. (b Ater analsis o a compan s cost structure the accountant ound that the ied cost was $ per annum with a constant variable cost o $9. per unit. I the product is selling or $ per unit, graphicall determine the break even point..3 The quadratic unction Functions come in all shapes and orms. Some are eas to represent and interpret others are more diicult. Linear unctions represent some o the easiest unctions to understand. Other unctions that are not linear, not too comple and occur requentl in business and science are quadratics. Let s return to the eamples rom section in module 3 and look at quadratic equations rom a unction perspective. All o the equations presented there are in act unctions For ever value o the independent (input variable there is onl one value o the dependent (output variable. The graph o a quadratic unction is called a parabola (pronounced pah-rab-o-la. Let s quickl look and see what shape the are beore we go into details. I ou have a graphing package ou ma want to sketch them oursel irst. ( Q 3 3 Q In economics, the average cost equation or a particular production plant is C ( Q.3Q 3.3Q.3 where C is the average cost in dollars o producing each unit and Q is the quantit produced.

39 Module B Relations and unctions.37 ( d d In engineering, the positioning o a suspension bridge cable can be approimated b the 3d quadratic equation H ( d where H, 7 is height above the road in metres and d is the distance rom the centre o the cable in metres. ( t In applied biolog, the relationship C ( t 3t t is used to predict the number o bacteria (C in a swimming pool, measured as count per cm 3, given the time in das rom treatment. t ( t In phsics, the height o a small object above the ground can be predicted rom the unction. H ( t.9t. t, where t is the amount o time in the air and H is the height above the ground. t

40 .38 TPP78 Mathematics tertiar preparation level B Recall then that a quadratic unction has a characteristic satellite dish shape. It can be olded in hal at its verte which is called the turning point and is thus said to be smmetrical. The straight line which bisects the parabola through its turning point is called the ais o smmetr. Ais o smmetr turning point turning point Ais o smmetr The general equation o this upward opening quadratic unction is ( a b c, where a is positive. The general equation o this downward opening quadratic unction is ( a b c, where a is negative..3. Sketching parabolas Previousl ou will have drawn quadratic unctions b plotting points rom a table o values. Although this is a useul wa to get an image o the curve it is oten diicult to get speciic details like the turning point or the points o intersection with the aes. Let s now ollow the procedure we developed or linear unctions to help us quickl sketch an accurate parabola. This technique will onl work i irst ou recognize that the unction ou have will have a parabolic shape. Once ou recognize this then ou can continue b irst inding the coordinates o the intercepts then the coordinates o the turning point. Return to section i ou are unsure how to pick a quadratic epression rom other epressions. Where does the parabola cut the aes? Recall that on a standard Cartesian plane, where is the independent variable and the dependent variable, along the -ais all values o are, while on the -ais all values o are. So i we want to determine the value o the -intercept we would irst put and i we wanted to ind the -intercept we would put. Let s look at an eample rom beore. A signwriting compan wants to manuacture a series o signs which their product suppliers (timber and paint sa must have a perimeter o metres and an area o 6 square metres. What length and width should the make these signs?

41 Module B Relations and unctions.39 In this problem or the signwriting compan we had to solve the equation l l 6 where l represents the length o the sign. Let s write the quadratic part o the equation as a unction and graph it. This means that we would graph the unction ( l l l 6. In this unction l is the independent variable and would be placed on the horizontal ais and the value o the unction (l on the vertical ais. Just as we did or linear unctions, to ind the vertical intercept or a quadratic unction we would put l, so ( 6 6. The vertical intercept is si. Note that 6 is the constant term in the quadratic unction. It will alwas be the case that the value o the constant term, c, in the general quadratic unction intercept. ( a b c is the value o the vertical To ind the horizontal intercept we would put ( l, producing l l 6. This is o course the quadratic equation that we spent so much time on in module 3. Quickl reresh that topic beore moving on. From this point we assume that ou are amiliar with actorization and the use o the quadratic ormula. To solve l l 6 we could actorize the epression on the let hand side to get ( l 3( l and then solve the equation to ind the horizontal intercepts o l 3 and l. I we had used the quadratic ormula we would have evaluated: b l b a ac ( 6 3 or. Although we wouldn t normall use the quadratic ormula to ind the solutions to a quadratic equation that is eas to actorize, it is useul because part o it ( b ac, commonl called the discriminant will tell us the number o solutions we would epect. This number o solutions is directl related to the number o times the curve will intersect the horizontal ais. Remember the table rom module 3? Let s etend it to include this concept. Quadratic equation Value o b ac Number o solutions Number o horizontal intercepts 3 3q q t 8 6 t So in terms o intersection with the horizontal ais quadratic unctions can be divided into three groups depending on the value o the discriminant, b ac.

42 . TPP78 Mathematics tertiar preparation level B I b ac is positive ( b ac then the equation will have two solutions and will cut the horizontal ais twice. or I b ac is zero ( ac then the equation will have one solution and will onl touch the horizontal ais at one point. b I b ac is negative ( ac then the equation has no solutions, cannot be solved and will not ever intersect the horizontal ais. b Where is the turning point? I we return to the signwriting unction ( l l l 6, we know it is a quadratic unction and so must be smmetrical about the ais o smmetr. We also know it will intersect the horizontal ais at and 3. Thus the value o l midwa between and 3 must be the l coordinate o the turning point. Thus the turning point will be (.,. (we get the nd coordinate b evaluating (..

43 Module B Relations and unctions. (l l The above sstem works well when we have a quadratic unction that touches or cuts the horizontal ais but proves diicult when the curve lies above or below the ais. In this case it b is useul to know that the irst coordinate o the turning point is (note we will not derive a the ormula at this stage and will return to it in the net level o Tertiar Preparation Mathematics. Eample Sketch point. ( indicating clearl the intercepts and the coordinates o the turning ( is a quadratic unction with a negative coeicient o downward opening shaped graph. so will be a To ind the vertical intercept evaluate (, (, the intercept is.

44 . TPP78 Mathematics tertiar preparation level B To ind the horizontal intercept solve To ind the turning point evaluate The value o the unction at this point is Coordinates o the turning point are ( We can now use these three points to get a quick sketch o the parabola. Note that b ac is positive so we would epect the unction to have two horizontal intercepts. ( or ( ( ( ( a b ( ( (,

45 Module B Relations and unctions.3 Eample Sketch the graph o the unction. ( 3. Use our graph to determine the domain and range o ( 3 is a quadratic unction with a positive coeicient o upward opening shaped graph. To ind the vertical intercept evaluate (, so will be an ( 3 3, the intercept is 3. To ind the horizontal intercept solve ( = ( 3 Note that b ac is negative so we would epect the unction to have no horizontal intercepts. There are no solutions to this equation. To ind the turning point evaluate b a The value o the unction at this point is ( ( ( 3 Coordinates o the turning point are (, Put these two points together to sketch the curve From the graph we can see that the etent o the values is ininite so the domain is all real values o. The range however, is restricted in that it has a minimum value at the turning point. So the range is (.

46 . TPP78 Mathematics tertiar preparation level B Activit.. The ollowing are parabolic curves Eamine the discriminant o each o the ollowing unctions to determine how man times (i an the intersect the -ais. (a (b (c (d ( 6. Sketch the graphs o the above unctions and clearl indicate the turning point, the vertical and horizontal intercepts. 3. A tug-o-war team can produce a tension on a rope ollowing the unction F 9(8t.t., where t is time in seconds and F is the tension orce. Sketch this graph and estimate when the tension is at its maimum value..3. Rate o change o a quadratic unction It s eas to determine the gradient or rate o change o a unction i it is a linear unction because linear unctions alwas have a constant gradient or rate o change. Quadratic unctions are not so eas because the rate o change o one variable with respect to the other is alwas changing. Let s look at part o a parabolic unction step b step ou ma have seen this beore i ou have studied the previous Tertiar Preparation Program Mathematics unit. A straight sided backard pool starts to be illed at a rate o cm per hour. The graph o that part would be something like this with a gradient or rate o change o. Depth o Pool Against Time Pool Depth (cm Time (hours