Obj: SWBAT Recall the many important types and properties of functions


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1 Obj: SWBAT Recll the mny importnt types nd properties of functions Functions Domin nd Rnge Function Nottion Trnsformtion of Functions Combintions/Composition of Functions OnetoOne nd Inverse Functions Even/Odd Functions nd Symmetry Types of Functions Algebric 1. Polynomil () Constnt (b) Liner (c) Qudrtic (d) Higher Power (Cubic, Qurtic, etc.). Root 3. Rtionl Trnscendentl 1. Eponentil. Logrithmic 3. Trigonometric 4. Absolute Vlue 5. Step 6. Piecewise Defined Mthemticl Modeling nd Functions Ctlog of Functions Constnt, Liner, Qudrtic, Root, Absolute Vlue, Rtionl, Eponentil, Logrithmic, Trigonometric, Piecewise, Step
2 Functions Definitions: 1. A reltion is correspondence between two sets (s in coordintes). A function is correspondence between two sets where ech member of the first set, inputs, corresponds to ectly one member of the second set, outputs (the grph must pss the verticl line test: no verticl line crosses the grph more thn once). Functions cn be represented in mny wys, s grphs, tbles, equtions, pictures, verblly, etc. Eercise: Determine which of the reltions below re functions: y y y = y
3 We usully lbel functions with for the input vlues nd y for the output vlues, corresponding to the nd y es in the coordinte plne. We hve mny different wys to refer to the vlues nd y represent (see the chrt below): Input Independent vrible Domin y Output Dependent vrible Rnge Here, little more needs to be sid bout domin nd rnge.
4 Domin nd Rnge Definitions: The domin of function is ll input vlues tht hve corresponding output vlue The rnge of function is ll output vlues tht correspond to these input vlues It is strightforwrd to find the domin of function from the grph, tble, nd even eqution. Here re some common functions nd their corresponding domins nd rnges. For the grph, look for the vlues (domin) nd yvlues (rnge), on the  nd yes respectively, for vlues tht correspond to the grph. With tbles, one column is the domin nd the other is the rnge. If you re given n eqution, you cn determine the domin by considering which vlues mke sense in the function (i.e. you cnnot hve the squre root of negtive number). Function Domin Rnge (, ) (, ) (, ) ( 0, ) ( 0, ) ( 0, ) 1 (,) (, ) (,0) (0, ) log ( 0, ) (, ) sin (, ) [1, 1] The most interesting domin questions re sked bout root functions (the domin is ll which llow the rdicnd to be nonnegtive) nd rtionl functions (the domin is ll ecept wht mkes the denomintor zero). Use the Rule of Four whenever it is helpful. Eercise: Find the domin nd rnge of #18 in the bove eercise
5 Function Nottion Function nottion is useful wy to represent functions, which mkes them esier to evlute. For emple, the eqution 3 cn be written in function nottion s f ( ) = 3. This gives our function nme, f, nd helps evlute: f (5) = (5) 3 = (5) 3 = 75 3 = 7 f (4c) = (4c) 3 = (16c ) 3 = 3c 3 f ( + 7) = ( + 7) 3 = ( ) 3 = ( ) 3 = Eercise: f ( + 7) f ( ) Evlute. 7
6 Trnsformtions of Functions Knowing the grphs of few bsic functions (see ctlog below), through different trnsformtions we cn esily grph mny, mny more functions. Assuming we know wht f() looks like nd tht c > 0, the following re some bsic trnsformtions: f () Originl function f ( ) + c Shift f() up c units f ( ) c Shift f() down c units f ( + c) Shift f() left c units f ( c) Shift f() right c units c f () Verticlly stretch/compress f() by fctor of c units f (c) Horizontlly stretch/compress f() by fctor of 1/c units f () Reflect f() over the is f ( ) Reflect f() over the yis f ( ) Reflect f() over the origin Here re some emples: ( + 3) 1 1 cos cos( π ) + 1 Eercise: Using s you bse function/grph, describe the trnsformtions to get the grph of Describe wht the new grph would look like.
7 Combintions/Composition of Functions One importnt concept in higher mthemtics is creting new functions from old ones. One importnt considertion we must be wre of when creting new function is their domins. Below is tble describing five wys to crete new function with descriptions nd their new domins. We will ssume we hve vilble the functions f(), with domin some set of numbers we will cll A; nd the function g(), with domin some set of numbers we will cll B. New Function New Domin Description f ( ) + g( ) (or ( f + g)( ) ) A B The new function hs output vlues tht re the sum of the outputs of f nd g f ( ) g( ) (or ( f g)( ) ) A B The new function hs output vlues tht re the difference of the outputs of f nd g f ( ) g( ) (or ( fg )( ) ) A B The new function hs output vlues tht re the product of the outputs of f nd g f ( ) / g( ) (or ( f / g)( ) ) A B, but The new function hs output vlues tht re the quotient of the outputs of f nd g g( ) 0 f ( g() ) (or ( f g)( ) ) All in the domin of I such tht g() is in the domin of f The outputs of g() become the inputs of f() to give new outputs f(g()). See the picture to the right Domin of g Domin of f Domin of fg Consider the functions f ( ) = 3 nd g( ) =. Let us investigte some combintions nd compositions: ( f + g)( ) = ( 3) + ( ) = 3 = 4 ( fg )( ) = ( 3)( ) = + 6 ( f g)( ) = f ( g( )) = f ( ) = ( ) 3 = 4 ( f f )( ) = f ( f ( )) = f ( 3) = ( 3) 3 = ( 6 + 9) 3 = 6 + 6
8 Function (grph color) Tble f ( ) = (blue) y g( ) = (red) y ( f + g)( ) = (green) y ( fg )( ) = + 6 (blck) y ( f g)( ) = 4 4 (purple) y ( f 4 f )( ) = 6 (ornge) y Grphs Eercise: 1. Construct tble of ( f g)( ) nd ( g f )( ). Sketch grph of ( f g)( ) nd ( g f )( ) 3. Determine the eqution of ( f g)( ) nd ( g f )( )
9 OnetoOne nd Inverse Functions Recll the definition of function: A function is correspondence between two sets where ech member of the first set corresponds to ectly one member of the second set. We cn specilize this even more nd define onetoone functions. Definition: A onetoone function is function from one set to nother set where ech member of the first set corresponds to ectly one member of the second set AND ech member of the second set corresponds to ectly one member of the first set (in other words, f ( ) f ( b) whenever b ). Just like we cn use the verticl line test to identify functions, we cn go one step further n do horizontl line test to identify onetoone functions (no horizontl line crosses the grph more thn once). Here re some emples nd nonemples of onetoone functions: OnetoOne Functions y Functions which re NOT onetoone y
10 Just like the unique properties of functions let us do more interesting mth with them thn mere reltions, onetoone functions let us do more interesting mth with them thn mere functions. One importnt property of onetoone functions is the fct tht they hve inverses. Definition: The inverse of function is obtined by switching the input with the output (we use the nottion function f). 1 f to represent the inverse of Eercise: Why is onetoone function gurnteed to hve n inverse? Why would function tht is NOT onetoone be unble to hve n inverse? There re mny wys to obtin nd check inverses of function: To obtin the inverse of function f: o Switch the input nd output o Reflect the grph over the line o Interchnge nd y in the eqution nd solve for y. To check tht functions f nd g re inverses: o Verify tht ( f g)( ) = ( g f )( ) = o Grph f nd g in the sme window nd verify they re reflected over the line.
11 Even/Odd Functions nd Symmetry Definitions: 1. The function f () is even if f ( ) = f ( ). The function f () is odd if f ( ) = f ( ) Here re some emples: y Even Odd y As you cn see, even functions re symmetric bout the yis nd odd functions re symmetric bout the origin. Eercise: Wht is interesting bout functions which re symmetric bout the is?
12 Types of Functions Clculus is relly just n nlyticl study of functions. There re mny different ctegories nd types of functions tht we need to be refmilirize ourselves with before beginning our clculus endevor. They fll into two bsic ctegories: lgebric nd trnscendentl. Algebric functions re ny function tht cn be constructed using lgebric opertions (ddition, subtrction, multipliction, division, nd tking roots) with polynomils. n n 1 Definition: A polynomil is n epression of the form n + n where is vrible nd the s re constnt (clled the coefficients). The degree of polynomil is the lrgest power of. Bsiclly, polynomil is the sum or difference of vribles to whole number powers with rel number coefficients. Emples of lgebric functions include liner, qudrtic, cubic, rtionl, root functions. Trnscendentl functions re just functions tht re NOT lgebric. Emples include trigonometric, eponentil, logrithmic, bsolute vlue, piecewise defined functions. Eercise: Why re eponentil functions (like f ( ) = ) NOT lgebric? Algebric Functions 1. Polynomil functions re functions which re polynomils. These type of functions re our best friends in clculus s we cn be gurnteed tht they lwys: o Are continuous (no breks) o Are smooth (no corners) o Hve domin of ll rel numbers (wht cn we sy bout the rnge?) o Hs t most n (the degree) roots/zeros/intercepts Let us consider some specil cses: () Constnt functions, like f ( ) = 17, re polynomils functions with degree zero. The hve the following properties: o The output is lwys the sme (constnt). o Their grphs re horizontl lines o Emple
13 (b) Liner functions, like f ( ) = 17 11, re polynomil functions with degree one (wht we cll lines). These were probbly the first type of functions you investigted. With functions such s these, there is lot of informtion we cn obtin. First, we need to define some new nottion tht will be helpful when studying clculus: Definition: 1. The increment = 1, is the chnge in (the difference of two vlues). We red this s delt of the chnge in. The increment y y1, is the chnge in y (the difference of two y vlues). We red this s delt y of the chnge in y We hve mny wys to represent line symboliclly: o Generl Form: A + By + C = 0 o PointSlope Form: y y1 = m( 1 ) where m is the slope nd (, y 1 1 ) is point on the line. o SlopeIntercept Form: m + b where m is the slope nd b is the yintercept. You should be fmilir with ech form, becuse they ech hve their own uses. The lst eqution bove tells you how to determine the slope of line symboliclly. We cn lso determine the slope numericlly, grphiclly, nd verblly: slope = m = rise run = chnge in chnge in y = y y 1 1 = y Verblly, it is little tougher. Look for the words rte of chnge or just nything bout quntities chnging. Some specil types of lines re prllel, perpendiculr, verticl, nd horizontl lines: Definitions: 1. Prllel lines re two distinct, nonverticl lines with the sme slope. Perpendiculr lines re two nonverticl lines with slopes tht re opposite reciprocls ( m 3. Verticl lines re lines of the form =, where is constnt. 4. Horizontl lines re lines of the form b, where b is constnt. 1 1 = ) m Eercise: Why cn verticl lines NEVER be functions? Click here for n emple of liner function.
14 (c) Qudrtic functions, like f ( ) = 4 5 = ( ) 9 = ( 5)( + 1), re polynomil functions with degree two (we cll their grphs prbols). These grphs hve verte, which is the mimum or minimum point. Using the form ( h) + k we cn identify the verte t (h, k), nd helps us determine if it opens up or down ( > 0 opens up, < 0 opens down). Using the form ( r1 )( r ) we cn identify roots/zeros/intercepts t r 1 nd r (qudrtic functions cn lso hve no roots). Here is n emple of qudrtic function. Eercise: Give n emple (symboliclly, grphiclly, or verblly) of qudrtic function with no roots (d) Higher power polynomil functions, like f ( ) = , sometimes hve specil nmes like cubic functions (polynomil functions of degree three) or qurtic functions (polynomil functions of degree four). Agin, they hve the properties of generl polynomil functions.. Root functions re similr to polynomil functions, but the eponents cn lso be frctions. Perhps the best known root function is 1 f ( ) = =. These functions re not gurnteed to hve domin of ll rel numbers (the domin of f ( ) = is ctully [ 0, ) 3 but the domin of g ( ) = IS ll rel numbers). This hs to do with the fct tht you cnnot tke the even root of negtive number (e: 6 3 does not eist). Here is n emple of root function Rtionl functions, like f ( ) =, re quotients of polynomil functions (the denomintor CANNOT equl zero). This type of 3 + function comes up often in clculus becuse they hve specil fetures, like symptotes (lines where the function pproches ner the ends ) nd holes. Becuse of these symptotes nd holes, we cnnot be gurnteed tht the domin of such functions is ll rel numbers. Here is n emple of rtionl function. Do you see the symptotes? Eercise: Grph f ( ) = using your clcultor nd ZOOM 4 (ZDeciml). Wht shpe does the grph hve? Wht do you + 1 notice hppens when = 1? Look bck t the function nd determine why this hppens.
15 Trnscendentl Functions 1. Eponentil functions, like f ( ) = re functions where the eponent is vrible nd the bse is positive constnt. These differ from polynomil functions becuse they increse (or decrese) t much fster rte thn polynomil functions. More specificlly if the bse of n eponentil grph is greter thn one, the grph increses t n incresing rte (the grph rises fster nd fster s increses). Fortuntely, these ll eponentil functions re lso gurnteed to hve domin of ll rel numbers AND horizontl symptote. Here re some grphs of eponentil functions: 1 (blck) 1 (green) (red) e (blue) 3 (ornge) 4 (purple) The nturl bse, e, is especilly useful, convenient, nd frnkly beutiful (which we will see lter on in clculus). It is n irrtionl number ( deciml tht never termintes or repets), pproimtely e = (note the connection to the grph bove). This is probbly good plce to remind you of the rules for eponents: y = + y y y y y = ( ) = ( b ) = b = b b n m = m n = m ( ) n Here is n emple of n eponentil function.. Logrithmic functions, like f ( ) = log 4 ( + 5), re simply inverses of eponentil functions. If you remember wht it mens for two functions to be inverses, it follows tht we re gurnteed tht the rnge of logrithmic functions is ll rel numbers AND verticl symptote. Understnding inverses, you should lso be ble to understnd the crucilly importnt sttement below (the definition of
16 y logrithm): log b b =. Logrithmic functions must hve positive bse which is NOT 1. Another importnt note: for b > 1, the grph of logrithmic function increses t decresing rte (rises more nd more slowly s increses). Here re some grphs of logrithmic functions: log 1 (blck) log (red) log e = ln (blue) log3 (ornge) log 4 (purple) The nturl logrithm, ln, is simply log e. Agin, the nturl logrithm function is especilly useful, convenient, nd beutiful. This is probbly good plce to remind you of the rules for logrithms (notice the reltionship with the rules for eponents): log + log y log ( y) log log y = Here is n emple of logrithmic function. r log ( ) r log log = log 1 = 0
17 3. Trigonometric functions, like f ( ) = tn( 3 π ) + 4, re functions involving sine, cosine, or tngent (including cosecnt, secnt, cotngent). In clculus, for inputs we lmost lwys use rdins insted of degrees (unless otherwise noted). Focusing just on sine, cosine, nd tngent (since ll other trigonometric functions cn be obtined from these); let us look t the grphs nd note some importnt properties: sin() cos() tn() o Any sine or cosine grph hs the domin ll rel numbers o Tngent grphs hve verticl symptotes nd lwys hve rnge of ll rel numbers o Trigonometric functions re periodic: f ( + c) = f ( ) for some constnt c, which we define to be the period (e: cos( π ) = cos( π + π ) = 1) o Sine nd cosine bove hve period π nd tngent hs period π o The mplitude of trigonometric grph is hlf of difference between the mimum vlue nd the minimum vlue (sine nd cosine bove hve n mplitude of 1, the mplitude of tngent is not defined). o Sine nd tngent re odd functions, wheres cosine is n even function
18 Here would be good plce to recll the importnt trigonometric identities: Reciprocl Identities Quotient Identities Cofunction Identities Pythgoren Identities Sum/Difference Formule DoubleAngle Formule csc = 1 sin tn = sin cos π sin = cos sin + cos = 1 π cos = sin sec = 1 cos cot = cos sin π sec = csc π csc = sec cot = 1 tn π tn = cot tn + 1 = sec 1 + cot = csc sin( ± y) = sin cos y ± cos sin y cos( ± y) = cos cos y sin sin y sin = sin cos cos = cos sin = cos = 1 sin 1 tn( ± tn y) = tn ± tn y 1 tn tn y tn tn = 1 π cot = tn Eercise: It is often esier to remember how to derive certin identities from other insted of just rote memoriztion of ech identity. A. Eplin how we cn derive the second Quotient Identity from the first. B. Eplin how we could derive the lst two Pythgoren Identities from the first one. C. Eplin we cn derive the DoubleAngle Formule from the Sum/Difference Formule. D. Eplin how we cn remember the cofunction identities. Here is n emple of trigonometric function. 4. Absolute vlue functions, like f ( ) = 4 + 3, re functions involving n bsolute vlue. Definition: The bsolute vlue of ny rel number is defined:, if 0 =., if < 0 Absolute vlue grphs re similr to prbols with very importnt difference. Insted of being nice, smooth grph, they will hve corner. This will become very importnt when studying differentil clculus.
19 Here is n emple of n bsolute vlue function 5. Step functions, like the gretest integer function: f ) = [[ ] ] (, re functions tht cn jump from one vlue to the net. These will become more interesting when we begin discussing limits. Definition: The gretest integer function f ) = [[ ] ] equl to the input (e: [[ 3.5]] = 3, [[ 17 ]] = 17, [[ 3.5] ] = 4 Here is n emple of step function. ( tkes in ll rel numbers s inputs nd outputs the lrgest number less thn or ) + 1, 6. Piecewise defined functions, like f ( ) = 3, < < 1, re just ptched together functions, defined in pieces. These functions 4 tn, 1, 1 cn be disconnected (like f() bove) or connected, like g( ) =, > 1 Eercise: 1. Evlute f(3), f(), f(0), f(1), f( π ), g(3), g(0), g(1), g(10). Check tht g() is connected in your clcultor by typing in Y 1 = ( )( 1) + ( ) ( > 1) (You cn find the symbol under the nd MATH or TEST key) 3. Determine the formul for the function grphed to the below: Here is n emple of piecewise defined function.
20 Mthemticl Modeling Using Functions Now, you my sk (nd probbly often do), Why do I need to lern this? One very prcticl nswer is tht these functions we re lerning bout cn be used to model rel world phenomen. You my hve herd of models like the Consumer Price Inde, predtorprey models, the shpe of the Golden Gte Bridge, the Popultion Growth model, supply/demnd models, Big Mc price models, etc. My wife, for emple, gets pid by the Nvy for modeling the lifetime nd costs of jet engines. Modeling is lso one of the big resons understnding the verbl representtion of function is so importnt. Models re bsiclly mthemticl tools to describe or represent rel world phenomen quntittively. Two key fetures of mthemticl models re simplifiction nd predictbility. Models need to hve enough detils to represent the essence of phenomenon, but simple enough to permit clcultions. Models lso cnnot just describe lredy observed behvior, they should help us predict nd understnd future behvior. Modeling is essentilly wy to mthemticlly predict the future! It gives new mening to the term mthemgicin Here re some emples using functions to model relworld phenomenon: o o o o The height of wter in your bthtub when filling it up cn be modeled by liner function with positive slope. The pth of bll thrown from pitching mound cn be modeled by prbol (qudrtic function) opening downwrds. The growth in the popultion of the world cn be modeled by n eponentil function. The temperture in Lusby, MD cn be modeled by trigonometric function.
21 A Ctlog of Functions Type Grphic Symbolic Numeric Verbl Constnt y A sitution with constnt output vlue Liner y A sitution with constnt rte of chnge (slope) nd n initil vlue (yintercept)
22 Qudrtic y A sitution where the chnge in y chnges t constnt rte (the chnge in y decreses by two in this emple) Root y 3 undef.  undef. 1 undef A sitution where the outputs re the squre roots of the inputs Absolute Vlue y A sitution where ech output vlue hs the sme sign
23 Rtionl 1 y undef A sitution where the output is the reciprocl of the input Eponentil e y A sitution where the outputs increse t n incresing rte Logrithmic log y 3 undef.  undef. 1 undef. 0 undef A sitution where the outputs increse t decresing rte
24 Trigonometric sin y  π 0  π / 1  π /4  / 0 0 π /4 / π / 1 π 0 A sitution tht is periodic in nture Piecewise Defined Function +, 1 6, 1 < 3, > 3 3 y A sitution where it is necessry to hve different functions on different intervls Step Function [[ ] ] y A sitution where the output vlues re discrete numbers
25 Liner 1 y A sitution with constnt rte of chnge (slope) nd n initil vlue (yintercept) Qudrtic or y + 4 = 3 ( 1) y A sitution where the chnge in y chnges t constnt rte (the chnge in y decreses by two in this emple)
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