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# Class Note for MATH 1310 at UH 3

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This 20 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 15 views.

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Date Created: 02/06/15

Seclinn 2 anhing a Func nn Page I 111 m Graphing a Function Let f be a funetlon wlth olomam A The graph ofthe funetlon l5 gwen by m l x e Agt The gaph is the set of all polnts xy m the eoorohnate plane whose xrcoordlnates are elements ofthe olomam A and whose conespondmgyrcoordmates are gwen by y fx For a functlony fx exaetly one yrvalue ean be asslgned to a gwen xrvalue Thls means that a vertleal hne ean lnterseet the gaph of a functlon at most onee Thls provldes a test for oletermlmng whether or not a eurve ln the xyrplane ls the gaph of a functlon The Vem39eal Line Test A eurve ln the xyrplane ls the graph ofa functlon lf and only lf no vertleal hne lnterseets that eurve more than onee For the gure shown below usmg the Vertleal Lme Test we see that the eurve ln part a oloes not represent a functlon whlle the eurve ln part b oloes represent a functlon The vemcal lme mterseets the gaph more then once Every vemcal lme mteseets a the grrphrtmostonee an Example 1 sketeh the graph of the funeta on fx J Snllltinn tn Example 1 Seclinn 2 anhing a Func nn Page 2 at m The fuhemhxs not de ned for those values ofx that are negauve Thus the domam of the fuhemh hem x2 0 The domam m interval hexamh xs0w To graphthe fuhehon make atable ofvalues m mam 0 lt0gt 0 0390 1 lt1gt 1 1391 2 lt2gtJ 245 3 lt3gt 245 4 lt4gtJ32 4392 The gaph Ts shown m the gure below U 0390 1 2 3 A 5 Example 2 Sketch the graph ofthe fuheh ahx x2 1 Snllltinn m Example 2 Seclinn 2 anhinngunc nn ragesurzn The fuheuohls de ned for all values ofx The domaln ofls the set ofreal numbers The domaln m lnterval holauoh l5 w The graph ls a parabola To graph the functlon make a table of values ND 1 2 e2e22l4l5 725 1 rlrl21112 712 0 002101 01 1 m 21ll2 12 2 2221415 25 Example 3 Sketch the graph ofthe fuheuOhr 2 lf x371 x2 lf xgtrl slurmm m Example 3 Seclinn 2 anhinngunc nn Pageurzn Forrg 1 the gaph eorherdes with the horizontal lmey2 Forxgt71 the gaph eorherdes thhthe graph ofthe parabola y x2 The gaph rs shown m the gure below The open dot at 711 mdxcates that this pomt rs excluded eorh the graph Example State whether or hot eaeh ofthe fouowrhg graphs represeth a fuhehoh y y a h Snllltinn m Example 4 Seclinn 2 anhinngunc nn ragesurzn Use the Vemcal Lme Test for the graph rh part a y a The rst graph does not represent afuncnon srhee we ear nd avemcal hhe that rhterseets the graph more than three Use the Vemcal Lme Test for the graph rh part b y h It rs easy to see om the 4 example hhes shown that every vemea1 hhe rhterseets the gaph at most same The second gaph represents a runetaor Example Seclinn 2 anhing a mm Page a am The graph ofy x 15 shown below a Fmdthe domam of b Fmd the range of e FmdLhefollowmg fuheuoh values fen71 01 d Forwhatvalues ofxxsx 27 Snllltinn m Example 5 Part a Determan the domam by mspeehhg the graph The domam m interval notation is 733 Seclinn 2 craphhrgarmm Pzge7uf2 Part h Deter mlne the range by rhspeehhg the graph The range m lnterval notatlonls 721 Part E Label the polnts on the graph that have rst coordlnates of 73 el 0 and l Flndthe fuhehor values73f71 0 and1 by seleehhg the second coordlnates of the pomts labeled on the graph Seclinn 2 anhinngunc nn ragesurzn ft fH AU 11 Part a Label the pomts on the graph that have a second coordlnate of 72 The value ofx forwhlch x215 found by seleehhg the rst coordlnate ofthe pomt labeled on the graph Example 5 State the domam of the fuhehOhx x 2 lh lhtemal hotahOh and then sketeh the gaph othe function Snllltinn tn Example 5 Seclinn 2 anhingzl unc nn P2ge uf2 For each rea1 number x VH2 rs area1 number Thus he domam m interval rotaaor rs emu To sketch the graph make atable of values 22224 71 71712123 0 002022 1 112123 2 222224 Plot the perms found m the table Seclinn 2 anhinngunc nn Page I uf2 Plot addmonal porhts xfnecessary to sketeh the graph Example 7 State the domam of the fuhetrOhr 2x3mmterva1 hotahor and ther sketeh the gaph othe runehor Snllltinn tn Example 7 For each real number x 2x3 xs areal number Thus the domam m mterval hotaaorrrs errata The graph rs ahhe W1thy1ntercept3 and slope 2 Use the yrmtercept 3 to plot the porht o 3 Seclinn 2 anhingzl unc nn Fig 11 uf2 Use the slope 2 to move 2 umts up and 1 umtnghtfrom the pomt 03 to locate another pomt on the hue 7 5 5 lumtnght 2 nuts upA my 2 2 1 D Label me new pomt umtnght 5 39Is 3 W4 Seclinn 2 anhinngunc nn Page lzurzn To eomplete the graph olraw alme through the two pomts Examples State the olomam of the fuhetrohr 7x2 2 m mterval hotauoh and then sketeh the gaph othe luheuoh Snlutinn tn Example 8 For eaeh real number x a a 2 ls areal humber Thus the olomam m mterval hotauor rs 7mm To sketeh the graph make atable of values 2 e2x722277427 new 4702 0 ltogtr lt0gt2 r2 1 1r1227 12 2 A2 22 774 71 Seclinn 2 cnphingarnnen39nn Page 1311 Plot the points found m the table an 41 t m t Plot addmonal points xfnecessary to sketeh the graph 31m 2 ak39 env 1 LI t 7 Example 9 State the domam of the functionx 1 2 tn interval notahon and then sketeh the gaph othe runehon Snllltinn tn Example 9 Seclinn 2 anhinngunc nn Pig unrzn For tx2 to be areal number x2 eannotbe negauve Solve the mequahty x22 0 to ndthe domam off x220 x2722072 x272 The domam tn interval notahon is 723 To sketeh the graph make atable of values 1 lt71gtm 1 0 0WJ 0J5 1 lt1gt412J 2 2 22JZ2 Plot the points found m the table Seclinn 2 anhing a Func nn Plot addmonal porhts xfnecessary to sketeh the graph Example 10 Sketeh the graph of the piecewiserde ned fuhetrorrf grveh by 3 1fxlt0 x 1 rf x20 Snllltinn tn Example 10 Forx lt0 the graph of eorherdes wrth the honzorrta1 lmey 3 Page IS uf2 Seclinn 2 anhingzl unc nn Pig mmn Seclinn 2 anhingzl unc nn Pig nmn Sketch the graph Exercise Set 32 Functions and Graphs Answer the following Determine whether or not each of the following graphs represents a function f x is shown below The graph of y 9 21 Find the domain of the function h Find the range of the function c Find the following function values f2 f0 f4 f6 d For What values of x is f x 7 gx is shown below 10 The graphof y 21 Find the domain of the function h Find the range of the function c Find the following function values g72 gH g0 g3 g5 d For What values ofx is gx1 7 Exercise Set 32 Functions and Graphs 11 The graph of y gx is shown below a Find the domain of the function h Find the range of the function c Find the following function values gH g0 g2 g3 g6 d Which is greater g72 or g3 12 The graph of y fx is shown below a Find the domain of the function h Find the range of the function c Find the following function values f3 f2 f1 f0 f4 d Which is smaller f0 or f3 For each of the following functions a State the domain of the function h Graph the function 13 x 7 6 14 x x7 4 15 hx3x75 71 xlt3 16 hx72x 73lt xg2 17 gxx73 18 gxx74 19 x 73 20 mo J5 21 Fx x2 7 4x 22 Gx x7 32 1 23 x 9 1 24 fxx4716 25 gx 2 x 26 hx 75 X Determine whether each of the following equations de nes y as a function of x Do not graph 27 3x75y8 28 2x796y2 29 c3y2 30 x2y3 31 3x472xy5x 32 7174 5 Exercise Set 32 Functions and Graphs 33 3x2y216 34 2y2 73x2 8x 35 x3y72y6 36 173y5 37 x35y 3s y377x 39 y72x 40 x3y4 Graph the following piecewisede ned functions 2x4 if 72 xlt1 7x3 if ISXSS 41 fx Lc2 if 73 x 0 42 fx 3 74x3 if xgt0 43 x 44 x 45 x x if x 73 x2 if 73ltxlt2 4 if x22 47 x x275 if xlt0 1 if ngg3 27x if xgt3 4s x 4 4 t 46 fx t

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