ࡱ> "$!M Cbjbj== 0WWClv v v v < $ 6       HJJJJJJ$ ln     n   r  H HhrT< 1Zv  < 0.  <<,Domain and Range Worksheet Given a function y = f(x), the Domain of the function is the set of permissible inputs and the Range is the set of resulting outputs. Domains can be found algebraically; ranges are often found algebraically and graphically. Domains and Ranges are sets. Therefore, you must use proper set notation. When finding the domain of a function, ask yourself what values can't be used. Your domain is everything else. There are simple basic rules to consider: The domain of all polynomial functions and exponential functions is the Real numbers R. Square root functions can not contain a negative underneath the radical. Set the expression under the radical greater than or equal to zero and solve for the variable. This will be your domain. Rational functions can not have zeros in the denominator. Determine which values of the input cause the denominator to equal zero, and set your domain to be everything else. Log functions must have a positive value in the argument position. Solve for the domain like you would for square root functions. Examples: Consider  EMBED Equation.3 . Answers: Since f(x) is a polynomial, the domain of f(x) is R. Since g(t) is a square root, set the expression under the radical to greater than or equal to zero: 2 - 3t ( 0 ( 2 ( 3t ( 2/3 ( t. Therefore, the domain of g(t) = { t | t ( 2/3 }. Confirm by graphing: you will see that the graph "lives" to the left of 2/3 on the horizontal axis. Since h(p) is a rational function, the bottom can not equal zero. Set p2 - 4 = 0 and solve: p2 - 4 = 0 ( (p + 2)(p - 2) = 0 ( p = -2 or p = 2. These two p values need to be avoided, so the domain of h(p) is { p | all R except p = -2 or 2 }. Comment on interval notation: The set of all reals can be abbreviated R, but not {R}. It can also be written  EMBED Equation.3  but not  EMBED Equation.3 . It can also be written { -( < x < ( }. For h(p), the domain could be written any of the following ways:  EMBED Equation.3  or {R\{-2,2}}. The backslash \ is read as "except". Whatever method you use, be consistent and correct. Domain Practice: Algebraically determine the following domains. Use correct set notation. d(y) = y + 3 g(k) = 2k2 + 4k - 6 h(x) = log (x - 6) b(n) =  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Answers at end. Challenging domain problems: these contain combinations of functions.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Range practice: Use your calculator to graph and determine the ranges of the functions numbered 1-10. Answers: R, since the function is a polynomial (line). R, since the function is a polynomial (parabola). { x | x > 6 } { n | n ( 4 } { t | t ( 3 } { x | R except x = -2 } or { x | R\{-2}} { r | R except r = 1 } or { r | R\{1}} R, since this is an exponential function. { c | R except c = 0 or c = -3 } or { c | R\{0,-3}} R. The denominator can not be solve for zero. No value of w causes the denominator to equal zero. { x | x > -3 }. In this case, the radical can not contain negatives, while the denominator can not contain zero (a zero under the radical is acceptable, but it makes the bottom zero, which is not acceptable). { r | R except r = 0 }. The expression 1/r in the exponent means that r can not be zero. { v | v ( -4 or v ( 2 }. The expression under the radical is a quadratic: it needs to be set greater than or equal to zero. Factor it and plot the points -4 and 2 (this is where the expression = 0, which is okay). Then test a point from each of the three regions to see if the result is greater than zero. { t | t < -1 or t ( 0 }. Set the expression  EMBED Equation.3 . Do NOT cross multiply!!!! Determine where the top = 0 (top = 0 at t = 0) and where bottom = 0 (bottom = 0 at t = -1). Plot these on a number line, and test a point in each region to determine when the expression is 0 or greater. Notice that it's okay for t = 0 but not okay for t = -1. Why? { y | y > 0 }. For the first two terms, all y is acceptable. For the third term that has the radical, y ( 0. But in the fourth term, y ( 0, so we have to exclude the 0. The only set of numbers for which all four terms are defined is y > 0. Ranges: R. It's a line. { g(k) | g(k) ( -8 }. It's a parabola. Find the vertex. R. The range for all log functions is R. { b(n) | b(n) ( 0 }. { m(t) | m(t) ( 0 }. { u(x) | R except u(x) = 1/2 }. Graph it and you'll see the graph level off horizontally along the line u(x) = 1/2. This is a horizontal asymptote (chapter 9). { a(r) | R except a(r) = 1} { e(x) | e(x) > 0 } { y(c) | R except y(c) = 0} { q(w) | R except q(w) = 0 }. In #7, 9 and 10, the functions "level" of horizontally. This is covered in Chapter 9. -.1234;A{;<IJ]^_`tuvw            " # $ % A B C D J K N O P Q       & ' ) * 0 1 ; CJH* jCJ jCJ jCJjCJEHUj3> UVmHnHu jCJU6CJCJ5CJLIJ>56bcmn   F G H $ & Fa$$a$$a$C; < = > G H   " # 6 7 8 9 B C V W X Y v w z { ~  I Z újCJEHUjz3> UVmHnHu jCJjCJEHUj3> UVmHnHujCJEHUj3> UVmHnHu jCJU5CJ6CJCJ jCJ>H I Z [  ) A Y q *BZ[$ & Fa$$a$      % & ' ( ) * = > ? @ A B U V W X Y Z m ȿzj(CJEHUj3> UVmHnHujCJEHUjֈ3> UVmHnHuj CJEHUjň3> UVmHnHuj| CJEHUj3> UVmHnHujZ CJEHUj3> UVmHnHu jCJUCJCJH*6CJ/m n o p q r &'()*+>?@ABCVߺߣߌuljCJEHUj3> UVmHnHujTCJEHUj3> UVmHnHuj)CJEHUj3> UVmHnHujCJEHUji3> UVmHnHujCJEHUj+3> UVmHnHuCJ jCJUj]CJEHUj 3> UVmHnHu$VWXY[j569:CDGHIJQRUVWX_`cdlmz{~  CDGHOQkl jCJ jCJ6CJ 5>*CJ5CJCJ jCJUj"CJEHUj)3> UVmHnHuN3AO] oA>23;<$ & Fa$$a$Z[01@ADEkl+,3;<=OPQRVWXY[\5CJ jCJjz$CJEHUjk3> UVmHnHu jCJU jCJCJ6CJP<M}@ABC$a$$ & Fa$DEFGC5CJCJ6CJ4/ =!"#$%DdB  S A? 2o ㇤̷X’KD`!C ㇤̷X’$ 8dxOSAgg_R #/H<#p0Bb1)ЄhB[1?hMo<p^I 1&>wvߛ/ټ|g̾O4R#4s@!Ey2*:%W0%լ H(f(5q(2w&^qD) nfx,ӁmR _}4F2cgZX~B؂َ p56b*1 ]%Bꎗ݉2Y !ޅhCn^#M՜=ă̯0v3$!.0t&|||>1bV_Bg?R_qN>b!^Ab^yt܉GYp'z,.a.K^_`D-O$R ¥*;%|pL՘$mh'~<'`ͱDؼsG͔"ZMuGY DdTB  S A? 2v; l;6iY`@|RI`!J; l;6iY`@|`  XJxQAJP}3IM+ ] k@c$H DoB}70d̛ U%q=*kK0UZɱk9⻠)q !KD?"9NWɰ>1gɇ`4=u~G;3RNԑ[ӷ+J=g/Cn݌rDSNuJ)neNߜӍI[8):--oKRA5E}}i9Z|KǓ8YgxI~DdTB  S A? 2xKuDv'STU`!LKuDv'S  ~ XJxcdd``ed``beV dX,XĐ IIMRcgb @P5< %! `}vL@(\_(jw0DHXAJc>?$37X/\!(?71a1&50MA|8_ jKQl"¥ -ݙPʌ .4\aNPLpl121)WB'f:L;Dd @B  S A? 2> oNi$#1c`!y> oNi$#1j` Gxcdd``6ed``beV dX,XĐ )APRcgb  QFnĒʂT ~35;a#L ! ~ Ay 9W'~#4ka|G>f|+/o5\ &UlNTsWƊj^ zwfT@ `2W&00AH R8?ݫ ʯeFOdGw)4q E@)cI)$5n@qq##3XM"DdHhB  S A? 2K2ٚ`Ŝ. h `!`K2ٚ`Ŝ. @@@"|.xJ@ҦVzhO=h&/P!PB 2< =R3 L,x> ev%f"WU:qqkDÝ]&&/M*>Mc1=\OWxȀX`öC@8"]jWTtԠѩa,w-lmI˗i|ӏ~!#LlMfOUf{t |^$XOPdI.VS=AGh:Y%V"97@oHu)1brSVKK GDd|B  S A? 2pC"xS `!pC"xS@ `PV0SxR=KP=&Zq*D "6VThЂ"gg7n wK s߹Bp pQ,"KD!X3ku]洮[7raKU&zd_N8v`tr/"۽]zp%ZzsSv#ǦeOMwЬgy6F6~$q2eGMu'ۙFiLp~i GQgh3÷ V3藬<(#d2A)fI'Fx`F@ޓ_U6DdxlB  S A? 22$n|`!t2$nb BxQ1K`}wIk  N(Bn:`;K+-qC?Y%.޽j`rJ&*,m8I٢i3ey5uM^^*)gd&bLMƩ?>Fpd|pZ!-J|kKdG;ޑga0WAؿ^.#ߝXOsB-N{|^7s:ga3N .?qY\߳u*q:WZ=~Y쫳K~\ݡ>m润[^uDqE~QP&/Dd  G#&'(*)+,-/.021345768:9;<=?>@BACDESFaHIJKLMNOPQRTVWXYZ[\]^_`Root EntryZ F@1%@Data &WordDocumentY0ObjectPool\'y1@1_1043563678Fy1"1Ole CompObjfObjInfo !$'*-0369<?BEHKNQTWZ]`cfhijklmopqrt FMicrosoft Equation 3.0 DS Equation Equation.39qkIvI f(x)=x 3 "6x 2 +5x"11,g(t)= 2"3t  ,h(p)=p"1p 2 "4Equation Native _1043564184 Fpi01pi01Ole CompObj f FMicrosoft Equation 3.0 DS Equation Equation.39q0kIvI {"","()} FMicrosoft Equation 3.0 DS EqObjInfo Equation Native  L_1043564201F6161Ole CompObjfObjInfoEquation Native L_1043564410 F%>1%>1uation Equation.39q0 II {"","[]} FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObjfObjInfoEquation Native dkIvI {("","2)*"("2,2)*"(2,")} FMicrosoft Equation 3.0 DS Equation Equation.39q$kIvI  2n"8 _1043564692F?D1?D1Ole CompObjfObjInfoEquation Native @_1043564710"FZJ1K1Ole CompObj f FMicrosoft Equation 3.0 DS Equation Equation.39q8II m(t)= 9"3t B FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo!"Equation Native #T_1043564741$F0Q10Q1Ole %CompObj#%&fObjInfo&(Equation Native )`_1043564758;)FPY1PY1DDII u(x)=x"52x+4O FMicrosoft Equation 3.0 DS Equation Equation.39q@II a(r)=r+Ole +CompObj(*,fObjInfo+.Equation Native /\1r"1 FMicrosoft Equation 3.0 DS Equation Equation.39qTvII e(x)=1.35(3.66) xu_1043564791.Fж_1ж_1Ole 1CompObj-/2fObjInfo04Equation Native 5p_1043564810,63FWg1Wg1Ole 7CompObj248f FMicrosoft Equation 3.0 DS Equation Equation.39qHI@mI y(c)=2c 2 +3c FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo5:Equation Native ;d_10435648438Fprm1prm1Ole =CompObj79>fObjInfo:@Equation Native Ah_10435649051E=Fu1u1L|II q(w)=w+4w 2 +1 FMicrosoft Equation 3.0 DS Equation Equation.39q@I@mI f(x)=xOle CCompObj<>DfObjInfo?FEquation Native G\ x+3  FMicrosoft Equation 3.0 DS Equation Equation.39q<8II h(r)=e 1r_1043564953BF.{1.{1Ole ICompObjACJfObjInfoDLEquation Native MX_1043564974@OGFH10ς1Ole OCompObjFHPf FMicrosoft Equation 3.0 DS Equation Equation.39qLJI t(v)= v 2 +2v"8  FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoIREquation Native Sh_1043565056LFPp1Pp1Ole UCompObjKMVfObjInfoNXEquation Native Y\_1043565097JTQFЊ1p1@8II n(t)= t1+t  FMicrosoft Equation 3.0 DS Equation Equation.39qdJ*J x(y)=y Ole [CompObjPR\fObjInfoS^Equation Native _4 +2y+ y  +1y FMicrosoft Equation 3.0 DS Equation Equation.39q,kIvI  t1+t e"0Oh+'0d_1043565931VFS1S1Ole aCompObjUWbfObjInfoXdEquation Native eH1TableU SummaryInformation([gDocumentSummaryInformation8n4lB  S A? 2f}Ur#u=`!mf}Ur#R `;xQ1K`}wiӦ ZAJi uqrqWvplĀsApݟ`fwݽw98-V(f &8eVÔΓ$ٺ5Z_d/}s&TO{a(/N*4LMIf˒x-g;Q}?xOroUcYɝ4k2\"MS'xpNt\=Xv﨨x#MGѿ%J)=Zsu>){Pb;_N[ebnt'qiqM5DdhB   S A ? 2hsIB{{l`!shsIB{ێ` @|Axcdd``vgd``beV dX,XĐ Ɂ)[Rcgb 77FnĒʂT ~3);a#&br<' 9E\ Pk#O&-`d3&3tdFI9A4ZP y=|,;o _e5$V~cSU o1bϘP 27)? WH02pA )J]@B+KRs!aF`u\^8DdlB   S A ?  2$IV:-><~`!v$IV:-><@ PVDxR=KP=&I,8H `'A[g%`l/`|jEK9w-2LCVl%rf"gYUeW]MD ""?NvLT4nl8oWȪ ݫ M|sHyĽh7:Ixͻ<gn70>[[bT(RNǤh. bɂU*Uf鑏KRi"lQ }uT8vo tjjTub5^?NQ>]X@DdlB   S A ?  2Zp@'.bԐ-`!~Zp@'.bԐ-@ LxRJBQ=3WMMH @\DJr~ |i(h骍iOk!ԮMf^.cpΝ3w!*ГȰf & qV٣5""rVFyeE`!"Z >VFyeE` d`xRJAf.bH!`@)> I N<I@+X v)| lUs 7.!TDLd2iLLfɱyTWk:Z`ݳ5S^L*g YNVӓhkj {[s,{IJܿ-~mnI-os'.2o~2Sb'q މgucwpy!neyA;H)*”Pz%xӿ'jrRspn'pi@ Q极y7\i$~dT+DdB   S A ?  21Lƣ#lqm`!i1Lƣ#l̖` :7xJ@MƂA+HOBуBx(b{[|7G<+ggRnef&`H._6$H;ykg6a*Nh3bNB^&v9 1 fHŪEjсѪ>˽ӡ3EzP^pPh:{ ݷ+|M#&=Z'7 M}ލ:wXf<lEم ނlO!;DuoK>'%qN0I'[Ⱥ gH!MQ (Jc؍'?WjxDd B  S A? 23J,9L=OSA~F"`!3J,9L=OSA~> jdxRKP%mMS Th7;؎P0ЂfNnnछB )߯R}pww߻w YdIq˒LnĘ0HFجTiM<56bcmnFGHIZ[ ) A Y q  * B Z [  3 A O ] o A >23;<M}@ABE000000 0 0 0 000000 0 0 0000 0 0000000000 0 0 0 0 0 0 0 0 0 0 00000 0  0  0  0  0000000000000 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0000 0 0 0 0 0 0 0 0 0 0 00000; m VC H <C CI]_"68BVX   % ' ) = ? A U W Y m o q    & ( * > @ B V X C:::::::::::::::::I`cj"9:=>ABY\^_b  & ( ) @ V X n p q   ' ) * A W Y [ ` E13T[6atvAC \sv Z 3 6 A D O R ] `  & o r A D >AMPDF} E:::::::::3333333333333333333333333333 Math.C:\My Documents\Domain and Range Worksheet.docMathCC:\WINDOWS\TEMP\AutoRecovery save of Domain and Range Worksheet.asdMathC:\WINDOWS\Desktop\domain.docMathC:\stefi\MAT170\domain.docela C:\My Web Page\Fall01\domain.doc6_p2S+txy6Hzpb60^`0o(.0^`0o(.0^`0o(.hh^h`o(-zpb6_xy6S+E@h&C@@UnknownGz Times New Roman5Symbol3& z Arial"1h UF UF!!02Domain and Range WorksheetMathelaCompObjsj