UNIVERSIDAD DE TARAPACFACULTAD DE CIENCIASDEPARTAMENTO DEMATEMTICAALGEBRAPROFESOR : FABIANSANTIAGOFUNCIONES REALESJUNIO 2015GRAFICAS DE FUNCIONESMuchas !"ac#$%!s a"&!'a#cas s$% ()c#"!s *! !%+!%*!, s# -$*!.$s /!u%a!-!s!%+ac#0%/#sua" *!!""as1 U%a&)2ca!%u%a#.a&!%3u!.u!s+a "a !"ac#0% !%+! *$s $ .)s /a#a'"!s !% u%a !cuac#0%1 A%+!s *!a-!%*! c$.$ c$%s+u# u%a &)2ca, *!'! c$%$c! !" s#s+!.a *!c$$*!%a*as ca+!s#a%$1E" s#s+!.a *! c$$*!%a*as ca+!s#a%$ 4$ +a.'#5% c$%$c#*$ c$.$!c+a%&u"a6, %$.'a*$ !% h$%$ *!" .a+!.)+#c$ 7 2"0s$($ (a%c5s R!%5D!sca+!s, c$%s#s+! !% *$s !8!s 4$ !c+as %u.5#cas6 !% u% -"a%$,*#'u8a*as *! ($.a -!-!%*#cu"au%a *!"a$+a1 O's5/!s! c0.$ "$s*$s !8!s *!+!.#%a% cua*a%+!s, !+#3u!+a*$s c$% %u.!a"!s $.a%$s I,II, II 7 IV1E" !8!h$#9$%+a" s!*!%$.#%a!8! X $!8!*!"asa'sc#sas1 E" !8!/!+#ca" *!%$.#%a*$ !8! Y $ !8! *! "as $*!%a*as1 E" -u%+$ *!#%+!s!cc#0% *! "$s !8!s s! ""a.a $#&!%1 I%#c#a%*$ !% !" $#&!% 7.$/#5%*$s! hac#a "a *!!cha, "$s %:.!$s c!c!%, .$/#5%*$s! hac#a "a#93u#!*a, "$s%:.!$s*!c!c!%1 M#s.as#+uac#0%s# !" .$/#.#!%+$!s*!s*!!" c!%+$hac#aa#'a7hac#aa'a8$1 U%-a4-a!8a6$*!%a*$(x, y)s! u+#"#9a -aa *a "as *$s c$$*!%a*as *! u% -u%+$1 E" $'8!+#/$ !s '$s3u!8a "a &)2ca *! a"&u%as (u%c#$%!s c$%h!a.#!%+as 3u!*!c#!+a.a%!as$%;"#.#+a*asac+#+u*1T!%#!%*$ -!s!%+! "$ a%+!#$, !a"#9a!.$s u% !s+u*#$ !s-a"*a*$ c$% !"D$.#%#$7R!c$#*$*!(u%c#$%!s, s#.!+=a7c$+!s!%"$s!8!s1 E%c#!+as $cas#$%!s %$ !s %!c!sa#$ $'+!%! +$*$s !s+$s !"!.!%+$s -aa-$*! !a"#9a !" '$s3u!8$ *! "a &)2ca, -!$ s# s!/#)% *! a7u*a -aac$%$c! caac+!=s+#cas *! !""a1a.- DominioE" D$.#%#$ *! u% (u%c#0% !a" *! /a#a'"! !a" $ +a.'#5% c$%$c#*$ c$.$Domf , !s !" c$%8u%+$ *! /a"$!s #%*!-!%*#!%+!s *!x -aa "$s 3u!!s+) *!2%#*a "a (u%c#0%, $ +a.'#5% c$%8u%+$ *! "as -!#.a&!%!s1 Es+$s/a"$!s s! !%cu!%+a !% !" !8! *! "as a'sc#sas1Domf ={ x R/ f ( x) R}S!&:% "$ !s+u*#a*$s !% !" a-u%+! a%+!#$, !" *$.#%#$ *! a"&u%as(u%c#$%!s !s+) *a*$ c$.$ s#&u!:Funcin Domf EjemploLinealf (x)=ax+b R f ( x)=2 x+3Domf : RRacionalf ( x)= p(x)q( x)Dom pDomq {q( x)=0}f ( x)=2 x1x1Domf =R{1}Exponencialf (x)=ep(x)Rf (x)=ex1Logartmicaf ( x)=loga( p(x)) p(x)>0 f (x)=log2( 2 x1)Domf =(12, )Irracionalf ( x)=np(x)4a6 S#n!s -a:p( x)04a6 f (x)=1xDomf : ( , 1]4'6 S#n!s #.-a:R4'6 f (x)=31xDomf : RValor Absolutof ( x)=|p(x)|R f (x)=|x2|Domf : Rb.- RecorridoC$%8u%+$ ($.a*$ -$ "as #.)&!%!s1 Es+$s /a"$!s s! !%cu!%+a% !% !"!8!*!"as$*!%a*as1 La($.a*!ca"cu"a!" R!c$#*$, !ss#.#"aa"*$.#%#$, *$%*!"a(u%c#0%!s#&ua"a*aa y , *!s-!8a%*$ x , -aa-$s+!#$.!%+!a%a"#9a "$s /a"$!s 3u!-u!*!%+$.a !s+a/a#a'"!*!-!%*#!%+!1Funcin Despeje de x Rec f f (x)=1x2y=1x2x2=1yx=1y1y 0Recf : (, 1](, 1]f (x)=ex1y=ex1/ lnln y=ln ex1/ ln e=1ln y=x1x=ln y+1y>0( 0 , )f ( x)=2 x1x1y=2x1x1y ( x1)=2x1xyy=2x1xy2 x=y1x( y2)=y1x= y1y2R{2}f (x)=ln(3 x1)y=ln(3 x1)Propiedaddeunlogartmo: logab=c ac=bconiderandoquelogartmonatural eunlogartmodebae eey=3 x1x=ey+13Rf (x)=|x2| y=|x2|+R0c.- !imetraLas &)2cas *! "as (u%c#$%!s -u!*!% -!s!%+a /a#$s +#-$s *! s#.!+=as,au%3u! s$"$ !s+u*#a!.$s *$s:'116 S#.!+=a c$%!s-!c+$*!" !8!*!$*!%a*as $!8!?, s# -aacua"3u#! -u%+$x *! su *$.#%#$ s! cu.-"! 3u!f (x)=f (x) 1 A "as(u%c#$%!scu7a&)2ca-!s!%+!!s+as#.!+=as!"!s""a.afuncionespares.'126 S#.!+=a c$% !s-!c+$ a" $#&!% *! c$$*!%a*as, s# -aa cua"3u#!-u%+$x*! su *$.#%#$ s! cu.-"! 3u! f (x)=f (x) 1 A "as (u%c#$%!scu7a &)2ca -!s!%+a !s+a s#.!+=a s! "!s ""a.a funciones impares.S# sa'!.$s 3u!u%a(u%c#0%-!s!%+acua"3u#!a*!!s+$s +#-$s *!s#.!+=a 'as+a c$% c$%s+u# su &)2ca !% "$s -u%+$s !% 3u!x0 1 P$s#.!+=a, -$*!.$s *#'u8a !" !s+$ *! "a &)2ca1E8!.-"$s c11: f ( x)=x2,f (x)=(x)2=x2=f ( x) ,(u%c#0% -a1E8!.-"$ c12:f (x)=x3f (x)=(x)3=x3=f (x) , (u%c#0% #.-a1Ms adelante analizaremos las grficas.E% cus$s.as a/a%9a*$ss!!s+u*#a%c$% $+ash!a.#!%+as!8!s*!s#.!+=as1d. "untos de #orte con los ejes#orteejeabscisas$Paa!%c$%+a !s+!4$s6 c$+!4s6 !s$"/!.$s "a!cuac#0%f ( x)=0 , *$%*! "a s!&u%*a c$$*!%a*a s! a%u"a, $'+!%#!%*$"a -!#.a&!% !% "a cua" "a #.a&!% !s c!$ 406, s# !s 3u! !>#s+!4%61E8!.-"$ *11: f ( x)=2 x3f (x)=0 2 x3=0x=32"u!&$ !" c$+! !% !" !8! *! "as a'sc#sas !s(32, 0)E8!.-"$ *12: f (x)=1x2f ( x)=0 1x2=0 1=x"u!&$ "$s c$+!s !% !" !8! *! "as a'sc#sas s!)% (1,0) y ( 1,0)E8!.-"$ *1@: f (x)=2 x1x1f (x)=0 2x1x1 =0 x=12"u!&$ !" c$+! !% !" !8! *! "as a'sc#sas !s(12, 0)E8!.-"$ *1A: f ( x)=ex1f (x)=0 ex1=0, !s+$ %u%ca suc!*!)"u!&$ !s+a (u%c#0% %$ +#!%! c$+! !% !" !8! *! "as a'sc#sas1E8!.-"$ *15:f (x)=ln(3 x1)f (x)=0 ln(3x1)=03 x1=1x=23"u!&$ !" c$+! !% !" !8! *! "as a'sc#sas !s (23, 0)E8!.-"$ *1B: f (x)=|x2|f (x)=0 |x2|=0x=2"u!&$ !" c$+! !% !" !8! *! "as a'sc#sas !s( 2,0)#orteeje%rdenadas$As= c$.$"as c$$*!%a*as !%!" !8!*!"asa'sc#sas +#!%! "a ($.a *!(a, 0) , $cu! a"&$ s#.#"a !% "$s -u%+$s *!c$$*!%a*as !% !" !8! *! "as $*!%a*as, c$% "a ($.a *!(0, b) , !% "acua" s! a%u"a "a -!#.a&!% *! "a (u%c#0%1 Paa !%c$%+a !s+ac$$*!%a*a, 'as+a c$% hac! c!$ "a -!#.a&!%1E8!.-"$ : f (x)=2 x3x=0y=2 x3y=3"u!&$ !" c$+! !% !" !8! *! "as $*!%a*as !s ( 0,3)E8!.-"$ 10: f ( x)=1x2x=0y=102=1y=1"u!&$ !" c$+! !% !" !8! *! "as $*!%a*as s!) ( 0,1)E8!.-"$ 11: f (x)=2 x1x1x=0y=2(0)1(0)1 y=1"u!&$ !" c$+! !% !" !8! *! "as $*!%a*as !s ( 0,1)E8!.-"$ 12: f ( x)=ex1x=0y=e(0)1y=e1

"u!&$ !" c$+! !% !" !8! *! "as $*!%a*as !s ( 0, e1)E8!.-"$ 1@:f ( x)=ln(3 x1)x=0y=ln(3(0)1)y=ln(1) %:.!$ 3u! %$ !s !a", "u!&$, "a (u%c#0%%$ +!%*) c$+! !% !" !8! *! "as $*!%a*as1E8!.-"$ 1A: f (x)=|x2|x=0y=|02|y=2"u!&$ !" c$+! !% !" !8! *! "as $*!%a*as !s( 0,2)e. &abla de 'aloresR!c$*a%*$ !" c$%c!-+$ *! (u%c#0% !a", 3u! !s u%a !"ac#0% !%+! *$s%:.!$s !a"!s, "a -#.!a 4-!#.a&!%6 ""a.a*a /a#a'"!#%*!-!%*#!%+!, "! c$!s-$%*! u% :%#c$ /a"$ 4#.a&!%6 ""a.a*a /a#a'"!*!-!%*#!%+! $ (u%c#0%1 La +a'"a *! /a"$!s !s u% a!&"$ 3u! s! -u!*!!sc#'# h$#9$%+a" $ /!+#ca" 7 *$%*! !% !" -#.! !%&"0% $ c$"u.%a, s!u'#ca%"$s/a"$!s*!"a/a#a'"!#%*!-!%*#!%+!4 x 6, 3u!-u!*!s!cua"!u# /a"$ 3u! s! !%cu!%+! !% !" Domf , 3u! -$ "$ &!%!a" s$%/a"$!s()c#"!s*!!!.-"a9a, 7!%!" s!&u%*$!%&"0%$c$"u.%as!u'#ca% !" /a"$ !su"+a%+! a" !/a"ua "a -!#.a&!%1 P!&u%+a #.-$+a%+!Ccu)%+$s /a"$!s *!'!.$s !!.-"a9aD, !s-u!s+a: "$s 3u! s!a%%!c!sa#$s -aa !%+!%*! "a +!%*!%c#a *! "a &)2ca1E8!.-"$ 15:f ( x)=2 x3"reimagen ( ) * -(Imagen 2(1)3=1 2(2)3=1 2(3)3=3 2(1)3=5E8!.-"$ 1B:f (x)=1x2"reimagen-) -* ) *Imagen1(2)2=3 1(3)2=8 1(2)2=3 1(3)2=8E8!.-"$ 1E:f (x)=2 x1x1"reimagen-( -) ) *Imagen 2(1)1(1)1 =322(2)1(2)1 =532(2)1(2)1 =32(3)1(3)1 =52E8!.-"$ 1F:f (x)=ex1"reimagen-( -) ( )Imagene(1)1=e2e21=e3e(1)1=1 e21=e1E8!.-"$ 1G:f (x)=ln(3 x1)"reimagen( ) * +Imagen ln(3(1)1)=ln 2=0.69 ln(3(2)1)=ln 5=1.61 ln(3(3)1)=ln 8=2.08 ln(3(4)1)=ln 11=2.40E8!.-"$ 20:f (x)=|x2|"reimagen-( -) ( *Imagen |(1)2|=3 |22|=4 |12|=1 |32|=1,R-FI#A % .%!/0E1% DE 02 F02#I32EjemplosFuncin D$.#%#$R!c$#*$S#.!+=a C$+!s Ta'"aLineal$f (x)=2 x3R R N$ +#!%!A'sc#sa:(32, 0)O*!%a*a:( 0,3)x y( -() (* *-( -4#uadr5ticaf (x)=1x2R (, 1] E8! ?A'sc#sa:(1,0)y( 1,0)O*!%a*a:( 0,1)x y-) -*-* -6) -** -6Racional$f ( x)=2 x1x1R{1} R{2}T#!%!,s!)%!s+u*#a*as!% cus$s-$s+!#$!sA'sc#sa:(12, 0)O*!%a*a:(0,1)x y-( *7)-) 47*) ** 47)Exponencial$f (x)=ex1R ( 0 , ) N$ +#!%!A'sc#sa:N$ +#!%!O*!%a*a:( 0, e1)x y-(e2-)e3( ()e1Logartmica$f (x)=ln(3 x1) (13, )R N$ +#!%!A'sc#sa:(23, 0)O*!%a*a:N$ +#!%!x y( 8.9:) (.9(* ).86+ ).+8ValorAbsolutof (x)=|x2|R+Recf : R0x=2A'sc#sa:( 2,0)O*!%a*a:( 0,2)x y-( *-) +( (* (Ejercicios propuestosD!+!.#%! !" *$.#%#$, !c$#*$, C$+!s !% "$s !8!s, !8!s *! s#.!+=a, 7 +a'"a *! /a"$!s, -aa "u!&$ '$s3u!8a "a &)2ca, !% "as s#&u#!%+!s (u%c#$%!s:4a6 L#%!a"!s:416 f (x)=2 x1 426 f (x)=3 x+1(3)f (x)=23 x24A6 f (x)=23 x 456 f (x)=2(1x) 4B6 f (x)=24'6 Cua*)+#cas:416 f (x)=4x2 426 f ( x)=1+x2 4@6f (x)=4 x294A6 f (x)=2 x2+5x+2 456 f (x)=x2+x+14B6f (x)=4 x2+4 x+14c6Rac#$%a":416 f (x)= x1x2426 f (x)=3 x12 x+1 4@6 f (x)=3 x2x+34A6 f (x)=x2 x1 456 f (x)=3x+34B6 f (x)=3 x+2x4*6E>-$%!%c#a":416 f (x)=ex426 f (x)=ex2 4@6 f (x)=2ex +14A6 f (x)=e2x 456 f (x)=e1x 4B6 f ( x)=e2x4!6 L$&a=+.#ca:416 f (x)=ln( x) 426 f ( x)=ln( x1)4@6 f (x)=ln( 2 x)4A6 f (x)=log( x2)456 f (x)=log( 1x)4A6f (x)=log( x21) 4(6 Va"$ A's$"u+$:416 f (x)=|x+2|426 f (x)=|3x+1| 4@6 f (x)=|1x|416 f (x)=|2x+1| 416 f (x)=|x2| 416 f ( x)=|32 x|