{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "2D Comment" -1 18 "Times" 0 1 0 0 0 0 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 0 1 0 0 255 1 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "Text" -1 201 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "_cstyle1" -1 204 "Times" 1 18 0 0 0 0 0 1 1 2 2 2 0 0 0 1 } {CSTYLE "_cstyle2" -1 205 "Times" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle3" -1 206 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }{CSTYLE "_cstyle4" -1 207 "Times" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle5" -1 208 "Times" 0 1 0 0 255 1 0 0 2 2 2 2 0 0 0 1 } {PSTYLE "_pstyle1" -1 200 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 2 2 2 0 0 0 1 }3 0 0 -1 12 12 1 0 1 0 2 2 -1 1 }{PSTYLE "_pstyle2" -1 201 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "_pstyle3" -1 202 1 {CSTYLE "" -1 -1 " Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "_pstyle4" -1 203 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }3 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "_pstyle5 " -1 204 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } 0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "_pstyle6" -1 205 1 {CSTYLE " " -1 -1 "Times" 0 1 0 0 255 1 0 0 2 2 2 2 1 0 0 1 }3 3 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle7" -1 206 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 200 "" 0 "" {TEXT 204 18 "Rational Functions" } {TEXT 204 0 "" }}{PARA 201 "" 0 "" {TEXT 205 113 "If f(x) =P(x)/Q(x) , where P(x) and Q(x) are polynomial functions, then f(x) is said to be a rational function" }{TEXT 205 0 "" }}{PARA 201 "" 0 "" {TEXT 205 23 "Example 1: If f(x) = " }{XPPEDIT 18 0 "2/(x-2);" "6#*&\"\"# \"\"\",&%\"xGF%F$!\"\"F(" }{TEXT 205 0 "" }}{PARA 201 "" 0 "" {TEXT 205 54 "(b) find the vertical and horizontal aysmptotes of f;" } {TEXT 205 0 "" }}{PARA 201 "" 0 "" {TEXT 205 22 "(c) graph function f ." }}{PARA 201 "" 0 "" {TEXT 205 27 "(d) solve f(x)>0 or f(x)<0." }}} {EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 14 "f:=x->2/(x-2);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 32 "plot(f(x),x=-1 0..30,y=-10..10); " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 205 94 "Notice that we have (a) a vertical aysmptote at x = 2. \+ (b) a horizontal aysmptote at y = 0. " }{TEXT 205 0 "" }}{PARA 201 " " 0 "" {TEXT 205 32 "Note the following values of f." }{TEXT 205 0 " " }}{PARA 201 "" 0 "" {TEXT 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 206 11 "f(1.9999); " }{TEXT 205 155 "(this is to say that w hen x is very close to 2 from the left, the y value is -large num ber; that is the reason why x = 2 is the verticcal asymptote.)" } {TEXT 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 206 11 "f(2.00 001);" }{TEXT 205 159 " (this is to say that when x is very close to 2 from the right, the y value is +large number ; that is the reas on why x = 2 is the verticcal asymptote.)" }{TEXT 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 206 17 "evalf(f(10000)); " }{TEXT 205 146 "(this is to say when x is a positive large number, the y value is very close to zero; that is the reason why y = is the horizontal \+ aysmptote.)" }{TEXT 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 206 18 "evalf(f(-10000)); " }{TEXT 205 145 "(this is to say when x i s a negative large number, the y value is very close to zero; that is \+ the reason why y = is the horizontal aysmptote.)" }{TEXT 205 0 "" }} {PARA 201 "" 0 "" {TEXT 205 0 "" }}{PARA 201 "" 0 "" {TEXT 205 0 "" }} {PARA 201 "" 0 "" {TEXT 205 22 "Exmaple 2. If f(x) = " }{XPPEDIT 18 0 "(x^2-2)/((x-1)*(x-2));" "6#*&,&*$%\"xG\"\"#\"\"\"F'!\"\"F(*&,&F&F(F (F)F(,&F&F(F'F)F(F)" }{TEXT 205 0 "" }}{PARA 201 "" 0 "" {TEXT 205 54 "(a) find the horizontal and vertical aysmptotes of f;" }{TEXT 205 0 "" }}{PARA 201 "" 0 "" {TEXT 205 22 "(b) graph function f." }}{PARA 201 "" 0 "" {TEXT 205 64 "(c) solve f(x)>0 or f(x)<0. Note that x^2=(x +sqrt(2)*(x-sqrt(2)." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 28 "f :=x->(x^2-2)/((x-1)*(x-2));" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 43 "plot(f(x),x=-2..5,y=-10..10, thickness=2); " }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }}{PARA 203 "" 0 "" {TEXT 207 0 "" }}{PARA 201 "" 0 "" {TEXT 205 26 "Example 3: If f(x) = " } {XPPEDIT 18 0 "2/((x-1)*(x-2));" "6#*&\"\"#\"\"\"*&,&%\"xGF%F%!\"\"F%, &F(F%F$F)F%F)" }{TEXT 205 0 "" }}{PARA 201 "" 0 "" {TEXT 205 0 "" }} {PARA 201 "" 0 "" {TEXT 205 54 "(a) find the horizontal and vertical a ysmptotes of f;" }{TEXT 205 0 "" }}{PARA 201 "" 0 "" {TEXT 205 22 "(b ) graph function f." }}{PARA 201 "" 0 "" {TEXT 205 4 "(c) " }{TEXT 205 23 "solve f(x)>0 or f(x)<0." }{TEXT 205 0 "" }}}{EXCHG {PARA 202 " > " 0 "" {MPLTEXT 1 0 22 "f:=x->2/((x-1)*(x-2));" }{MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 49 "plot(f(x), x = -1..3, y \+ = -50..50, thickness=3); " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 205 0 "" }}{PARA 201 "" 0 "" {TEXT 205 0 "" }}{PARA 201 "" 0 "" {TEXT 205 122 "** Exercise: Find two rational functions f which s atisfy all the following conditions and use Maple to check your answer s." }{TEXT 205 0 "" }}{PARA 201 "" 0 "" {TEXT 205 57 "(1) the vertical asymptotes of f are at x=1 and x= -3," }{TEXT 205 0 "" }{TEXT 205 49 "\n(2) the horizontal asymptote of f is at y = 2," }{TEXT 205 0 " " }}{PARA 201 "" 0 "" {TEXT 205 12 "(3) f(0)=3." }{TEXT 205 0 "" }} {PARA 201 "" 0 "" {TEXT 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 206 18 "evalf(f(-10000)); " }{TEXT 205 145 "(this is to say when x is a negative large number, the y value is very close to zer o; that is the reason why y = is the horizontal aysmptote.)" }{TEXT 205 0 "" }}{PARA 201 "" 0 "" {TEXT 205 0 "" }}{PARA 201 "" 0 "" {TEXT 205 0 "" }}{PARA 204 "" 0 "" {TEXT 201 47 "Example 3: If f(x) = ( x+2)^2/((x-1)*(x-2))" }}{PARA 201 "" 0 "" {TEXT 205 40 "(b) find the h orizontal aysmptote of f;" }{TEXT 205 0 "" }}{PARA 201 "" 0 "" {TEXT 205 22 "(c) graph function f." }{TEXT 205 0 "" }}}{EXCHG {PARA 204 " " 0 "" {XPPEDIT 2 0 "%?" 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