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2 2 2 1 1 1 1 } 3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item " -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 44 "More on exponential and logarithm functions " }}{PARA 3 "" 0 "" {TEXT 263 44 " .. equations, graphs and inverse functions " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Na naimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 23.3.2007 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 68 "Solving equations which involve e xponential and logarithm functions " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 \+ " }}{PARA 0 "" 0 "" {TEXT 287 8 "Question" }{TEXT -1 21 ": Solve the e quation " }{XPPEDIT 18 0 "exp(4-3*x)=7" "6#/-%$expG6#,&\"\"%\"\"\"*&\" \"$F)%\"xGF)!\"\"\"\"(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 288 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 13 "The equation " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(4-3*x)=7" "6#/-%$expG6#,&\"\"%\"\"\"*&\"\"$F)% \"xGF)!\"\"\"\"(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "is eq uivalent to " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "4-3*x =log[exp(1)]*7" "6#/,&\"\"%\"\"\"*&\"\"$F&%\"xGF&!\"\"*&&%$logG6#-%$ex pG6#F&F&\"\"(F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "which \+ in turn is equivalent to " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "3*x=4-log[exp(1)]*7" "6#/*&\"\"$\"\"\"%\"xGF&,&\"\"%F&* &&%$logG6#-%$expG6#F&F&\"\"(F&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 11 "This gives " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x= (4-log[exp(1)]*7)/3" "6#/%\"xG*&,&\"\"%\"\"\"*&&%$lo gG6#-%$expG6#F(F(\"\"(F(!\"\"F(\"\"$F1" }{TEXT -1 1 " " }{TEXT 286 1 " ~" }{TEXT -1 15 " 0.6846966170. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "exp(4-3*x)=7;\nsolve(%,x);\nevalf(evalf[14](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#,&\"\"%\"\"\"*&\"\"$F)%\"xGF)!\"\"\"\"(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"\"%\"\"$\"\"\"*&#F'F&F'-%#lnG6# \"\"(F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+qh'p%o!#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 289 8 "Question" }{TEXT -1 21 ": Solve the equation " }{XPPEDIT 18 0 "log[4](x+3)+log[4](x-3) \+ = 2" "6#/,&-&%$logG6#\"\"%6#,&%\"xG\"\"\"\"\"$F-F--&F'6#F)6#,&F,F-F.! \"\"F-\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 290 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 13 "The equation " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "log[4](x+3)+log[4](x-3) = 2" "6#/,&-&%$logG6#\"\"%6#,&% \"xG\"\"\"\"\"$F-F--&F'6#F)6#,&F,F-F.!\"\"F-\"\"#" }{TEXT -1 12 " ---- --- (i)" }}{PARA 0 "" 0 "" {TEXT -1 14 "implies that " }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log[4](x^2-9)=2" "6#/-&%$logG6# \"\"%6#,&*$%\"xG\"\"#\"\"\"\"\"*!\"\"F-" }{TEXT -1 15 " ------- (ii), \+ " }}{PARA 0 "" 0 "" {TEXT -1 23 "which is equivalent to " }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "x^2-9=4^2" "6#/,&*$%\"xG\"\"#\"\"\"\"\"*!\"\"*$ \"\"%F'" }{TEXT -1 2 ", " }}{PARA 259 "" 0 "" {TEXT -1 9 "that is, " } }{PARA 257 "" 0 "" {XPPEDIT 18 0 "x^2=25" "6#/*$%\"xG\"\"#\"#D" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "or " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "x=``" "6#/%\"xG%!G" }{TEXT 291 1 "+" }{TEXT -1 4 " 5. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Both of these values for " }{TEXT 292 1 "x" }{TEXT -1 141 " provide solutions for equation (ii). However, since equation (ii ) is not equivalent to equation (i), we should check these solutions i n (i)." }}{PARA 0 "" 0 "" {TEXT -1 10 "The value " }{XPPEDIT 18 0 "x=5 " "6#/%\"xG\"\"&" }{TEXT -1 29 " satisfies equation (i), but " } {XPPEDIT 18 0 "x=-5" "6#/%\"xG,$\"\"&!\"\"" }{TEXT -1 1 " " }{TEXT 261 8 "does not" }{TEXT -1 7 " since " }{XPPEDIT 18 0 "log[4](-2)" "6# -&%$logG6#\"\"%6#,$\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "log [4](-8)" "6#-&%$logG6#\"\"%6#,$\"\")!\"\"" }{TEXT -1 33 " do not exist (as real numbers). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 293 10 "Conclusion" }{TEXT -1 45 ": Equation (i) has the singl e real solution: " }{XPPEDIT 18 0 "x=5" "6#/%\"xG\"\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "'log[4](x+3)+log[4](x-3)'=2;\nsolve(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-&%$logG6#\"\"%6#,&%\"xG\"\"\"\"\"$F-F--F &6#,&F,F-F.!\"\"F-\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 297 8 "Question" }{TEXT -1 21 ": Solve the equation " }{XPPEDIT 18 0 "2*ln*x = ln(2*x+1)+1;" " 6#/*(\"\"#\"\"\"%#lnGF&%\"xGF&,&-F'6#,&*&F%F&F(F&F&F&F&F&F&F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 298 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 40 "The equat ion can be written in the form " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "2*ln(x)-ln(2*x+1) = 1;" "6#/,&*&\"\"#\"\"\"-%#lnG6#%\"x GF'F'-F)6#,&*&F&F'F+F'F'F'F'!\"\"F'" }{TEXT -1 13 " ------- (i)," }} {PARA 0 "" 0 "" {TEXT -1 19 "which implies that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln(x^2/(2*x+1)) = 1;" "6#/-%#lnG6#*&%\" xG\"\"#,&*&F)\"\"\"F(F,F,F,F,!\"\"F," }{TEXT -1 15 " ------- (ii). " } }{PARA 0 "" 0 "" {TEXT -1 35 "The equation (ii) is equivalent to " }} {PARA 257 "" 0 "" {XPPEDIT 18 0 "x^2/(2*x+1) = exp(1);" "6#/*&%\"xG\" \"#,&*&F&\"\"\"F%F)F)F)F)!\"\"-%$expG6#F)" }{TEXT -1 2 ", " }}{PARA 259 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 257 "" 0 "" {XPPEDIT 18 0 " x^2 = 2*exp(1)*x+exp(1);" "6#/*$%\"xG\"\"#,&*(F&\"\"\"-%$expG6#F)F)F%F )F)-F+6#F)F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2-2*exp(1)*x = exp( 1);" "6#/,&*$%\"xG\"\"#\"\"\"*(F'F(-%$expG6#F(F(F&F(!\"\"-F+6#F(" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "This quadratic equation may be solved by completing the s quare to give the equation: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x^2-2*exp(1)+exp(2)=exp(1)+exp(2)" "6#/,(*$%\"xG\"\"#\" \"\"*&F'F(-%$expG6#F(F(!\"\"-F+6#F'F(,&-F+6#F(F(-F+6#F'F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 24 "which is equivalent to: " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x-exp(1))^2=exp(1)+e xp(2)" "6#/*$,&%\"xG\"\"\"-%$expG6#F'!\"\"\"\"#,&-F)6#F'F'-F)6#F,F'" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x-exp(1) =`` " "6#/,&%\"xG \"\"\"-%$expG6#F&!\"\"%!G" }{TEXT 300 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(exp(1)+exp(2));" "6#-%%sqrtG6#,&-%$expG6#\"\"\"F*-F(6#\"\"# F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=exp(1)" "6#/%\"xG-%$expG6 #\"\"\"" }{TEXT -1 1 " " }{TEXT 301 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(exp(1)+exp(2));" "6#-%%sqrtG6#,&-%$expG6#\"\"\"F*-F(6#\"\"# F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=exp(1)+sqrt(exp(1)+exp(2))" "6 #/%\"xG,&-%$expG6#\"\"\"F)-%%sqrtG6#,&-F'6#F)F)-F'6#\"\"#F)F)" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "x=exp(1)-sqrt(exp(1)+exp(2))" "6#/%\"xG,&- %$expG6#\"\"\"F)-%%sqrtG6#,&-F'6#F)F)-F'6#\"\"#F)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Both of these values for " }{TEXT 302 1 "x" }{TEXT -1 141 " provide soluti ons for equation (ii). However, since equation (ii) is not equivalent \+ to equation (i), we should check these solutions in (i)." }}{PARA 0 " " 0 "" {TEXT -1 19 "The negative value " }{XPPEDIT 18 0 "x=exp(1)-sqrt (exp(1)+exp(2))" "6#/%\"xG,&-%$expG6#\"\"\"F)-%%sqrtG6#,&-F'6#F)F)-F'6 #\"\"#F)!\"\"" }{TEXT -1 35 " cannot satisfy equation (i) since " } {XPPEDIT 18 0 "ln*x)" "6#*&%#lnG\"\"\"%\"xGF%" }{TEXT -1 49 " does exi st (as a real number) for this value of " }{TEXT 303 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 299 10 "Conclusion" }{TEXT -1 45 ": Equation (i) has the single real solution : " }{XPPEDIT 18 0 "x=exp(1)+sqrt(exp(1)+exp(2))" "6#/%\"xG,&-%$expG6# \"\"\"F)-%%sqrtG6#,&-F'6#F)F)-F'6#\"\"#F)F)" }{TEXT 304 1 "~" }{TEXT -1 14 " 5.897485805. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "2*ln(x)-ln(2*x+1)=1;\nsolve(%,x);\nevalf( evalf[14](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&\"\"#\"\"\"-%#l nG6#%\"xGF'F'-F)6#,&*&F&F'F+F'F'F'F'!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#\"\"\"F'*$,&*$)F$\"\"#F'F'F$F'#F'F,F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+0e[(*e!\"*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 261 4 "Note" }{TEXT -1 12 ": The value " }{XPPEDIT 18 0 " x=exp(1)+sqrt(exp(1)+exp(2))" "6#/%\"xG,&-%$expG6#\"\"\"F)-%%sqrtG6#,& -F'6#F)F)-F'6#\"\"#F)F)" }{TEXT 305 1 "~" }{TEXT -1 23 " 5.897485805 g ives the " }{TEXT 306 1 "x" }{TEXT -1 58 " coordinate of the point of \+ intersection of the graphs of " }{XPPEDIT 18 0 "f(x)=2*ln*x" "6#/-%\"f G6#%\"xG*(\"\"#\"\"\"%#lnGF*F'F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 " g(x)=ln(2*x+1)+1" "6#/-%\"gG6#%\"xG,&-%#lnG6#,&*&\"\"#\"\"\"F'F/F/F/F/ F/F/F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "In the following picture the graph of of " } {XPPEDIT 18 0 "f(x)=2*ln*x" "6#/-%\"fG6#%\"xG*(\"\"#\"\"\"%#lnGF*F'F* " }{TEXT -1 13 " is drawn in " }{TEXT 258 3 "red" }{TEXT -1 19 ", and \+ the graph of " }{XPPEDIT 18 0 "g(x)=ln(2*x+1)+1" "6#/-%\"gG6#%\"xG,&-% #lnG6#,&*&\"\"#\"\"\"F'F/F/F/F/F/F/F/" }{TEXT -1 13 " is drawn in " } {TEXT 256 4 "blue" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "f := x -> 2*ln(x):\n'f(x)'= f(x);\ng := x -> ln(2*x+1)+1:\n'g(x)'=g(x);\nplot([f(x),g(x)],x=-.7..8 ,y=-3..4.25,color=[red,blue]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"fG6#%\"xG,$*&\"\"#\"\"\"-%#lnGF&F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,&-%#lnG6#,&*&\"\"#\"\"\"F'F/F/F/F/F/F/F/" }} {PARA 13 "" 1 "" {GLPLOT2D 510 286 286 {PLOTDATA 2 "6&-%'CURVESG6$7fn7 $$\"3-+++]H%zj$!#?$!3y![ShPnK7\"!#;7$$\"3o*****\\tW^Z*F*$!3&y9P`]m\"=$ *!#<7$$\"32+++_YBJ:!#>$!3%R>Ayi\">e$)F37$$\"3')*****4$[&\\6#F7$!3'3A&F37$$\"3m+++3`-q'*F7$!3SP'>(=& yAn%F37$$\"3-+++u3Q*>\"!#=$!3023N:#f:C%F37$$\"3/+++hv8k;Fjn$!3KQMGqhb' e$F37$$\"38+++\\U*)G@Fjn$!3#3X')*fZ'R4$F37$$\"3u*****R8s1*HFjn$!38(zZP (Q<9CF37$$\"3/+++?+X_QFjn$!3LM*>lb^x!>F37$$\"3z*******4rZu%Fjn$!3S,tp/ Q3\"\\\"F37$$\"3c******z@4PcFjn$!3%G!priMVY6F37$$\"3]+++g;x#[(Fjn$!3Ak '=s`O'*z&Fjn7$$\"3f+++??`A$*Fjn$!32jAc/l,.9Fjn7$$\"3!*******o^\\@6F3$ \"3'ff-pMbKH#Fjn7$$\"31+++(fz\")G\"F3$\"3KC=_K6gk]Fjn7$$\"3/+++$yEeZ\" F3$\"3o=,,ssO%y(Fjn7$$\"3)******fYWUm\"F3$\"33_Ja#\\U(=5F37$$\"3)***** *p\")>e%=F3$\"3QZc6[q%eA\"F37$$\"3$)*****4j22,#F3$\"3QJfnIM(pR\"F37$$ \"36+++!>vn?#F3$\"3Kh'f'>X1$e\"F37$$\"3++++J,(GP#F3$\"3@10P'\\+#G_tv%)=F37$$\"3))*****Rw*4PFF3$\"3%)eU/fxz8?F37 $$\"3))*****zqPZ#HF3$\"3_S(o$3'4k9#F37$$\"3z*****HS8M5$F3$\"3#H4Rp?0]E #F37$$\"3!)*****puU)*G$F3$\"31M5uK&z;Q#F37$$\"39+++\"3V5Y$F3$\"3C#\\oU -SJ[#F37$$\"3#)*****4x0dk$F3$\"3/#fbx%**4(e#F37$$\"30+++1%=v$QF3$\"374 s]==l*o#F37$$\"3w******o=\\/SF3$\"3iCwi'RL[x#F37$$\"3K+++%yF[=%F3$\"3U yij/6$H'GF37$$\"3G+++IC8rVF3$\"3s)*RnDU/]HF37$$\"3s*****z<&R`XF3$\"3#R nK`HY<.$F37$$\"3&)******fPuHZF3$\"3Vz(\\`0Ux5$F37$$\"3#******Hz[b#\\F3 $\"3a$*Q$pRr))=$F37$$\"3'******RK)[,^F3$\"3m%*Ry]Y1fKF37$$\"3u*****fKT $*G&F3$\"3@u'[,X(QJLF37$$\"3k*****p1k&faF3$\"3y'3@2*yt%R$F37$$\"3!**** **\\[ick&F3$\"3ic!4r4v'3(F3$\"3EkLP\"o&H;RF37$$\"3e*****pp4*ysF3$\"3[o@'*pRF 37$$\"3!******pN!G^uF3$\"3#)f/Db#F37$$!38+++a(3Fu%Fjn$!39*4 \\_\\%)p'>F37$$!3A+++kg9\"p%Fjn$!3sC]IDZK%y\"F37$$!3w*****Ho?!)e%Fjn$! 3ScJyW+A'\\\"F37$$!3')*****>I&*[[%Fjn$!3#[\"f;hg$*Fjn7$$!3C+++yPRsS Fjn$!3KpvC%yee%oFjn7$$!3))*****f,Vh'QFjn$!3^%)*R[JG\"Q[Fjn7$$!33+++aA* )fOFjn$!3k15TU*yo;$Fjn7$$!3y*****H\\TOX$Fjn$!3Pn**ye4NN'f#)))F77$$!3u*****>GRe_#Fjn$\"3!fh%RAKjkHFjn7$$ !3'******f1#Fjn$\"3[R]#)z?7$o%Fjn7$$!3)******42P!)f\"Fjn$\"3C>![l%o 9\\hFjn7$$!3m+++]y%3k'F7$\"3[c!fx(z%\\d)Fjn7$F@$\"3ykyn2tc_5F37$Fhn$\" 3BU%)f::,:7F37$Fco$\"3!octcA=ZN\"F37$F]p$\"3oK6pYjDr:F37$Fgp$\"3SF^xU# 4\\v\"F37$F\\q$\"3O'G!*R^6\\\">F37$Faq$\"3=:s]]gR_?F37$Ffq$\"3MD1)z%e \\w@F37$F[r$\"3'=P;!)HXVF#F37$F`r$\"3g?\"=c6MTP#F37$Fer$\"3uIXm/&=_Y#F 37$Fjr$\"3%yz$\\B@yXDF37$F_s$\"3zJ:n==r8EF37$Fds$\"3IK#*3S^!*)o#F37$Fi s$\"3%ybjive%[FF37$F^t$\"3D:?w6ka8GF37$Fct$\"3/%4C\\(\\#y'GF37$Fht$\"3 (QxNx1sT#HF37$F]u$\"3kW^HW(G](HF37$Fbu$\"3j`nB2$F37 $F[w$\"3lelbOjZPKF37$F`w$\"37C1XAOZwKF37$Few$\"30u;U5v?8LF37$Fjw$\"3Ic EmBxn()p`$F37 $F]z$\"3%)\\m-JR`lNF37$Fbz$\"3%>z\"f+on#f$F37$Fgz$\"3hN[#=$\\K?OF37$F \\[l$\"3]>prb'\\ik$F37$Fa[l$\"3a,YMBT3sOF37$Ff[l$\"3U(>EJ*>1(p$F37$F[ \\l$\"3Q'RB%*ow%>PF37$F`\\l$\"3h!pbrOjXu$F37$Fe\\l$\"3+.QXZ,[mPF37$Fj \\l$\"33L?u=2L*y$F37$F_]l$\"3f$*R*G=B2\"QF37$Fd]l$\"3;;i0WL@LQF3-Fj]l6 &F\\^lF`^lF`^lF]^l-%+AXESLABELSG6$Q\"x6\"Q\"yFc]m-%%VIEWG6$;$!\"(!\"\" Fd]l;$!\"$Ff]l$\"$D%!\"#" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 10 "Example 4 " }}{PARA 0 "" 0 "" {TEXT 284 8 "Question" } {TEXT -1 22 ": Solve the equation: " }{XPPEDIT 18 0 "2*exp(-x^2)=exp(x )" "6#/*&\"\"#\"\"\"-%$expG6#,$*$%\"xGF%!\"\"F&-F(6#F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 285 8 "Solu tion" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 30 "The equation is \+ equivalent to " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2 = exp(x^2)*exp(x);" "6#/\"\"#*&-%$expG6#*$%\"xGF$\"\"\"-F'6#F*F+" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "which in turn is equival ent to " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(x^2+x) = 2;" "6#/-%$expG6#,&*$%\"xG\"\"#\"\"\"F)F+F*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 53 "This equation can be converted to the log arithm form " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2+x =ln*2" "6#/,&*$%\"xG\"\"#\"\"\"F&F(*&%#lnGF(F'F(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 57 "Completing the square for this quadratic \+ equation gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x ^2+x+1/4 = ln*2+1/4;" "6#/,(*$%\"xG\"\"#\"\"\"F&F(*&F(F(\"\"%!\"\"F(,& *&%#lnGF(F'F(F(*&F(F(F*F+F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 24 "which is equivalent to: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "(x+1/2)^2 = (1+4*ln*2)/4;" "6#/*$,&%\"xG\"\"\"*&F'F'\" \"#!\"\"F'F)*&,&F'F'*(\"\"%F'%#lnGF'F)F'F'F'F.F*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 11 "This gives " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x+1/2 = ``;" "6#/,&%\"xG\"\"\"*&F&F&\"\"#!\" \"F&%!G" }{TEXT 294 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(1+4*ln* 2)/2;" "6#*&-%%sqrtG6#,&\"\"\"F(*(\"\"%F(%#lnGF(\"\"#F(F(F(F,!\"\"" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = -1/2;" "6#/%\"xG,$*&\"\"\"F' \"\"#!\"\"F)" }{TEXT -1 1 " " }{TEXT 295 1 "+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "sqrt(1+4*ln*2)/2;" "6#*&-%%sqrtG6#,&\"\"\"F(*(\"\"%F(%# lnGF(\"\"#F(F(F(F,!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = -1/2+sq rt(1+4*ln*2)/2;" "6#/%\"xG,&*&\"\"\"F'\"\"#!\"\"F)*&-%%sqrtG6#,&F'F'*( \"\"%F'%#lnGF'F(F'F'F'F(F)F'" }{TEXT -1 1 " " }{TEXT 308 1 "~" }{TEXT -1 17 " 0.4711576497 or " }{XPPEDIT 18 0 "x = -1/2-sqrt(1+4*ln*2)/2" " 6#/%\"xG,&*&\"\"\"F'\"\"#!\"\"F)*&-%%sqrtG6#,&F'F'*(\"\"%F'%#lnGF'F(F' F'F'F(F)F)" }{TEXT -1 1 " " }{TEXT 307 1 "~" }{TEXT -1 1 " " } {XPPEDIT 18 0 "-1.471157650" "6#,$-%&FloatG6$\"+]w:r9!\"*!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "2*exp(-x^2)=exp(x);\nsolve(%,x);\nevalf(evalf[14 ](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&\"\"#\"\"\"-%$expG6#,$* $)%\"xGF&F'!\"\"F'F'-F)6#F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&#\"\" \"\"\"#!\"\"*&#F%F&F%*$,&F%F%*&\"\"%F%-%#lnG6#F&F%F%#F%F&F%F',&#F%F&F' *&F1F%F*F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$!+]w:r9!\"*$\"+(\\w:r %!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 13 ": T he values " }{XPPEDIT 18 0 "x = -1/2-sqrt(1+4*ln*2)/2" "6#/%\"xG,&*&\" \"\"F'\"\"#!\"\"F)*&-%%sqrtG6#,&F'F'*(\"\"%F'%#lnGF'F(F'F'F'F(F)F)" } {TEXT -1 1 " " }{TEXT 310 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-1.471 157650" "6#,$-%&FloatG6$\"+]w:r9!\"*!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "x = -1/2+sqrt(1+4*ln*2)/2;" "6#/%\"xG,&*&\"\"\"F'\"\"#!\"\"F)*&- %%sqrtG6#,&F'F'*(\"\"%F'%#lnGF'F(F'F'F'F(F)F'" }{TEXT -1 1 " " }{TEXT 309 1 "~" }{TEXT -1 23 " 0.4711576497 give the " }{TEXT 296 1 "x" } {TEXT -1 60 " coordinates of the points of intersection of the graphs \+ of " }{XPPEDIT 18 0 "f(x) = 2*exp(-x^2);" "6#/-%\"fG6#%\"xG*&\"\"#\"\" \"-%$expG6#,$*$F'F)!\"\"F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(x) = exp(x);" "6#/-%\"gG6#%\"xG-%$expG6#F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "In the following pic ture the graph of of " }{XPPEDIT 18 0 "f(x) = 2*exp(-x^2);" "6#/-%\"fG 6#%\"xG*&\"\"#\"\"\"-%$expG6#,$*$F'F)!\"\"F*" }{TEXT -1 13 " is drawn \+ in " }{TEXT 258 3 "red" }{TEXT -1 19 ", and the graph of " }{XPPEDIT 18 0 "g(x) = exp(x);" "6#/-%\"gG6#%\"xG-%$expG6#F'" }{TEXT -1 13 " is \+ drawn in " }{TEXT 256 4 "blue" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "f := x -> 2 *exp(-x^2):\n'f(x)'=f(x);\ng := x -> exp(x):\n'g(x)'=g(x);\nplot([f(x) ,g(x)],x=-2..2,y=0..3,color=[red,blue]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,$*&\"\"#\"\"\"-%$expG6#,$*$)F'F*F+!\"\"F+F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG-%$expGF&" }}{PARA 13 " " 1 "" {GLPLOT2D 510 286 286 {PLOTDATA 2 "6&-%'CURVESG6$7Y7$$!\"#\"\"! $\"3WOouxx7jO!#>7$$!3MLLL$Q6G\">!#<$\"3q)4*p%)yV_^F-7$$!3bmm;M!\\p$=F1 $\"3![U3E,cy%oF-7$$!3MLLL))Qj^%FC7$$!3gmmmc4`i6F1$\"3E 'f\\^cYr<&FC7$$!3KLLLQW*e3\"F1$\"3G60'=#yp]hFC7$$!3w++++()>'***FC$\"3A p5!oc$=jtFC7$$!3E++++0\"*H\"*FC$\"3K_]UA!e+p)FC7$$!35++++83&H)FC$\"3l` Vr$Gw]+\"F17$$!3\\LLL3k(p`(FC$\"3d3RE%4ZK8\"F17$$!3Anmmmj^NmFC$\"3G\"p G4[&o(G\"F17$$!3)zmmmYh=(eFC$\"3O#e5AjTnT\"F17$$!3+,++v#\\N)\\FC$\"3o^ oA?=;g:F17$$!3commmCC(>%FC$\"3!)>nSrM&pn\"F17$$!39*****\\FRXL$FC$\"3_2 ]=IZ`*y\"F17$$!3t*****\\#=/8DFC$\"3%>)*G`;)fx=F17$$!3=mmm;a*el\"FC$\"3 M'zaIB0f%>F17$$!3_mm;H9Li7FC$\"3'>%p&>'HQo>F17$$!3komm;Wn(o)F-$\"3+17R &oh\\)>F17$$!3$G++]7bDW%F-$\"3ouZWLm0'*>F17$$!3IqLLL$eV(>!#?$\"3EQN$Q? #****>F17$$\"3qbm;/rI2?F-$\"3OGs'fI%>**>F17$$\"3V[mmT+07UF-$\"3#3C*es[ X'*>F17$$\"3:Tm;zHz;kF-$\"3?gVK%)=y\"*>F17$$\"3)Qjmm\"f`@')F-$\"3%eQYy $*)=&)>F17$$\"3mILLL1+Y7FC$\"35-T*pV*=p>F17$$\"3%z****\\nZ)H;FC$\"3'Ho T?Prv%>F17$$\"3ckmm;$y*eCFC$\"3-!fy(H?l#)=F17$$\"3f)******R^bJ$FC$\"3_ `dXAez\"z\"F17$$\"3'e*****\\5a`TFC$\"395U%Ra%3$o\"F17$$\"3'o****\\7RV' \\FC$\"3w0/xYh9j:F17$$\"3Y'*****\\@fkeFC$\"3$\\_G_&3&zT\"F17$$\"3_ILLL &4Nn'FC$\"3u\\>N22>\"G\"F17$$\"3A*******\\,s`(FC$\"3dU$GIk3K8\"F17$$\" 3%[mm;zM)>$)FC$\"3EQX)H7^4+\"F17$$\"3M*******pfa<*FC$\"3&zv.1*>!zh)FC7 $$\"39HLLeg`!)**FC$\"39oV'p>eiQ(FC7$$\"3w****\\#G2A3\"F1$\"3B0oT^V1+iF C7$$\"3;LLL$)G[k6F1$\"3A\"4i)fWo`^FC7$$\"3#)****\\7yh]7F1$\"3sb4#FC7$$\"3!*******Rv&)z:F1$\"3!>nbAfi$[;F C7$$\"3ILLLGUYo;F1$\"3e1u+*o(4O7FC7$$\"3_mmm1^rZF1$\"39qSC!))zp9&F -7$$\"\"#F*F+-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*F^]l-F$6$7S7$F($\"3- FhOKGN`8FC7$F/$\"3mL4@0mkw9FC7$F5$\"3q(zuc7FIf\"FC7$F:$\"33E,e![-\\t\" FC7$F?$\"3%*R$Rw4#[!*=FC7$FE$\"3A\"pB8%G;Te'f>CFC7$FT$\"3>&e(=NU)Qj#FC7$FY$\"3OD#)Hb(pj'GFC7$Fhn$\"3Dq R_Tw$p7$FC7$F]o$\"3=$*H3()o*fP$FC7$Fbo$\"3'*[&ew4$>!o$FC7$Fgo$\"31/%os _?K,%FC7$F\\p$\"3e#z1k=QEO%FC7$Fap$\"3%*3#z8OJiq%FC7$Ffp$\"3'Hf#)HA*=] ^FC7$F[q$\"3o![.*pF*)ebFC7$F`q$\"3KmY(yn#HvgFC7$Feq$\"372'*[w-GslFC7$F jq$\"397`(G1\\W;(FC7$F_r$\"3smIfnu&yx(FC7$Fdr$\"3))R?QF)RRZ)FC7$F^s$\" 3Kv$eR)3!z;*FC7$Fhs$\"3A!4+V*eF!)**FC7$F]u$\"3I4?O`5/!4\"F17$Fgu$\"3'R k81w=q<\"F17$F\\v$\"3e$Ryr\"*o(y7F17$Fav$\"3g:?^hH8$R\"F17$Ffv$\"3RF\" )*=32\\^\"F17$F[w$\"3@9IJdA&Gk\"F17$F`w$\"3O(f&Qv@h(z\"F17$Few$\"31MZg 0t1\\>F17$Fjw$\"3Rf[`T-*[7#F17$F_x$\"3i$[nV+syH#F17$Fdx$\"3B(Qnc1SJ]#F 17$Fix$\"3)QXji7'*Hr#F17$F^y$\"3K'Go$pk=^HF17$Fcy$\"3MP8ivaE/KF17$Fhy$ \"3%o$fA,+]#\\$F17$F]z$\"3+9#RDb)e%z$F17$Fbz$\"3))**e)[!)e08%F17$Fgz$ \"3YrqL[\"=J\\%F17$F\\[l$\"3WGb\\BUEa[F17$Fa[l$\"3uo!4=y:SI&F17$Ff[l$ \"3\\_g\"H.p9u&F17$F[\\l$\"3qQjuyzpZiF17$F`\\l$\"3;#Gn\"Gt(Rx'F17$Fe\\ l$\"3S]1$*)4c!*Q(F1-Fh\\l6&Fj\\lF^]lF^]lF[]l-%+AXESLABELSG6$Q\"x6\"Q\" yF[gl-%%VIEWG6$;F(Fe\\l;F^]l$\"\"$F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 93 "Graphs of functions which involve exponential and logarit hm functions .. inverse functions " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Example 1: \+ " }{XPPEDIT 18 0 "g(x)=ln(x+1)+2" "6#/-%\"gG6#%\"xG,&-%#lnG6#,&F'\"\" \"F-F-F-\"\"#F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 21 "Consider the functin " }{XPPEDIT 18 0 "g( x)=ln(x+1)+2" "6#/-%\"gG6#%\"xG,&-%#lnG6#,&F'\"\"\"F-F-F-\"\"#F-" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 52 "The graph of g is obtai ned by translating the graph " }{XPPEDIT 18 0 "y=ln*x" "6#/%\"yG*&%#ln G\"\"\"%\"xGF'" }{TEXT -1 43 " one unit to the left and 2 units upward s. " }}{PARA 0 "" 0 "" {TEXT -1 9 "The line " }{XPPEDIT 18 0 "x=-1" "6 #/%\"xG,$\"\"\"!\"\"" }{TEXT -1 46 " is a verical asympote for the gra ph, and the " }{TEXT 266 1 "y" }{TEXT -1 23 " intercept is the point" }{XPPEDIT 18 0 "``(0,2)" "6#-%!G6$\"\"!\"\"#" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 196 "g := x -> ln(x+1)+2:\n'g(x)'=g(x);\np1 := plot(g(x),x=-1.5..5,y=- 2..4):\np2 := plots[implicitplot](x=-1,x=-1.5..5,y=-2..4,numpoints=2, \n color=black,linestyle=2):\nplots[display]([p1,p2]);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,&-%#lnG6#,&F'\"\"\"F-F -F-\"\"#F-" }}{PARA 13 "" 1 "" {GLPLOT2D 486 293 293 {PLOTDATA 2 "6&-% 'CURVESG6$7gn7$$!31+++%H\"*4'**!#=$!3'ze*)*>bbYN!#<7$$!3Y+++=+Q<**F*$! 3Yp%=F-7$$!3U+++2\\$Hu*F*$!3K E`**Q5,h;F-7$$!3))*****>N7dl*F*$!3cd]'z(G')o8F-7$$!3Y+++(z*[o&*F*$!3X1 $4j@\\I9\"F-7$$!3E+++#zt[R*F*$!3[tbOuK.\\!)F*7$$!3=+++)yd7A*F*$!3'\\\" emDIgEbF*7$$!33+++%yTw/*F*$!3npW[18*R^$F*7$$!31+++\"yDS())F*$!3#*)G5jf k$R=F*7$$!3')*****Hx$zE&)F*$\"3oH.*Q%Rg&[)!#>7$$!3u*****fwh&z\")F*$\"3 s(*)>-n@\\'HF*7$$!3%******ps/d`(F*$\"3#oP!>.y?$*fF*7$$!3/+++(oZ=*oF*$ \"30-Kn]MV9$)F*7$$!3c*****f,r^A'F*$\"3**3\"Hk)*pd-\"F-7$$!3;+++XV\\ebF *$\"3]p7XP$3%)=\"F-7$$!3#*******4%Q&zTF*$\"3a84gwWze9F-7$$!3C+++5[+0GF *$\"3Wc_#)y0!3n\"F-7$$!31+++S!G6R\"F*$\"3kVOw,#3-&=F-7$$!3/++++i%yX\"F do$\"3sCF+GUJ&)>F-7$$\"3#*******>rIi>L=@F-7$$\"3y******Ra* Qm#F*$\"3*G`vI(*phB#F-7$$\"3w*******)G\\?SF*$\"3[a)*>W\\$zL#F-7$$\"3[* *****RLT__F*$\"3!z:U$\\E:ACF-7$$\"3T+++5fG^AezF*$\"3+PJ%49jae#F-7$$\"3Z+++ICt,%*F*$\"37ZOgpsxiEF-7$$\"3'* *******4[z1\"F-$\"3/c%>'fobEFF-7$$\"3/+++ot837F-$\"3%)oxCS$\\@z#F-7$$ \"3&******RqI;M\"F-$\"3/\\8rov%3&GF-7$$\"3%******f*p\"4[\"F-$\"3KLaG[# G'3HF-7$$\"30+++\"HD)3;F-$\"3QM:%*R+!*eHF-7$$\"31+++p;zY$F-7$$\"3A+++A/&\\U#F-$\"3>`4G'*p3JKF-7$$\"39+++4^qcDF-$\"3mDd6&eM )oKF-7$$\"39+++Di*Hq#F-$\"3hhs[#GU\"4LF-7$$\"30++++`WMGF-$\"3N+u*GyCSM $F-7$$\"39+++X_zuHF-$\"3!*\\x3OK(*zLF-7$$\"3/+++bJ(>5$F-$\"3dt,E:\"o9T $F-7$$\"3%)*****>?75C$F-$\"3'*H*Rw'>![W$F-7$$\"31+++6r$=P$F-$\"3;;4I7L =vMF-7$$\"3,+++Noe3NF-$\"3!RZeSw$)f]$F-7$$\"33+++(o%GUOF-$\"3.\"\\c$F-7$$\"3-+++!)41k?\\0%F-$\"3APsV`@O?OF-7$$\"3o*****fvQ;>%F-$\"3m\"fR **R\\qk$F-7$$\"3H+++/&osJ%F-$\"3I?3SK(f4n$F-7$$\"31+++tVDhWF-$\"3'3i:? ]ywp$F-7$$\"3G+++]q.!f%F-$\"39??0:f)4s$F-7$$\"3f*****RCYts%F-$\"3AYZw* G__u$F-7$$\"3I+++k'o(e[F-$\"3Gv@\"fXRzw$F-7$$\"\"&\"\"!$\"3'\\0G#p%f " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "The domain of g is" }{XPPEDIT 18 0 "``(-1,infinity)" "6#- %!G6$,$\"\"\"!\"\"%)infinityG" }{TEXT -1 46 " and the range is the set of all real numbers " }{TEXT 269 1 "R" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "To find the " } {TEXT 267 1 "x" }{TEXT -1 16 " intercept, let " }{XPPEDIT 18 0 "y=0" " 6#/%\"yG\"\"!" }{TEXT -1 17 " in the equation " }{XPPEDIT 18 0 "y=ln(x +1)+2" "6#/%\"yG,&-%#lnG6#,&%\"xG\"\"\"F+F+F+\"\"#F+" }{TEXT -1 10 ", \+ so that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln(x+1)+2 =0" "6#/,&-%#lnG6#,&%\"xG\"\"\"F*F*F*\"\"#F*\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln(x+1)=-2" "6#/-%#lnG6#,&%\"xG\"\"\"F)F),$\"\"#!\"\" " }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 12 "which gives " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x+1=exp(-2)" "6#/,&% \"xG\"\"\"F&F&-%$expG6#,$\"\"#!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 8 "so that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=exp(-2)-1" "6#/%\"xG,&-%$expG6#,$\"\"#!\"\"\"\"\"F,F+" }{TEXT -1 1 " " }{TEXT 268 1 "~" }{TEXT -1 16 " -0.8646647168. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "g := \+ x -> ln(x+1)+2:\nsolve(g(x)=0,x);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#!\"#\"\"\"F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+orkY')!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 " To find the inverse function " }{XPPEDIT 18 0 "g^(-1)" "6#)%\"gG,$\"\" \"!\"\"" }{TEXT -1 24 ", we solve the equation " }{XPPEDIT 18 0 "y=ln( x+1)+2" "6#/%\"yG,&-%#lnG6#,&%\"xG\"\"\"F+F+F+\"\"#F+" }{TEXT -1 5 " f or " }{TEXT 270 1 "x" }{TEXT -1 13 " in terms of " }{TEXT 271 1 "y" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=ln(x+1)+2" "6#/%\"yG,&-%#lnG6#,&%\"xG \"\"\"F+F+F+\"\"#F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 17 "i s equivalent to " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y -2=ln(x+1)" "6#/,&%\"yG\"\"\"\"\"#!\"\"-%#lnG6#,&%\"xGF&F&F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 33 "which, in turn, is equivalent to " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x+1=exp(y-2) " "6#/,&%\"xG\"\"\"F&F&-%$expG6#,&%\"yGF&\"\"#!\"\"" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x=exp(y-2)-1" "6#/%\"xG,&-%$expG6#,&%\"yG\"\"\"\"\"#!\" \"F+F+F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "This means t hat " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g^(-1)" "6#)% \"gG,$\"\"\"!\"\"" }{XPPEDIT 18 0 "``(x) = exp(x-2)-1;" "6#/-%!G6#%\"x G,&-%$expG6#,&F'\"\"\"\"\"#!\"\"F-F-F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "g := x -> ln(x+1)+2:\nsolve(g(x)=y,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#,&\"\"#!\"\"%\"yG\"\"\"F+F+ F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "In the following picture \+ the graph of " }{XPPEDIT 18 0 "g(x)=ln(x+1)+2" "6#/-%\"gG6#%\"xG,&-%#l nG6#,&F'\"\"\"F-F-F-\"\"#F-" }{TEXT -1 13 " is drawn in " }{TEXT 258 3 "red" }{TEXT -1 42 ", while the graph of the inverse function " }} {PARA 257 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "g^(-1)" "6#)%\"gG,$ \"\"\"!\"\"" }{XPPEDIT 18 0 "``(x) = exp(x-2)-1;" "6#/-%!G6#%\"xG,&-%$ expG6#,&F'\"\"\"\"\"#!\"\"F-F-F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 12 "is drawn in " }{TEXT 256 4 "blue" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 391 "g := x -> ln(x+1)+2:\n'g(x)'=g(x);\ng@@(-1) := x -> exp(x-2)-1:\n '(g@@(-1))(x)'=(g@@(-1))(x);\np1 := plot(g(x),x=-1.5..5,y=-2..4,color= red):\np2 := plot(x,x=-1.5..5,y=-1.5..5,color=black,linestyle=3):\np3 \+ := plot((g@@(-1))(x),x=-1.5..5,y=-2..4,color=blue):\np4 := plots[impli citplot](\{x=-1,y=-1\},x=-1.5..5,y=-2..4,numpoints=2,\n c olor=black,linestyle=2):\nplots[display]([p1,p2,p3,p4]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,&-%#lnG6#,&F'\"\"\"F-F-F-\"\"#F- " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/--%#@@G6$%\"gG!\"\"6#%\"xG,&-%$ex pG6#,&F+\"\"\"\"\"#F)F1F1F)" }}{PARA 13 "" 1 "" {GLPLOT2D 477 477 477 {PLOTDATA 2 "6)-%'CURVESG6$7gn7$$!31+++%H\"*4'**!#=$!3'ze*)*>bbYN!#<7$ $!3Y+++=+Q<**F*$!3Yp%=F-7$$!3 U+++2\\$Hu*F*$!3KE`**Q5,h;F-7$$!3))*****>N7dl*F*$!3cd]'z(G')o8F-7$$!3Y +++(z*[o&*F*$!3X1$4j@\\I9\"F-7$$!3E+++#zt[R*F*$!3[tbOuK.\\!)F*7$$!3=++ +)yd7A*F*$!3'\\\"emDIgEbF*7$$!33+++%yTw/*F*$!3npW[18*R^$F*7$$!31+++\"y DS())F*$!3#*)G5jfk$R=F*7$$!3')*****Hx$zE&)F*$\"3oH.*Q%Rg&[)!#>7$$!3u** ***fwh&z\")F*$\"3s(*)>-n@\\'HF*7$$!3%******ps/d`(F*$\"3#oP!>.y?$*fF*7$ $!3/+++(oZ=*oF*$\"30-Kn]MV9$)F*7$$!3c*****f,r^A'F*$\"3**3\"Hk)*pd-\"F- 7$$!3;+++XV\\ebF*$\"3]p7XP$3%)=\"F-7$$!3#*******4%Q&zTF*$\"3a84gwWze9F -7$$!3C+++5[+0GF*$\"3Wc_#)y0!3n\"F-7$$!31+++S!G6R\"F*$\"3kVOw,#3-&=F-7 $$!3/++++i%yX\"Fdo$\"3sCF+GUJ&)>F-7$$\"3#*******>rIi>L=@F- 7$$\"3y******Ra*Qm#F*$\"3*G`vI(*phB#F-7$$\"3w*******)G\\?SF*$\"3[a)*>W \\$zL#F-7$$\"3[******RLT__F*$\"3!z:U$\\E:ACF-7$$\"3T+++5fG^AezF*$\"3+PJ%49jae#F-7$$\"3Z+++ICt,%*F*$\"37ZO gpsxiEF-7$$\"3'********4[z1\"F-$\"3/c%>'fobEFF-7$$\"3/+++ot837F-$\"3%) oxCS$\\@z#F-7$$\"3&******RqI;M\"F-$\"3/\\8rov%3&GF-7$$\"3%******f*p\"4 [\"F-$\"3KLaG[#G'3HF-7$$\"30+++\"HD)3;F-$\"3QM:%*R+!*eHF-7$$\"31+++p;z Y$F-7$$\"3A+++A/&\\U#F-$\"3>`4G'*p3JKF-7$$\"39+++4^q cDF-$\"3mDd6&eM)oKF-7$$\"39+++Di*Hq#F-$\"3hhs[#GU\"4LF-7$$\"30++++`WMG F-$\"3N+u*GyCSM$F-7$$\"39+++X_zuHF-$\"3!*\\x3OK(*zLF-7$$\"3/+++bJ(>5$F -$\"3dt,E:\"o9T$F-7$$\"3%)*****>?75C$F-$\"3'*H*Rw'>![W$F-7$$\"31+++6r$ =P$F-$\"3;;4I7L=vMF-7$$\"3,+++Noe3NF-$\"3!RZeSw$)f]$F-7$$\"33+++(o%GUO F-$\"3.\"\\c$F-7$$\"3-+++!)41k?\\0%F-$\"3APsV`@O?OF-7$$\"3o*****fv Q;>%F-$\"3m\"fR**R\\qk$F-7$$\"3H+++/&osJ%F-$\"3I?3SK(f4n$F-7$$\"31+++t VDhWF-$\"3'3i:?]ywp$F-7$$\"3G+++]q.!f%F-$\"39??0:f)4s$F-7$$\"3f*****RC Yts%F-$\"3AYZw*G__u$F-7$$\"3I+++k'o(e[F-$\"3Gv@\"fXRzw$F-7$$\"\"&\"\"! $\"3'\\0G#p%f+]P4[+ 0GF*Fc`l7$$!3;ML$e/G6R\"F*Ff`l7$$!3_qm;Hi%yX\"FdoFi`l7$$\"3)4++D6xhD\" F*F\\al7$$\"3-****\\Pa*Qm#F*F_al7$$\"36++]()G\\?SF*Fbal7$$\"3II$ekL8CD &F*Feal7$$\"3\\lm;/fG5$F-F^el7$$\"3B++D,A,TKF-Fael7$$\"3[l\"z%4r$=P$F- Fdel7$$\"3;+D1Moe3NF-Fgel7$$\"3slmT&o%GUOF-Fjel7$$\"3E*\\7`%RD#y$F-F]f l7$$\"3mLLLy41%F-Fffl7$ $\"3;++]-&osJ%F-Fifl7$$\"37m;/rVDhWF-F\\gl7$$\"3gKLL[q.!f%F-F_gl7$$\"3 r*\\7GCYts%F-Fbgl7$$\"3#**\\(=i'o(e[F-Fegl7$Fg]lFg]l-F]^l6&F_^lFi]lFi] lFi]l-%*LINESTYLEG6#\"\"$-F$6$7W7$Fh^l$!3?:oxlh-)p*F*7$F[_l$!3_TN6XG1_ '*F*7$F^_l$!3^S#)e_XT1'*F*7$Fa_l$!3;t.ltb)ya*F*7$Fd_l$!3%p9&[GaFDq3@FS*F*7$Fj_l$!3*RD#4%*pj?$*F*7$F]`l$!3m0])QMPPA*F*7$F``l $!3QODI&[i*3\"*F*7$Fc`l$!3OR8s<(pw(*)F*7$Ff`l$!3z\"fI#ysSA))F*7$Fi`l$! 3?l&*4_QBm')F*7$F\\al$!3K6<.[F]l%)F*7$F_al$!3!Q&QRN^aL#)F*7$Fbal$!3Q9) =^8$*o(zF*7$Feal$!3yvp(Gag;r(F*7$Fhal$!3/Hy![q^1N(F*7$F[bl$!3NlJ=*)Rh+ qF*7$F^bl$!3]Ab\"\\jT[`'F*7$Fabl$!3wa:*oW]D1'F*7$Fdbl$!3;CyXQw**paF*7$ Fgbl$!3Oy'GoY^I#[F*7$Fjbl$!3Q;IZ(HS$\\SF*7$F]cl$!37F\\AN+QPKF*7$F`cl$! 39`RgY'QpB#F*7$Fccl$!3<0!3'QJwS5F*7$Ffcl$\"3SRL6'*f5'\\\"Fdo7$Ficl$\"3 \")RUgG*4Nh\"F*7$F\\dl$\"3oDO$3tTzM$F*7$F_dl$\"3s$)G1ae9&H&F*7$Fbdl$\" 3W\\?\\cr8\\uF*7$Fedl$\"3L9Va5az>5F-7$Fhdl$\"3/&yN0#f`.8F-7$F[el$\"3YC y\\BWi];F-7$F^el$\"351/0d&*45?F-7$Fael$\"3k2a&=,8\"fCF-7$Fdel$\"3$**=` /,(eUHF-7$Fgel$\"3o&pX/HQ._$F-7$Fjel$\"3'y`Ov\"4'p;%F-7$F]fl$\"3qaqikq BV\\F-7$F`fl$\"3[2`#\\&4%4!eF-7$Fcfl$\"35P<>[$=i!oF-7$Fffl$\"3uk#*fRx' )\\zF-7$Fifl$\"3_-(eLS2\"!#;7$$\"3\"*)\\(o4d kDXF-$\"3%*zF-Oj*)\\6Fh`m7$F_gl$\"3-dl\"eaEIB\"Fh`m7$$\"3;;HdX;peYF-$ \"3._Tjr'fxK\"Fh`m7$Fbgl$\"3E#fT.7D#H9Fh`m7$$\"3O****\\_u0$z%F-$\"3SI8 Fh`m-F]^l6&F_^lFc^lFc^lF`^l-F$6(7$7$Fh^l$!3A+++ ++++5F-7$$\"3SnmmmmmmmF*F_cm7$7$$\"3+++++++] " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 11 "Example 2: " }{XPPEDIT 18 0 "g(x) = 3-2*exp(1-x); " "6#/-%\"gG6#%\"xG,&\"\"$\"\"\"*&\"\"#F*-%$expG6#,&F*F*F'!\"\"F*F1" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Consider the functin " }{XPPEDIT 18 0 "g(x) = 3-2*exp(1-x );" "6#/-%\"gG6#%\"xG,&\"\"$\"\"\"*&\"\"#F*-%$expG6#,&F*F*F'!\"\"F*F1 " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 52 "The graph of g is ob tained by translating the graph " }{XPPEDIT 18 0 "y = -2*exp(-x);" "6# /%\"yG,$*&\"\"#\"\"\"-%$expG6#,$%\"xG!\"\"F(F." }{TEXT -1 44 " one uni t to the right and 3 units upwards. " }}{PARA 0 "" 0 "" {TEXT -1 9 "Th e line " }{XPPEDIT 18 0 "y = 3;" "6#/%\"yG\"\"$" }{TEXT -1 50 " is a h orizontal asymptote for the graph, and the " }{TEXT 272 1 "y" }{TEXT -1 23 " intercept is the point" }{XPPEDIT 18 0 "``(0,3-2*exp(1));" "6# -%!G6$\"\"!,&\"\"$\"\"\"*&\"\"#F)-%$expG6#F)F)!\"\"" }{TEXT -1 8 ", wh ere " }{XPPEDIT 18 0 "3-2*exp(1)" "6#,&\"\"$\"\"\"*&\"\"#F%-%$expG6#F% F%!\"\"" }{TEXT -1 1 " " }{TEXT 276 1 "~" }{TEXT -1 15 " -2.436563657. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "g := x -> 3-2*exp(1-x):\n'g(x)'=g(x);\np1 := plot(g( x),x=-1..5,y=-6..4):\np2 := plot(3,x=-1..5,y=-2..4,color=black,linesty le=2):\nplots[display]([p1,p2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %\"gG6#%\"xG,&\"\"$\"\"\"*&\"\"#F*-%$expG6#,&F*F*F'!\"\"F*F1" }}{PARA 13 "" 1 "" {GLPLOT2D 486 293 293 {PLOTDATA 2 "6&-%'CURVESG6$7U7$$!\"\" \"\"!$!33Ihy>7\"y<\"!#;7$$!3[*****\\P&3Y$*!#=$!3#ol[k\"oE%3\"F-7$$!3/+ ++]2<#p)F1$!3g.+wvqVm**!#<7$$!3#3+]78.K7)F1$!3wGVU!4%G\\#*F97$$!3[++]7 bBavF1$!3'=JEYp&zr&)F97$$!3++++D$3XF'F1$!39c_278w\"=(F97$$!3c*****\\F) H')\\F1$!31@)*Rw^5^fF97$$!3J++]i3@/PF1$!3A'zjHL;S([F97$$!3V++]7*eWQ(3vF17$$\"32,++]>q0]F1$! 3u\"4Y3#*Gc&HF17$$\"3h******\\U80jF1$\"3e`I(R&*p,1\"F17$$\"3'4+++0ytb( F1$\"3c16m!\\>kY%F17$$\"3w****\\(QNXp)F1$\"3d@@[?C)4@(F17$$\"3.+++XDn/ 5F9$\"3Ef#)G\"HB$45F97$$\"3.+++!y?#>6F9$\"3(el;cPwZA\"F97$$\"3'****\\( 3wY_7F9$\"3q$o=oCPiW\"F97$$\"3#)******HOTq8F9$\"3HjjWPE5>;F97$$\"37++v 3\">)*\\\"F9$\"3Ke%[&G#>ny\"F97$$\"3:++DEP/B;F9$\"3JmBTd3QF>F97$$\"3=+ +](o:;v\"F9$\"3YOPcb?zc?F97$$\"3=++v$)[op=F9$\"3-Np,BG$=;#F97$$\"3%*** **\\i%Qq*>F9$\"3![s'Q5*e?E#F97$$\"3&****\\(QIKH@F9$\"3G.%*3xf\\`BF97$$ \"3#****\\7:xWC#F9$\"3'yJAf+\"\\#F 97$$\"3y******4FL(\\#F9$\"3C]e:&yZDb#F97$$\"3#)****\\d6.BEF9$\"3D$*Qk4 0S0EF97$$\"3(****\\(o3lWFF9$\"31b$HF0)e]EF97$$\"3!*****\\A))ozGF9$\"39 7R\\3[s%p#F97$$\"3e******Hk-,IF9$\"3=Z9Q:rgHFF97$$\"36+++D-eIJF9$\"3C! oKqFjCw#F97$$\"3u***\\(=_(zC$F9$\"3OVys(Hu()y#F97$$\"3M+++b*=jP$F9$\"3 3&3*oBi@9GF97$$\"3g***\\(3/3(\\$F9$\"3F(o*3$**\\`$GF97$$\"33++vB4JBOF9 $\"3K)o*fmb([&GF97$$\"3u*****\\KCnu$F9$\"3AOX4xZsrGF97$$\"3s***\\(=n#f (QF9$\"3Ae`MOAF()GF97$$\"3P+++!)RO+SF9$\"3)))\\j'*4i/!HF97$$\"30++]_!> w7%F9$\"3PQ_!z(fN7HF97$$\"3O++v)Q?QD%F9$\"3q9uA%GZF#HF97$$\"3G+++5jypV F9$\"3l'Q*HEg?JHF97$$\"3<++]Ujp-XF9$\"38ciyeywRHF97$$\"3++++gEd@YF9$\" 3%>&p*3#)=l%HF97$$\"39++v3'>$[ZF9$\"3E^Y\"QS&)G&HF97$$\"37++D6Ejp[F9$ \"3QPL2KzEeHF97$$\"\"&F*$\"3cJDAA(oL'HF9-%'COLOURG6&%$RGBG$\"#5F)$F*F* F[\\l-F$6%7S7$F($\"\"$F*7$F5F`\\l7$F@F`\\l7$FEF`\\l7$FJF`\\l7$FOF`\\l7 $FTF`\\l7$FYF`\\l7$FhnF`\\l7$F^oF`\\l7$FcoF`\\l7$FhoF`\\l7$F]pF`\\l7$F bpF`\\l7$FgpF`\\l7$F\\qF`\\l7$FaqF`\\l7$FfqF`\\l7$F[rF`\\l7$F`rF`\\l7$ FerF`\\l7$FjrF`\\l7$F_sF`\\l7$FdsF`\\l7$FisF`\\l7$F^tF`\\l7$FctF`\\l7$ FhtF`\\l7$F]uF`\\l7$FbuF`\\l7$FguF`\\l7$F\\vF`\\l7$FavF`\\l7$FfvF`\\l7 $F[wF`\\l7$F`wF`\\l7$FewF`\\l7$FjwF`\\l7$F_xF`\\l7$FdxF`\\l7$FixF`\\l7 $F^yF`\\l7$FcyF`\\l7$FhyF`\\l7$F]zF`\\l7$FbzF`\\l7$FgzF`\\l7$F\\[lF`\\ l7$Fa[lF`\\l-Ff[l6&Fh[lF*F*F*-%*LINESTYLEG6#\"\"#-%+AXESLABELSG6%Q\"x6 \"Q\"yF\\`l-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(Fa[l;$!\"'F*$\"\"%F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2 " }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "The domain of g is the se t of all real numbers " }{TEXT 277 1 "R" }{TEXT -1 18 ", and the range is" }{XPPEDIT 18 0 "``(-infinity, 3);" "6#-%!G6$,$%)infinityG!\"\"\" \"$" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "To find the " }{TEXT 273 1 "x" }{TEXT -1 16 " intercep t, let " }{XPPEDIT 18 0 "y=0" "6#/%\"yG\"\"!" }{TEXT -1 17 " in the eq uation " }{XPPEDIT 18 0 "y=3-2*exp(1-x)" "6#/%\"yG,&\"\"$\"\"\"*&\"\"# F'-%$expG6#,&F'F'%\"xG!\"\"F'F/" }{TEXT -1 10 ", so that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "0 = 3-2*exp(1-x);" "6#/\"\" !,&\"\"$\"\"\"*&\"\"#F'-%$expG6#,&F'F'%\"xG!\"\"F'F/" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*exp(1-x) = 3;" "6#/*&\"\"#\"\"\"-%$expG6#,&F&F&% \"xG!\"\"F&\"\"$" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 12 "whic h gives " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(1-x) \+ = 3/2;" "6#/-%$expG6#,&\"\"\"F(%\"xG!\"\"*&\"\"$F(\"\"#F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1-x=ln(3/2)" "6#/,&\"\"\"F%%\"xG!\"\"-% #lnG6#*&\"\"$F%\"\"#F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=1-ln(3/2) " "6#/%\"xG,&\"\"\"F&-%#lnG6#*&\"\"$F&\"\"#!\"\"F-" }{TEXT -1 1 " " } {TEXT 278 1 "~" }{TEXT -1 15 " 0.5945348919. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "g := x -> 3- 2*exp(1-x):\nsolve(g(x)=0,x);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$-%#lnG6##\"\"$\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+>*[`%f!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "To find \+ the inverse function " }{XPPEDIT 18 0 "g^(-1)" "6#)%\"gG,$\"\"\"!\"\" " }{TEXT -1 24 ", we solve the equation " }{XPPEDIT 18 0 "y = 3-2*exp( 1-x);" "6#/%\"yG,&\"\"$\"\"\"*&\"\"#F'-%$expG6#,&F'F'%\"xG!\"\"F'F/" } {TEXT -1 5 " for " }{TEXT 274 1 "x" }{TEXT -1 13 " in terms of " } {TEXT 275 1 "y" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 3-2*exp(1-x);" "6 #/%\"yG,&\"\"$\"\"\"*&\"\"#F'-%$expG6#,&F'F'%\"xG!\"\"F'F/" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 17 "is equivalent to " }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*exp(1-x) = 3-y;" "6#/*&\"\"#\" \"\"-%$expG6#,&F&F&%\"xG!\"\"F&,&\"\"$F&%\"yGF," }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 33 "which, in turn, is equivalent to " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(1-x) = (3-y)/2;" "6#/-%$expG6#,&\"\"\"F(%\"xG!\"\"*&,&\"\"$F(%\"yGF*F(\"\"#F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1-x = ln((3-y)/2);" "6#/,&\"\"\"F%%\"xG!\"\"- %#lnG6#*&,&\"\"$F%%\"yGF'F%\"\"#F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 8 "that is " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=1-ln((3-y)/2)" "6#/%\"xG,&\"\"\"F&-%#lnG6#*&,&\"\"$F&%\"yG!\" \"F&\"\"#F.F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "This me ans that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g^(-1)" "6#)%\"gG,$\"\"\"!\"\"" }{XPPEDIT 18 0 "``(x) = 1-ln((3-x)/2);" "6#/-% !G6#%\"xG,&\"\"\"F)-%#lnG6#*&,&\"\"$F)F'!\"\"F)\"\"#F0F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "g := x -> 3-2*exp(1-x):\nso lve(g(x)=y,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$-%#lnG6#,& #\"\"$\"\"#F$*&F+!\"\"%\"yGF$F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "In the following picture the graph of " }{XPPEDIT 18 0 "g (x) = 3-2*exp(1-x);" "6#/-%\"gG6#%\"xG,&\"\"$\"\"\"*&\"\"#F*-%$expG6#, &F*F*F'!\"\"F*F1" }{TEXT -1 13 " is drawn in " }{TEXT 258 3 "red" } {TEXT -1 42 ", while the graph of the inverse function " }}{PARA 257 " " 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "g^(-1)" "6#)%\"gG,$\"\"\"!\"\" " }{XPPEDIT 18 0 "``(x) = 1-ln((3-x)/2);" "6#/-%!G6#%\"xG,&\"\"\"F)-%# lnG6#*&,&\"\"$F)F'!\"\"F)\"\"#F0F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 12 "is drawn in " }{TEXT 256 4 "blue" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 385 "g := x -> 3-2*exp(1-x):\n'g(x)'=g(x);\ng@@(-1) := x -> 1-ln((3-x) /2):\n'(g@@(-1))(x)'=(g@@(-1))(x);\np1 := plot(g(x),x=-1..5,y=-6..4,co lor=red):\np2 := plot(x,x=-6..5,y=-6..5,color=black,linestyle=3):\np3 \+ := plot((g@@(-1))(x),x=-6..4,y=-1..5,color=blue):\np4 := plots[implici tplot](\{x=3,y=3\},x=-6..5,y=-6..5,numpoints=2,\n color=b lack,linestyle=2):\nplots[display]([p1,p2,p3,p4]); " }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"gG6#%\"xG,&\"\"$\"\"\"*&\"\"#F*-%$expG6#,&F*F*F' !\"\"F*F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/--%#@@G6$%\"gG!\"\"6#%\" xG,&\"\"\"F--%#lnG6#,&#\"\"$\"\"#F-*&F4F)F+F-F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 477 477 477 {PLOTDATA 2 "6)-%'CURVESG6$7U7$$!\"\"\"\"!$!33Ih y>7\"y<\"!#;7$$!3[*****\\P&3Y$*!#=$!3#ol[k\"oE%3\"F-7$$!3/+++]2<#p)F1$ !3g.+wvqVm**!#<7$$!3#3+]78.K7)F1$!3wGVU!4%G\\#*F97$$!3[++]7bBavF1$!3'= JEYp&zr&)F97$$!3++++D$3XF'F1$!39c_278w\"=(F97$$!3c*****\\F)H')\\F1$!31 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{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Consider \+ the function g given by " }{XPPEDIT 18 0 "g(x) = 1/(exp(x)-1);" "6#/-% \"gG6#%\"xG*&\"\"\"F),&-%$expG6#F'F)F)!\"\"F." }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 28 "The domain of g is the set \{" }{TEXT 279 1 "x" }{TEXT -1 1 " " }{TEXT 280 1 ":" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x in R" "6#-%#inG6$%\"xG%\"RG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 " x<>0" "6#0%\"xG\"\"!" }{TEXT -1 1 "\}" }{XPPEDIT 18 0 "`` = R minus \{ 0\}" "6#/%!G-%&minusG6$%\"RG<#\"\"!" }{XPPEDIT 18 0 "``=``(-infinity,0 ) union ``(0,infinity)" "6#/%!G-%&unionG6$-F$6$,$%)infinityG!\"\"\"\"! -F$6$F-F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "The graph of g may be sketched by first drawing t he graph " }{XPPEDIT 18 0 "y=exp(x)-1" "6#/%\"yG,&-%$expG6#%\"xG\"\"\" F*!\"\"" }{TEXT -1 42 ", and then checking what happens when the " } {TEXT 281 1 "y" }{TEXT -1 45 " values along this curve are divided int o 1. " }}{PARA 0 "" 0 "" {TEXT -1 22 "For example, positive " }{TEXT 282 1 "y" }{TEXT -1 89 " values which are greater than 1, give positiv e numbers less than 1 on division into 1. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "h := x -> exp(x)- 1:\ng := x -> 1/(exp(x)-1):\n'g(x)'=g(x);\nplot([g(x),h(x),-1],x=-3..3 ,y=-3..3,color=[red,brown,black],\n linestyle=[1,2,4],discont =true);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG*&\"\"\"F), &-%$expGF&F)F)!\"\"F-" }}{PARA 13 "" 1 "" {GLPLOT2D 325 347 347 {PLOTDATA 2 "6'-%'CURVESG6&7gn7$$!\"$\"\"!$!3)fD\"\\'p&R_5!#<7$$!3*GyI w`3Y$HF-$!3-oT+$>Nh0\"F-7$$!3Ew&pex6x(GF-$!33'oVzk<'f5F-7$$!3;\\Di;as8 GF-$!35)oO;U3Q1\"F-7$$!3?q8D9\\J\\FF-$!3E:oJNPMo5F-7$$!3yyXvV0@&o#F-$! 3W$oO2a*>t5F-7$$!3sWMP'exdi#F-$!3ErH#4lJ!y5F-7$$!3[Bl\\,#QUc#F-$!3w*Hb :c(R$3\"F-7$$!3p6Qi\"3%f+DF-$!3IQEBLwO*3\"F-7$$!3*=p]#pS:PCF-$!3ih&pMW \"y&4\"F-7$$!3m.iv=#)*=P#F-$!3*o\"3WA[!H5\"F-7$$!3ie67I3U9BF-$!3&p\"yr =/m46F-7$$!3Sq0+/\\r\\AF-$!3/P\\Y.Y&y6\"F-7$$!3S808*GVZ=#F-$!3I$>#[d*o n7\"F-7$$!3%ytb#*4J@7#F-$!3?t!p[eug8\"F-7$$!3Q`W\\KKFl?F-$!3-.JZf:>X6F -7$$!3ksY]HPm(*>F-$!3GpO<$>Tp:\"F-7$$!3y?#>@h*QS>F-$!3%pW')o1Wx;\"F-7$ $!3ynu(y>mP(=F-$!3y\\@HDnR\"=\"F-7$$!3t8/P(=$z9=F-$!3Kezxi)fX>\"F-7$$! 3=>[7[/4]C;F-$!3;SQ%>%)RaC\"F-7$$!3%[o%*4cd^c\"F-$!3=Y*G!G0Jk7F-7$$!3lQ qur2[,:F-$!3+(>&[(4qmG\"F-7$$!30BVv$[Q`V\"F-$!3`wU hxP\"F-$!3Fk3.&)G:P8F-7$$!3pY)='Hmd:8F-$!39X9Ykorm8F-7$$!3)HL(\\[OL^7F -$!3Dm\"Q+X,3S\"F-7$$!3IJI([U%[)=\"F-$!3RAUM&G#>Q9F-7$$!30^'p$pXnF6F-$ !3aUnE5.#)y9F-7$$!31*oHEfb,1\"F-$!3iD5xFA**H:F-7$$!3Ko-,!*y'[***!#=$!3 'G`!>p%\\Ce\"F-7$$!3q-eI;*)4Z$*Ffu$!3yye$[-Kmk\"F-7$$!3Ua(H([R7g()Ffu$ !3'G<&R]/i8>w()y/>O%Ffu$!3(3z>n'3\")GGF-7$$!3;.#)y3\")*3t$Ff u$!3911m2(Q8@$F-7$$!3]iyp.&o5:$Ffu$!3?\")>-UAu*p$F-7$$!3Okp_U$=l[#Ffu$ !3UeTGrsQUXF-7$$!3%)Hd@cn8#*=Ffu$!3%)QL-f')y+eF-7$$!3G&>LP,-%e7Ffu$!3# *e'[g0qqX)F-7$$!3/7w*33&>^&*!#>$!3MZ!)>6_y(4\"!#;7$$!3TrKYC+P=lFbz$!3e hd.;!pYe\"Fez7$$!3[`uMowx))[Fbz$!3o)p7=Y3f4#Fez7$$!3cN;B7`=fKFbz$!39!y Nb-B&=JFez7$$!3eEP9\\Fj\\l7$$!3'******* ********f!#E$!3w!pjHommm\"!\"*7gn7$$\"3'***************fF]^l$\"3KiT7_m mm;F`^l7$$\"3NL!Q!*>*[V?Fg\\l$\"3ig!f,`#f))[Fj\\l7$$\"3Qmg2)Rsp3%Fg\\l $\"3J3^_6C!=W#Fj\\l7$$\"3U*49rfb/8'Fg\\l$\"3#)zt4W_?E;Fj\\l7$$\"3YK@:' zQR<)Fg\\l$\"3j!R!\\7sS=7Fj\\l7$$\"3')>GU>04E7Fbz$\"3x6Y6Uo51\")Fez7$$ \"3YE/BfryM;Fbz$\"3eL;8\"[Sr1'Fez7$$\"3/Sc%)Q/=_CFbz$\"3e/(3#yw?GSFez7 $$\"3C`3Y=PdpKFbz$\"3rDgiX^x3IFez7$$\"3O!G\"px-O/\\Fbz$\"3C;?q&f5%*)>F ez7$$\"3z1<#p$o9RlFbz$\"3Ii)*Qtjzz9Fez7$$\"3kI?)\\#\\)RQ*Fbz$\"39KxnWs U;5Fez7$$\"3XNUI,B)GA\"Ffu$\"3IN'HD+!f(o(F-7$$\"353Xx$*eui=Ffu$\"3!)4` CsE$R)[F-7$$\"3^)H'[<4&o]#Ffu$\"3J4x3lq$*4NF-7$$\"3O5UXAY*y9$Ffu$\"3!y a%e[c\"Hq#F-7$$\"39^bE'>CAu$Ffu$\"3b1I\\s-K.AF-7$$\"3YjZ.X!=wN%Ffu$\"3 )zF,(p2.J=F-7$$\"3O\")=wV#fS*\\Ffu$\"3&zv5&G](Ffu$\"3)Q3un$erZ*)Ffu7$$\"3I l[porc_\")Ffu$\"3X?'z\"Rp/QzFfu7$$\"35AEWn!*oy()Ffu$\"36m**4NFb8rFfu7$ $\"3[ka0NxEZ$*Ffu$\"3g&>U&*\\ShY'Ffu7$$\"3UF`\\wiL-5F-$\"3Wu\"\\@$=J)z &Ffu7$$\"3Ez2)QR5'f5F-$\"3z#3N4pXVI&Ffu7$$\"3FKD73QBE6F-$\"33\"HeJQ>%) z%Ffu7$$\"3K'eH'=o?&=\"F-$\"3'f8L!y@k-WFfu7$$\"3)3=vyb4*\\7F-$\"3#=^n' H--;SFfu7$$\"3Nc>]m=_68F-$\"3)GdJZoivo$Ffu7$$\"3K%Q)*p%y!eP\"F-$\"3%=* Hv.0N!Q$Ffu7$$\"3@:`+XC%[V\"F-$\"3uKOu-J*f7$Ffu7$$\"3ShHDM#>&)\\\"F-$ \"3k,p)>5]w(GFfu7$$\"3+xcCA:mk:F-$\"3#\\TAX_jZk#Ffu7$$\"3)G_!Qy&QAi\"F -$\"3I16$G%)y.Y#Ffu7$$\"3O`6QwLU%o\"F-$\"3iC)z/fX#yAFfu7$$\"33nE]djm[< F-$\"3WU%f(yKi1@Ffu7$$\"3wop7\"e::\"=F-$\"3c,N6)fLK&>Ffu7$$\"3+\\.jOaK s=F-$\"3PCAXfy0<=Ffu7$$\"3*4JqLTW)R>F-$\"3SX&RZ+4&y;Ffu7$$\"3At*)*p@80 +#F-$\"3=8SaQxCk:Ffu7$$\"3y>%pV6!Hl?F-$\"3%oc!=Dv)=X\"Ffu7$$\"3hCq76w) R7#F-$\"3oWfkO#zyN\"Ffu7$$\"3O\"oB\"z%f\")=#F-$\"3qG$3EM?GE\"Ffu7$$\"3 O>z(e?S&[AF-$\"3-l/ZcZ4!=\"Ffu7$$\"3.*o^KYb;J#F-$\"3I-$*RF\\(**4\"Ffu7 $$\"3evKvj@OtBF-$\"3)Qh\\o>)QF5Ffu7$$\"39t!*\\gL'zV#F-$\"3en*R=[a'p&*F bz7$$\"37O'**4*>=+DF-$\"3FqlTthxS*)Fbz7$$\"3'3QAr_4Qc#F-$\"3DKwp8.jV$) Fbz7$$\"3uz67&>5pi#F-$\"3%H\\H_vVOz(Fbz7$$\"3!Q@Ic:$*[o#F-$\"33&Hne&yW AtFbz7$$\"3i.tur\"[8v#F-$\"3*4_#*R#\\a>oFbz7$$\"32F%y.L'y5GF-$\"3#)GDW id#3S'Fbz7$$\"3_!oEY!)fT(GF-$\"3Q:O&H%zC%)fFbz7$$\"3Mn`v0j\"[$HF-$\"3T \\\")HhwG7cFbz7$$\"\"$F*$\"37&fD\"\\'p&R_Fbz-%'COLOURG6&%$RGBG$\"*++++ \"!\")$F*F*F`am-%*LINESTYLEG6#\"\"\"-F$6%7V7$F($!3\\g8K;$H@]*Ffu7$$!3! ******\\2<#pGF-$!3U*46.lmDV*Ffu7$$!3#)***\\7bBav#F-$!3)\\MiI;zTO*Ffu7$ $!36++]K3XFEF-$!3[zg'HjvtF*Ffu7$$!3%)****\\F)H')\\#F-$!3?O0\"o`C!y\"*F fu7$$!3#****\\i3@/P#F-$!3]p(Q))G'el!*Ffu7$$!3;++Dr^b^AF-$!3mY%4*=ckZ*) Ffu7$$!3$****\\7Sw%G@F-$!35E[$z2:)4))Ffu7$$!3*****\\7;)=,?F-$!3'fiAT@a #[')Ffu7$$!3/++DO\"3V(=F-$!3+U`+d*)Ql%)Ffu7$$!3#******\\V'zViUC \"F-$!38m,P#pl%=rFfu7$$!3-++DhkaI6F-$!3E'[:^?K9x'Ffu7$$!3s******\\XF`* *Ffu$!3m4^EMg(RI'Ffu7$$!3u*******>#z2))Ffu$!3Il#ocp,a&eFfu7$$!3S++]7RK vuFfu$!3U)eY4#Rmk_Ffu7$$!3s,+++P'eH'Ffu$!3QPmzu&y=n%Ffu7$$!3q)***\\7*3 =+&Ffu$!3cL#QXY!zNRFfu7$$!3[)***\\PFcpPFfu$!3AM^5H$R09$Ffu7$$!3;)**** \\7VQ[#Ffu$!3S\"*)>=,*R*>#Ffu7$$!32)***\\i6:.8Ffu$!3W5oEwA\"=A\"Ffu7$$ !3Wb+++v`hHFg\\l$!3;sJutk:dHFg\\l7$$\"3]****\\(QIKH\"Ffu$\"3o()**G:qd! 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" }}{PARA 0 "" 0 "" {TEXT -1 37 "It is apparent that the range of g is" }{XPPEDIT 18 0 " ` `(-infinity,-1) union ``(0,infinity)" "6#-%&unionG6$-%!G6$,$%)infinity G!\"\",$\"\"\"F+-F'6$\"\"!F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 22 "Note also that g is a " }{TEXT 261 19 "one-to-one functio n" }{TEXT -1 106 ". This can be deduced from the fact that it is the c omposite function formed from the one-to-one function " }{XPPEDIT 18 0 "x->exp(x)-1" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\",&-%$expG6#F%\" \"\"F/!\"\"F*F*F*" }{TEXT -1 26 " followed by the function " } {XPPEDIT 18 0 "x->1/x" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"*&\"\"\" F,F%!\"\"F*F*F*" }{TEXT -1 28 ", which is also one-to-one. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "To find the inv erse function " }{XPPEDIT 18 0 "g^(-1)" "6#)%\"gG,$\"\"\"!\"\"" } {TEXT -1 24 ", we solve the equation " }{XPPEDIT 18 0 "y = 1/(exp(x)-1 );" "6#/%\"yG*&\"\"\"F&,&-%$expG6#%\"xGF&F&!\"\"F," }{TEXT -1 5 " for \+ " }{TEXT 264 1 "x" }{TEXT -1 13 " in terms of " }{TEXT 265 1 "y" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/(exp(x)-1);" "6#/%\"yG*&\"\"\"F&, &-%$expG6#%\"xGF&F&!\"\"F," }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 17 "is equivalent to " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "1/y = exp(x)-1;" "6#/*&\"\"\"F%%\"yG!\"\",&-%$expG6#%\" xGF%F%F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 33 "which, in tu rn, is equivalent to " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(x) = 1+1/y;" "6#/-%$expG6#%\"xG,&\"\"\"F)*&F)F)%\"yG!\"\"F) " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = ln(1+1/y);" "6#/%\"xG-%#lnG6#,& \"\"\"F)*&F)F)%\"yG!\"\"F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "This means that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g^(-1)" "6#)%\"gG,$\"\"\"!\"\"" }{XPPEDIT 18 0 "``(x) = ln(1+1/y );" "6#/-%!G6#%\"xG-%#lnG6#,&\"\"\"F,*&F,F,%\"yG!\"\"F," }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "g := x -> 1/(exp(x)-1):\nsol ve(g(x)=y,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#*&,&\"\"\"F(% \"yGF(F(F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "In the fo llowing picture the graph of " }{XPPEDIT 18 0 "g(x) = 1/(exp(x)-1);" " 6#/-%\"gG6#%\"xG*&\"\"\"F),&-%$expG6#F'F)F)!\"\"F." }{TEXT -1 13 " is \+ drawn in " }{TEXT 258 3 "red" }{TEXT -1 42 ", while the graph of the i nverse function " }}{PARA 257 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "g^(-1)" "6#)%\"gG,$\"\"\"!\"\"" }{XPPEDIT 18 0 "``(x) = ln(1+1/y);" " 6#/-%!G6#%\"xG-%#lnG6#,&\"\"\"F,*&F,F,%\"yG!\"\"F," }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 12 "is drawn in " }{TEXT 256 4 "blue" }{TEXT -1 2 ". 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