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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 33 "The natural logarithms function  \+
" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, 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 30 "The natural logarithm function" }}{PARA 0 "" 0 "" 
{TEXT -1 22 "The function f where  " }{XPPEDIT 18 0 "f(x)=exp(x)" "6#/
-%\"fG6#%\"xG-%$expG6#F'" }{TEXT -1 85 " is called the exponential fun
ction and the corresponding logarithm functiong where  " }{XPPEDIT 18 
0 "g(x)= log[exp(1)]*x" "6#/-%\"gG6#%\"xG*&&%$logG6#-%$expG6#\"\"\"F/F
'F/" }{TEXT -1 11 " is called " }}{PARA 0 "" 0 "" {TEXT -1 4 "the " }
{TEXT 262 17 "natural logarithm" }{TEXT -1 49 " function and it is usu
ally denoted by ln.  Thus " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "y=ln*x " "6#/%\"yG*&%#lnG\"\"\"%\"xGF'" }{TEXT -1 17 " \+
 exactly when   " }{XPPEDIT 18 0 "exp(y)=x" "6#/-%$expG6#%\"yG%\"xG" }
{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 303 12 "____
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" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "C
urve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve
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 66" "Curve 67" }}{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
}{PARA 0 "" 0 "" {TEXT -1 10 "The point " }{XPPEDIT 18 0 "``(1,exp(1))
" "6#-%!G6$\"\"\"-%$expG6#F&" }{TEXT -1 21 " is on the graph of  " }
{XPPEDIT 18 0 "y=exp(x)" "6#/%\"yG-%$expG6#%\"xG" }{TEXT -1 18 ", whil
e the point " }{XPPEDIT 18 0 "``(exp(1),1)" "6#-%!G6$-%$expG6#\"\"\"F)
" }{TEXT -1 21 " is on the graph of  " }{XPPEDIT 18 0 "y=ln*x" "6#/%\"
yG*&%#lnG\"\"\"%\"xGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 
14 "The graph of  " }{XPPEDIT 18 0 "y=exp(x)" "6#/%\"yG-%$expG6#%\"xG
" }{TEXT -1 48 " has a special property. Zooming in towards the " }
{TEXT 304 1 "y" }{TEXT -1 11 " intercept " }{XPPEDIT 18 0 "``(0,1)" "6
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y=exp(x)" "6#/%\"yG-%$expG6#%\"xG" }{TEXT -1 44 ", the graph nearly co
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F__l$\"#?F__lQ*y~=~x~-~1Fc`n-%%FONTG6$%*HELVETICAG\"\")-%+AXESLABELSG6
%%!GFfbn-F_bn6#%(DEFAULTG-%*AXESTICKSG6$7&/F__l%#-1G/Fa_l%\"1G/Ff[l%\"
2G/F^_l%\"3GF]cn-%%VIEWG6$;$!$R\"F`[lF^`nFicn" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve
 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17
" "Curve 18" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 1 ":" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example
 1" }}{PARA 0 "" 0 "" {TEXT 305 8 "Question" }{TEXT -1 17 ": Simplify \+
 (a)  " }{XPPEDIT 18 0 "( exp(2*x)-1)/(exp(x)-1) " "6#*&,&-%$expG6#*&
\"\"#\"\"\"%\"xGF*F*F*!\"\"F*,&-F&6#F+F*F*F,F," }{TEXT -1 8 "    (b) \+
" }{XPPEDIT 18 0 "(exp(3*x)+1)/(exp(x)+1);" "6#*&,&-%$expG6#*&\"\"$\"
\"\"%\"xGF*F*F*F*F*,&-F&6#F+F*F*F*!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 8 "Solution" }{TEXT 
-1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 10 "(a)  Let  " }{XPPEDIT 18 0 "u
=exp(x)" "6#/%\"uG-%$expG6#%\"xG" }{TEXT -1 7 ". Then " }}{PARA 256 "
" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "(exp(2*x)-1)/(exp(x)-1) = (u^2-
1)/(u-1)" "6#/*&,&-%$expG6#*&\"\"#\"\"\"%\"xGF+F+F+!\"\"F+,&-F'6#F,F+F
+F-F-*&,&*$%\"uGF*F+F+F-F+,&F4F+F+F-F-" }{XPPEDIT 18 0 "``=(u-1)*(u+1)
/(u-1)" "6#/%!G*(,&%\"uG\"\"\"F(!\"\"F(,&F'F(F(F(F(,&F'F(F(F)F)" }
{XPPEDIT 18 0 "``=u+1" "6#/%!G,&%\"uG\"\"\"F'F'" }{TEXT -1 1 " " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=exp(x)+1" "6#/%!G,
&-%$expG6#%\"xG\"\"\"F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 
-1 9 "(b) Let  " }{XPPEDIT 18 0 "u=exp(x)" "6#/%\"uG-%$expG6#%\"xG" }
{TEXT -1 7 ". Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "(exp(3*x)+1)/(exp(x)+1)=(u^3+1)/(u+1)" "6#/*&,&-%$expG6#*&\"\"$\"\"
\"%\"xGF+F+F+F+F+,&-F'6#F,F+F+F+!\"\"*&,&*$%\"uGF*F+F+F+F+,&F4F+F+F+F0
" }{XPPEDIT 18 0 "``=(u+1)*(u^2-u+1)/(u+1)" "6#/%!G*(,&%\"uG\"\"\"F(F(
F(,(*$F'\"\"#F(F'!\"\"F(F(F(,&F'F(F(F(F," }{XPPEDIT 18 0 "``=u^2-u+1" 
"6#/%!G,(*$%\"uG\"\"#\"\"\"F'!\"\"F)F)" }{TEXT -1 1 " " }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=exp(2*x)-exp(x)+1" "6#/%!G,(-
%$expG6#*&\"\"#\"\"\"%\"xGF+F+-F'6#F,!\"\"F+F+" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" 
{TEXT 318 8 "Question" }{TEXT -1 10 ":  Given  " }{XPPEDIT 18 0 "f(x) \+
= 1-exp(-x);" "6#/-%\"fG6#%\"xG,&\"\"\"F)-%$expG6#,$F'!\"\"F." }{TEXT 
-1 10 ",  find  f" }{XPPEDIT 18 0 "``^(-1)" "6#)%!G,$\"\"\"!\"\"" }
{TEXT -1 1 "(" }{TEXT 306 1 "x" }{TEXT -1 35 ")  and sketch the graphs
 of f and f" }{XPPEDIT 18 0 "``^(-1)" "6#)%!G,$\"\"\"!\"\"" }{TEXT -1 
28 " with the same set of axes. " }}{PARA 0 "" 0 "" {TEXT -1 50 "     \+
              State the domain and range of f" }{XPPEDIT 18 0 "``^(-1)
" "6#)%!G,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 258 
8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 40 "To find t
he inverse function of f let   " }{XPPEDIT 18 0 "y = 1-exp(-x);" "6#/%
\"yG,&\"\"\"F&-%$expG6#,$%\"xG!\"\"F," }{TEXT -1 16 ", and solve for \+
" }{TEXT 307 1 "x" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "y = 1-exp(-x);" "6#/%\"yG,&\"\"\"F&-%$expG6#,$%\"xG!
\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "is equivalent t
o " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-x) = 1-y;
" "6#/-%$expG6#,$%\"xG!\"\",&\"\"\"F+%\"yGF)" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 40 "The corresponding logarithm equation is \+
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-x = ln(1-y);" "6
#/,$%\"xG!\"\"-%#lnG6#,&\"\"\"F+%\"yGF&" }{TEXT -1 2 ", " }}{PARA 0 "
" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "x=-ln(1-y)" "6#/%\"xG,$-%#lnG6#,&\"\"\"F*%\"yG!\"\"F," 
}{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }}{PARA 256 "" 0 
"" {TEXT -1 2 " f" }{XPPEDIT 18 0 "``^(-1)" "6#)%!G,$\"\"\"!\"\"" }
{TEXT -1 1 "(" }{TEXT 310 1 "y" }{TEXT -1 1 ")" }{XPPEDIT 18 0 "`` = -
ln(1-y);" "6#/%!G,$-%#lnG6#,&\"\"\"F*%\"yG!\"\"F," }{TEXT -1 3 ".  " }
}{PARA 0 "" 0 "" {TEXT -1 23 "Replacing the variable " }{TEXT 311 1 "y
" }{TEXT -1 4 " by " }{TEXT 312 1 "x" }{TEXT -1 8 " gives: " }}{PARA 
256 "" 0 "" {TEXT -1 2 " f" }{XPPEDIT 18 0 "``^(-1)" "6#)%!G,$\"\"\"!
\"\"" }{TEXT -1 1 "(" }{TEXT 309 1 "x" }{TEXT -1 1 ")" }{XPPEDIT 18 0 
"`` = -ln(1-x);" "6#/%!G,$-%#lnG6#,&\"\"\"F*%\"xG!\"\"F," }{TEXT -1 2 
". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 308 9 "_________" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 39 "In the following picture the graph of  " }{XPPEDIT 18 0 "
y = 1-exp(-x);" "6#/%\"yG,&\"\"\"F&-%$expG6#,$%\"xG!\"\"F," }{TEXT -1 
15 " is plotted in " }{TEXT 267 3 "red" }{TEXT -1 22 ", while the grap
h of  " }{XPPEDIT 18 0 "y = -ln(1-x);" "6#/%\"yG,$-%#lnG6#,&\"\"\"F*%
\"xG!\"\"F," }{TEXT -1 15 " is plotted in " }{TEXT 260 4 "blue" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 268 "" 0 "" 
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45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve
 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu
rve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" }}
{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 14 "The graph of  " }{XPPEDIT 18 0 "y=exp(-x)" "6#/%\"yG-%$ex
pG6#,$%\"xG!\"\"" }{TEXT -1 42 "  is obtained by reflecting the graph \+
of  " }{XPPEDIT 18 0 "y=exp(x)" "6#/%\"yG-%$expG6#%\"xG" }{TEXT -1 9 "
  in the " }{TEXT 313 1 "y" }{TEXT -1 6 " axis." }}{PARA 0 "" 0 "" 
{TEXT -1 14 "The graph of  " }{XPPEDIT 18 0 "y=-exp(-x)" "6#/%\"yG,$-%
$expG6#,$%\"xG!\"\"F+" }{TEXT -1 41 " is obtained by reflecting the gr
aph of  " }{XPPEDIT 18 0 "y=exp(-x)" "6#/%\"yG-%$expG6#,$%\"xG!\"\"" }
{TEXT -1 8 " in the " }{TEXT 314 1 "x" }{TEXT -1 7 " axis (" }{TEXT 
258 8 "dark red" }{TEXT -1 14 " dotted curve)" }}{PARA 0 "" 0 "" 
{TEXT -1 14 "The graph of  " }{XPPEDIT 18 0 "y=1-exp(-x)" "6#/%\"yG,&
\"\"\"F&-%$expG6#,$%\"xG!\"\"F," }{TEXT -1 42 " is obtained by transla
ting the graph of  " }{XPPEDIT 18 0 "y=-exp(-x)" "6#/%\"yG,$-%$expG6#,
$%\"xG!\"\"F+" }{TEXT -1 30 " upwards by 1 unit. The line  " }
{XPPEDIT 18 0 "y = 1;" "6#/%\"yG\"\"\"" }{TEXT -1 28 " is a horizontal
 asymptote. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 14 "The graph of  " }{XPPEDIT 18 0 "y = ln(-x);" "6#/%\"yG-%#lnG6#,
$%\"xG!\"\"" }{TEXT -1 42 "  is obtained by reflecting the graph of  \+
" }{XPPEDIT 18 0 "y = ln*x;" "6#/%\"yG*&%#lnG\"\"\"%\"xGF'" }{TEXT -1 
9 "  in the " }{TEXT 315 1 "y" }{TEXT -1 6 " axis." }}{PARA 0 "" 0 "" 
{TEXT -1 14 "The graph of  " }{XPPEDIT 18 0 "y = -ln(-x);" "6#/%\"yG,$
-%#lnG6#,$%\"xG!\"\"F+" }{TEXT -1 41 " is obtained by reflecting the g
raph of  " }{XPPEDIT 18 0 "y = ln(-x);" "6#/%\"yG-%#lnG6#,$%\"xG!\"\"
" }{TEXT -1 8 " in the " }{TEXT 316 1 "x" }{TEXT -1 7 " axis (" }
{TEXT 264 9 "dark blue" }{TEXT -1 14 " dotted curve)" }}{PARA 0 "" 0 "
" {TEXT -1 14 "The graph of  " }{XPPEDIT 18 0 "y = -ln(-(x-1));" "6#/%
\"yG,$-%#lnG6#,$,&%\"xG\"\"\"F,!\"\"F-F-" }{TEXT -1 12 ", that is,  " 
}{XPPEDIT 18 0 "y=-ln(1-x)" "6#/%\"yG,$-%#lnG6#,&\"\"\"F*%\"xG!\"\"F,
" }{TEXT -1 42 " is obtained by translating the graph of  " }{XPPEDIT 
18 0 "y = -ln(-x);" "6#/%\"yG,$-%#lnG6#,$%\"xG!\"\"F+" }{TEXT -1 24 " \+
to the right by 1 unit." }}{PARA 0 "" 0 "" {TEXT -1 10 "The line  " }
{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 26 " is a vertical asym
ptote. " }}{PARA 0 "" 0 "" {TEXT -1 46 "Note that both curves pass thr
ough the origin." }}{PARA 0 "" 0 "" {TEXT -1 10 "The point " }
{XPPEDIT 18 0 "``(-1,1-exp(1))" "6#-%!G6$,$\"\"\"!\"\",&F'F'-%$expG6#F
'F(" }{TEXT -1 19 "  is on the curve  " }{XPPEDIT 18 0 "y=1-exp(-x)" "
6#/%\"yG,&\"\"\"F&-%$expG6#,$%\"xG!\"\"F," }{TEXT -1 18 "  while the p
oint " }{XPPEDIT 18 0 "``(1-exp(1),-1)" "6#-%!G6$,&\"\"\"F'-%$expG6#F'
!\"\",$F'F+" }{TEXT -1 21 " is on the graph of  " }{XPPEDIT 18 0 "y=-l
n(1-x)" "6#/%\"yG,$-%#lnG6#,&\"\"\"F*%\"xG!\"\"F," }{TEXT -1 3 ".  " }
{XPPEDIT 18 0 "1-exp(1)" "6#,&\"\"\"F$-%$expG6#F$!\"\"" }{TEXT -1 1 " \+
" }{TEXT 317 1 "~" }{TEXT -1 11 " -1.71828. " }}{PARA 0 "" 0 "" {TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 15 "The domain of f" }{XPPEDIT 18 
0 "``^(-1)" "6#)%!G,$\"\"\"!\"\"" }{TEXT -1 63 " is the set of all rea
l numbers that are less than 1, that is, " }{XPPEDIT 18 0 "``(-infinit
y,1)" "6#-%!G6$,$%)infinityG!\"\"\"\"\"" }{TEXT -1 38 ". This is the s
ame as the range of f. " }}{PARA 0 "" 0 "" {TEXT -1 14 "The range of f
" }{XPPEDIT 18 0 "``^(-1)" "6#)%!G,$\"\"\"!\"\"" }{TEXT -1 42 " is the
 set of all real numbers, that is, " }{XPPEDIT 18 0 "``(-infinity,infi
nity);" "6#-%!G6$,$%)infinityG!\"\"F'" }{TEXT -1 39 ". This is the sam
e as the domain of f. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ":" }}}}{PARA 0 "" 0 "" {TEXT -1 2 " \+
 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "
" 0 "" {TEXT -1 18 "Code for pictures " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 19 "Code for pi
ctures  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1280 "f := x -> exp(x):\np1 := plot([x,f(x),x=-4.4..3.4],
thickness=2,color=COLOR(RGB,.93,0,0)):\np2 := plot([f(x),x,x=-4.4..3.4
],thickness=2,color=COLOR(RGB,.3,.1,.86)):\np3 := plot([[[0,1],[1,exp(
1)]]$4],style=point,symbol=[circle$2,diamond,cross],\n    symbolsize=[
15,10$3],color=[black,COLOR(RGB,0,1,0)$3]):\np4 := plot(x,x=-4.4..8.4,
color=COLOR(RGB,.3,.6,.3),linestyle=1,thickness=2):\np5 := plot([[[1,0
],[exp(1),1]]$4],style=point,symbol=[circle$2,diamond,cross],\n    sym
bolsize=[15,10$3],color=[black,COLOR(RGB,1,.5,0)$3]):\np6 := plot([seq
([[i/2,-4.4],[i/2,8.4]],i=-8..16),seq([[-4.4,j/2],[8.4,j/2]],j=-8..16)
],color=COLOR(RGB,.6,.6,.6)):\nt1 := plots[textplot]([[8.39,-.25,`x`],
[.23,8.39,`y`],[6,5.2,`y = x`]]):\nt2 := plots[textplot]([3.2,7,`y = e
`],color=COLOR(RGB,.75,0,0),font=[HELVETICA,8]):\nt3 := plots[textplot
]([3.7,7.2,`x`],color=COLOR(RGB,.75,0,0),font=[HELVETICA,7]):\nt4 := p
lots[textplot]([7,2.4,`y = ln x`],color=COLOR(RGB,0,0,.7),font=[HELVET
ICA,8]):\nplots[display]([p||(1..6),t||(1..4)],view=[-4.39..8.39,-4.39
..8.39],\n  scaling=unconstrained,font=[HELVETICA,8],\n   xtickmarks=[
-4=`-4`,-3=`-3`,-2=`-2`,-1=`-1`,1=`1`,2=`2`,3=`3`,4=`4`,5=`5`,6=`6`,7=
`7`,8=`8`],\n   ytickmarks=[-4=`-4`,-3=`-3`,-2=`-2`,-1=`-1`,1=`1`,2=`2
`,3=`3`,4=`4`,5=`5`,6=`6`,7=`7`,8=`8`],labels=[``,``]);" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 443 "p1 := plot(exp(x),x=-.11..0.13,color=COLOR
(RGB,.93,0,0),thickness=2):\np2 := plot(x+1,x=-.11..0.13,color=COLOR(R
GB,.01,.5,.01),thickness=2):\np3 := plot([seq([[i/100,0.87],[i/100,1.1
29]],i=-11..13),\n seq([[-.11,j/100],[.13,j/100]],j=88..112)],color=CO
LOR(RGB,.6,.6,.6)):\nt1 := plots[textplot]([[0.129,0.88,`x`],[-.005,1.
129,`y`],[6,5.2,`y = x`]]):\nplots[display]([p1,p2,p3,t1],font=[HELVET
ICA,8],view=[-.11..0.129,0.87..1.129],\n labels=[``,``]);" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 871 "f :
= x -> exp(x):\np1 := plot([x,f(x),x=-1.4..1.4],thickness=2,color=COLO
R(RGB,.93,0,0)):\np2 := plot([f(x),x,x=-1.4..1.4],thickness=2,color=CO
LOR(RGB,.3,.1,.86)):\np3 := plot([[[0,1],[1,exp(1)]]$4],style=point,sy
mbol=[circle$2,diamond,cross],\n    symbolsize=[15,10$3],color=[black,
COLOR(RGB,0,1,0)$3]):\np4 := plot([x+1,x,x-1],x=-4.4..8.4,linestyle=[1
,2,1],\n   color=[brown,COLOR(RGB,.1,.1,.1),COLOR(RGB,.1,.3,.1)],thick
ness=1):\np5 := plot([[[1,0],[exp(1),1]]$4],style=point,symbol=[circle
$2,diamond,cross],\n    symbolsize=[15,10$3],color=[black,COLOR(RGB,1,
.5,0)$3]):\nt1 := plots[textplot]([[3.49,-.12,`x`],[-.12,3.49,`y`],[2.
2,2.8,`y = x + 1`],\n   [2.8,2.4,`y = x`],[3.4,2.0,`y = x - 1`]]):\npl
ots[display]([p||(1..5),t1],view=[-1.39..3.49,-1.39..3.49],font=[HELVE
TICA,8],\n   xtickmarks=[-1=`-1`,1=`1`,2=`2`,3=`3`],\n  ytickmarks=[-1
=`-1`,1=`1`,2=`2`,3=`3`],labels=[``,``]);" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1014 "f := x -> 1-exp(
-x):\np1 := plot([x,f(x),x=-3.4..4.4],thickness=2,color=COLOR(RGB,.93,
0,0)):\np2 := plot([f(x),x,x=-3.4..4.4],thickness=2,color=COLOR(RGB,.3
,.1,.86)):\np3 := plot(-exp(-x),x=-3.4..4.4,thickness=1,color=COLOR(RG
B,.4,.1,.1),linestyle=2):\np4 := plot(-ln(-x),x=-3.4..4.4,thickness=1,
color=COLOR(RGB,.1,.1,.4),linestyle=2):\np5 := plot(x,x=-0..4.5,color=
black,linestyle=2):\np6 := plots[implicitplot](\{x=1\},x=-2.3..4,y=-2.
3..4,color=COLOR(RGB,.1,.1,.4),linestyle=3):\np7 := plot(1,x=-2.3..4,c
olor=COLOR(RGB,.4,.1,.1),linestyle=3):\np8 := plot([[[-1,1-exp(1)]]$4]
,style=point,symbol=[circle$2,diamond,cross],\n       symbolsize=[15,1
0$3],color=[black,COLOR(RGB,0,1,0)$3]):\np9 := plot([[[1-exp(1),-1]]$4
],style=point,symbol=[circle$2,diamond,cross],\n    symbolsize=[15,10$
3],color=[black,COLOR(RGB,1,.5,0)$3]):\nt1 := plots[textplot]([[3.49,-
.15,`x`],[-.15,3.49,`y`],[3,2.6,`y = x`]],color=COLOR(RGB,.01,.01,.01)
):\nplots[display]([p||(1..9),t1],font=[HELVETICA,8],labels=[``,``],\n
  view=[-2.49..3.49,-2.49..3.49]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 6 "      " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "7 0 0" 9 
}{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }