Exponential Functions In this chapter, a will always be a positive number. For any positive number a>0, there is a function f : R (0, )called an exponential function that is defined as f(x)=ax. ! 1 x 4 x For example, f(x)=3 is an exponential function, and g(x)=(17) is an exponential function. There is a big di↵erence between an exponential function and a polynomial. The function p(x)=x3 is a polynomial. Here the “variable”, x, is being raised to some constant power. The function f(x)=3x is an exponential function; the variable is the exponent. Rules for exponential functions Here are some algebra rules for exponential functions that will be explained in class. If n N, then an is the product of na’s. For example, 34 =3 3 3 3=81 2 · · · a0 =1 If n, m N, then 2 n m m n a m = pan =(pa) x 1 a− = ax The rules above were designed so that the following most important rule of exponential functions holds: 178 axay = ax+y Another variant of the important rule above is x a x y = a − ay And there is also the following slightly related rule (ax)y = axy Examples. 1 2 4 2 = p4=2 • 2 6 4 2+6 4 0 7− 7 7− =7− − =7 =1 • · · 3 1 1 10− = 3 = • 10 1000 156 6 5 1 5 =15− =15 =15 • 15 (25)2 =210 =1024 • 20 1 2 (3 ) 10 =3 =9 • 2 3 1 1 1 1 8− = 2 = 3 2 = 22 = 4 • (8)3 ( p8) ************* 179 The base of an exponential function If f(x)=ax, then we call a the base of the exponential function. The base must always be positive. Base 1 If f(x)isanexponentialfunctionwhosebaseequals1–thatisiff(x)=1x –thenforn, m N we have 2 n n m m f =1m = p1n = p1=1 m In fact, for any real number⇣ ⌘x,1x =1,sof(x)=1x is the same function as the constant function f(x)=1.Forthisreason,weusuallydon’ttalkmuch about the exponential function whose base equals 1. ************* Graphs of exponential functions It’s really important that you know the general shape of the graph of an exponential function. There are two options: either the base is greater than 1, or the base is less than 1 (but still positive). Base greater than 1. If a is greater than 1, then the graph of f(x)=ax grows taller as it moves to the right. To see this, let n Z.Weknow that 1 <a, and we know from our rules of inequalities that2 we can multiply both sides of this inequality by a positive number. The positive number we’ll multiply by is an, so that we’ll have an(1) <ana Because an(1) = an and ana = an+1, the inequality above is the same as an <an+1 Because the last inequality we found is true for any n Z,weactuallyhave an entire string of inequalities: 2 3 2 1 0 1 2 3 <a− <a− <a− <a <a <a <a < ··· ··· Keeping in mind that ax is positive for any number x, and that a0 =1, we now have a pretty good idea of what the graph of f(x)=ax looks like if a>1. The y-intercept is at 1; when moving to the right, the graph grows 180 tallertaller and and taller; taller; and and when when moving moving to to the the left, left, the the graph graph becomes becomes shorter shorter and and shorter,taller and shrinking taller; and towards, when moving but never to thetouching, left, the the graphx-axis. becomes shorter and shorter,shorter, shrinking shrinking towards, towards, but but never never touching, touching, the thex-axis.x-axis. Not only does the graph grow bigger as it moves to the right, but it gets Not only does the graph grow bigger as it moves to the right, but it gets bigNot in a only hurry. does For the example, graph grow if we bigger look at as the it moves exponential to the function right, but whose it gets basebig in is a 2, hurry. then For example, if we look at the exponential function whose basebig is in 2, a then hurry. For example, if we look at the exponential function whose base is 2, thenf(64) = 264 =18, 446, 744, 073, 709, 525, 000 f(64) = 264 =18, 446, 744, 073, 709, 525, 000 And 2 isn’t evenf a(64) very = big 264 number=18, 446 to, 744 be, 073 using, 709 for, 525 a, base000 (any positive numberAnd 2 isn’t can be even a base, a very and big plenty number of numbers to be using are much, for a base much (any bigger positive than And 2 isn’t even a very big number to be using for a base (any positive 2).number The bigger can be the a base, base of and an plenty exponential of numbers function, are the much, faster much its graph bigger grows than number can be a base and plenty of numbers are much, much bigger than 2). as2). it The moves bigger to the the right. base of an exponential function, the faster its graph grows asThe it moves bigger to the the base right. of an exponential function,x the faster it grows. Moving to the left, the graph of f(x)=a growsx small very quickly if a>1. Moving to the left, the graph of f(x)=xa grows small very quickly if a>1. AgainMoving if we to look the left, at the the exponential graph of f(x function)=a grows whose small base very is 2, quickly then if a>1. AgainAgain if if we we look look at at the the exponential exponential function function whose whose base base is is 2, 2, then then 10 1 1 f( 10) = 2− 10= 1= 1 10 101 1 f−( 10) = 2− =2 10 =1024 f( −10) = 2− = 102 = 1024 The bigger the base, the faster− the graph2 of an exponential1024 function shrinks The bigger the base, the faster the graph of an exponential function shrinks asThe it biggermoves theto the base, left. the faster the graph of an exponential function shrinks asas it we moves move to to the the left. left. 181 4 145 Base less than 1 (but still positive). If a is positive and less than 1, Base less than 1 (but still positive). If a is positiven and+1 lessn than 1, thenBase we less we can than show 1 (but from still our positive).rules of inequalitiesIf a is that positivean+1 and<a lessn for than any 1, then we we can show from our rules of inequalities that a n+1<a forn any nthenZ. we That we means can show that from our rules of inequalities that a <a for any n ⌅Z. That means that 2n Z. That means that3 2 1 0 1 2 3 ⌅ >a−3 >a−2 >a−1 >a0 >a1 >a2 >a3 > ···>a− >a3 − >a2 − >a1 >a0 >a1 >a2 >3 ··· ··· >a− >ax − >a− >a >a >a >a >··· So the graph of···f(x)=ax when the base is smaller than 1··· slopes down as SoSo the the graph graph of off(fx()=x)=a awhenx when the the base base is is smaller smaller than than 1 1slopes slopes down down as as itit moves moves to to the the right, right, but but it it is is always always positive. positive. As As it it moves moves to to the the left, left, the the graphit moves grows to tall the very right, quickly. but it is always positive. As it moves to the left, the graphgraph grows grows tall tall very very quickly. quickly. One-to-oneOne-to-one and and onto onto One-to-oneRecall that an exponential and onto function f : (0, ) has as its domain the Recall that an exponential function f : RR (0, ) has as its domain the set Recalland that has as an its exponential target the function set (0, f).: R!⇥ (01⇤, ) has as its domain the set RRand has as its target the set (0, ). ⇥ ⇤ set R and has as its target the set (0x ⇤, ). WeWe see see from from the the graph graph of offf((xx)=)=aax,1, if if either eithera>a>1or01or0<a<<a<1,1, that thatff((xx)) We see from the graph of f(x)=ax,⇤ if either a>1or0<a<1, that f(x) isis one-to-one one-to-one and and onto. onto. Remember Remember that that to to check check if if ff((xx)isone-to-one,we)isone-to-one,we is one-to-one and onto. Remember that to check if f(x)isone-to-one,we cancan use use the the horizontal horizontal line line test test (which (whichff((xx)passes).Tocheckwhattherange)passes).Tocheckwhattherange can use the horizontal line test (which f(x)passes).Tocheckwhattherange ofofff((xx)) is, is, we we think think of of compressing compressing the the graph graph of offf((xx)ontothe)ontotheyy-axis.-axis. If If we we of f(x) is, we think of compressing the graph of f(x)ontothey-axis. If we diddid that, that, we we would would see see that that the the range range of offf((xx)isthesetofpositivenumbers,)isthesetofpositivenumbers, (0did, that,). Since we the would range see and that target the range of f( ofx)arethesameset,f(x)isthesetofpositivenumbers,f(x)isonto.