Exponential Functions. Christopher Thomas

Transcription

1 Mathematics Learning Centre Eponential Functions Christopher Thomas c 1998 Universit of Sdne

2 Mathematics Learning Centre, Universit of Sdne 1 1 Eponential Functions 1.1 The functions =2 and =2 From our work on eponents we know that, if b is a positive number, we can make sense of the epression b for all real numbers. Itturns out that functions of the tpe = f() =b, where b is a positive number, are of great importance in mathematics and in all branches of the sciences. To get an indication of how these functions behave we have graphed the function f() =2 in Figure 1. 1 Figure 1: Graph of the function f() =2 You should be aware of several important features of this graph. The function f() =2 is alwas positive (the graph of the function never cuts the - ais), although the value of the function gets ver close to zero for values of ver large negative (ie a long wa to the left along the -ais). For eample, when = wehave 2 = The function 2 increases ver rapidl for large values of. From the rules of eponents, we know that 2 +1 =2 2.Inwords, the value of 2 doubles if is increased b 1. The graph of =2 intercepts the -ais at =1. We should epect this because we know from the rules of eponents that 2 0 =1. Figure 2 displas the graph of the function f() =2. How is the graph of =2 related to the graph of =2? Well, if we set =1then 2 =2 1 = 1, which is the value which would have been obtained b setting = 1 in 2 the function =2.Inthe same wa, if we set = 7 inthe function =2 then we obtain the same value as we would b setting =7in the function =2. Proceeding like this we see that the graph of the function =2 is the reflection in the -ais of the graph of =2. Compare Figure 1 with Figure 2. From the rules of eponents, it follows that 2 =(2 1 ) =( 1 2 ). The function =2 is the same as the function =( 1 2 ), and so 2 (+1) =( 1 2 )+1 = 1 2 (1 2 ) =( 1 2 ) 2.

3 Mathematics Learning Centre, Universit of Sdne 2 1 Figure 2: Graph of the function =2 In words, the value of the function =2 is decreased b a factor of 1 2 b 1. if is increased 1.2 The functions = b and = b An function of the form = b where b>0 and b 1behaves like one of the functions =2 or =( 1 2 ) =2. If b>1 then the function = b is increasing and behaves like =2. If b<1 then the function is decreasing and behaves like =( 1 2 ) =2. If b =1then =1 =1for all. Notice that regardless of the value of b, providing alwas that b>0, the function = b intercepts the -ais at =1. This is because b 0 =1for all numbers b. Figure 3 shows the graphs of the functions = b for various values of b. = 1 0 = e 1 - = 1. = (2/3) = 1. = 1 = 1 Figure 3: Graphs of = b for various values of b.

4 Mathematics Learning Centre, Universit of Sdne The functions = e and = e There is a number called e which has a special importance in mathematics. Like the number π, the number e is an irrational number, which is equivalent to saing that it has a non-terminating, non-repeating decimal representation. In other words we can never write down eactl what e is. To decimal places it is equal to , but this is just an approimation of the correct value. Unless ou reall need to write down an approimate value for e it is more convenient and accurate to leave the smbol e in epressions involving this number. For eample, it is preferable to write 2e rather than or In mathematics the functions e and e are particularl important. Because of this we have graphed them in Figure 4. You can see how similar these functions are to the other eponential functions. The function = e is often referred to as the eponential function, and is even given another special smbol, ep, so that ep() =e and ep( ) =e. = e 1 = e Figure 4: Graphs of e and e. 1.4 Summar Functions of the form f() =b, where b>0 and b 1are called eponential functions. If b<1 then b is a decreasing function, and if b>1 then b is an increasing function. The function b is equal to the function ( 1 b ). The number e and the functions e and e are of special importance in mathematics. The function e is often given the special name ep, so that ep() =e and ep( ) =e.

5 Mathematics Learning Centre, Universit of Sdne 4 1. Eercises 1. Make a careful sketch of the graphs of the functions =2. and =. Indicate where (if at all) these functions intercept the aes. 2. Which of the following functions are increasing and which are decreasing? You should be able to decide without graphing the functions or substituting an values, though ou ma do so if ou wish. a. f() =2.7 b. f() =( ) c. f() =3 d. f() = Which of the following functions are increasing and which are decreasing? If ou have understood this section full ou will be able to answer this question without graphing the functions or substituting an values. a. f() =( 3 ) b. f() =( 3 ) c. f() =( 3 ) d. f() =( 3 ) 4. Sketch the graphs of the functions f() =3 and f() =3.Onthe same diagrams mark in roughl the graphs of f() =2.9 and It is true that e Tr it for ourself on a calculator if ou have one. How do ou think the functions =3 and = e compare? Wh? If ou cannot solve this otherwise, ou might like to tr substituting in a few numbers for in both of the functions and comparing the values.

6 Mathematics Learning Centre, Universit of Sdne 1.6 Solutions to eercises 1. The graphs of =2. and = appear below. In both cases the graph intercepts the -ais at =1. In neither case does the graph intercept the -ais, though the graph does get etremel close to the -ais in both cases. 1 Figure : Graph of =2.. 1 Figure 6: Graph of =. 2. Remember that the function f() =b is increasing if b>1 and is decreasing if b<1. a. f() =2.7 is increasing since 2.7 > 1. b. f() =( ) =2.7,sothis function is also increasing. c. f() =3 =( 1 3 ) is decreasing since 1 < 1. 3 d. f() =0.22 is decreasing since 0.22 < 1.

7 Mathematics Learning Centre, Universit of Sdne 6 3. Again, remember that the function f() =b is increasing if b>1 and is decreasing if b<1. a. f() =( 3 ) is increasing because > 1. 3 b. f() =( 3 ) =( 3 ) is decreasing because 3 < 1. c. f() =( 3 ) =( 3 ) is increasing because > 1. 3 d. f() =( 3 ) is decreasing because 3 < The graphs are drawn in Figures 21 and 22 below. Notice that the graph of f() = 2.9 is ver close to the graph of f() =3, and similarl for the other pair of graphs. 40 = 3 = Figure 7: Graphs of =3 and =2.9. = 3-40 = Figure 8: Graphs of =3 and =2.9.. On m calculator I get e = Now e = (e ) 3 Thus the functions 3 and e agree ver closel with each other.