1. Section 4.1 Exponential Functions

3. Recall. Domain and Range of a Relation
Relation: a set of ordered pairs
Example. R = {(-3, 3), (0,-7), (1, -5), (2, 4)}
Domain: the set of all x-coordinates from the ordered pairs
Example. Dom(R) = {-3, 0, 1, 2}
Range: the set of all y-coordinates from the ordered pairs
Example. Ran(R) = {-7, -5, 3, 4}
Domain Range -3 1 2 -7 -5 3 4

4. Exponential Functions
Definition: An exponential function is a function of the form 𝑓 𝑥 = 𝑏 𝑥 or 𝑦= 𝑏 𝑥 , where the base b is any positive constant not equal to 1, and the independent variable x is a real number in the exponent.
Examples
𝑓 𝑥 = 2 𝑥
𝑔 𝑥 = 𝑥+1
Nonexamples
𝐹 𝑥 = 𝑥 2
𝐺 𝑥 = 1 𝑥
𝐻 𝑥 = 𝑥 𝑥
𝐽 𝑥 = (−3) 𝑥

5. Types of Exponential Functions
Exponential Growth
Exponential Decay
b > 1
0 < b < 1

6. Example 1. Evaluating an Exponential Function
Observe Figure 4.1, which is modeled by the exponential function 𝑓 𝑥 = 42.2 (1.56) 𝑥 . What is the average amount spent, to the nearest dollar, after four hours?

7. Graphing Exponential Functions

8. Recall. Exponential Expressions where x is a Rational Number
𝑏 1.7 = 𝑏 = 10 𝑏 17
𝑏 1.73 = 𝑏 = 100 𝑏 173
Observe that the exponent can be any real number, which includes irrational numbers in the domain.
As a result, exponential functions have no holes, or points of discontinuity, at the irrational values of x.

9. Example 2. Graphing an Exponential Function
First, create a table of coordinates: x
𝑓 𝑥 = 2 𝑥 -3 -2 -1 1 2 3
Asymptote?

10. Example 3. Graphing an Exponential Function
Graph g 𝑥 = ( ) 𝑥
First, create a table of coordinates: x
𝑔 𝑥 = ( ) 𝑥 -3 -2 -1 1 2 3
Asymptote?

11. 𝑓 𝑥 = 2 𝑥
𝑔 𝑥 = ( ) 𝑥

12. Characteristics of Exponential Functions of the form 𝑓 𝑥 = 𝑏 𝑥
Dom(f(x)) = (-∞, ∞) and Ran(f(x)) = (0, ∞)
The graphs of all exponential functions of the form 𝑓 𝑥 = 𝑏 𝑥 pass through the point (0, 1) because 𝑓 0 = 𝑏 0 =1. The is y-intercept is 1; however there is no x-intercept.
If b > 1, 𝑓 𝑥 = 𝑏 𝑥 is an increasing function. The greater the value or b, of the steeper the increase.
If 0 < b < 1, 𝑓 𝑥 = 𝑏 𝑥 is a decreasing function. The smaller the value of b, the steeper the decrease.
𝑓 𝑥 = 𝑏 𝑥 is one-to-one and has an inverse that is a function.
The graph of 𝑓 𝑥 = 𝑏 𝑥 approaches, but does not touch, the x-axis. The x-axis, or y = 0, is a horizontal asymptote.

13. Recall. Inverse of a Function
Definition: Let f and g be two functions such that f(g(x)) = x, for every x in Dom(g) and g(f(x)) = x for every x in Dom(f). Then g is the inverse of the function f.
Consider the function 𝑓 −1 , which is the inverse of the function f. So, Dom( 𝑓 −1 ) = Ran(f) and Ran( 𝑓 −1 ) = Dom(f)

14. Transformations of Exponential Functions

15. Table 4.1 Transformations Involving Exponential Functions

16. Example 4. Transformations Involving Exponential Functions
Use the graph of 𝑓 𝑥 = 3 𝑥 to obtain the graph of g 𝑥 = 3 𝑥+1
First, create a table of coordinates: x
𝑓 𝑥 = 3 𝑥 -2 -1 1 2

17. Example 5. Transformations Involving Exponential Functions
Use the graph of 𝑓 𝑥 = 2 𝑥 to obtain the graph of g 𝑥 = 2 𝑥 −3
First, create a table of coordinates: x
𝑓 𝑥 = 2 𝑥 -2 -1 1 2

18. Page 421 # 30
Begin by graphing 𝑓 𝑥 = 2 𝑥 . Then use transformations of this graph to graph ℎ 𝑥 = 2 𝑥+2 −1. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each function’s domain and range. If applicable, use a graphing utility to confirm your hand- drawn graphs.

19. 𝑓 𝑥 = 3 𝑥 𝑔 𝑥 = 3 𝑥−1 ℎ 𝑥 = 3 𝑥 −1 𝐹 𝑥 = −3 𝑥 𝑓 𝑥 = 3 −𝑥 𝐻 𝑥 = −3 −𝑥
The graphs of an exponential function are given. Select the function for each graph from the following options:
𝑓 𝑥 = 3 𝑥 𝑔 𝑥 = 3 𝑥−1 ℎ 𝑥 = 3 𝑥 −1 𝐹 𝑥 = −3 𝑥 𝑓 𝑥 = 3 −𝑥 𝐻 𝑥 = −3 −𝑥
Page 420 # 19
Page 420 # 20

20. The Natural Base e
e is an irrational number, which appears as the base in many applied exponential functions
Defined as the value that (1+ 1 𝑛 ) 𝑛 approaches as n gets infinitely larger
𝑒≈2.72 is called the natural base
𝑓 𝑥 = 𝑒 𝑥 is called the natural exponential function

21. Example 6. Grey Wolf Population
The exponential function 𝑓 𝑥 = 1.26 𝑒 0.247𝑥 models the grey wolf population of the Northern Rocky Mountains, f(x), x years after If the wolf is not removed from the endangered species list and trends shown in Figure 4.6 continue, project its population in the recovery area in 2010.

22. Compounding Interest
Compound interest: the interest computed on an original investment plus any accumulated interest.
Principal: a sum of money, denoted by P
The accumulated value A of an investment at an annual percent rate r compounded once per year is given by 𝐴=𝑃+𝑃𝑟=𝑃(1+𝑟)

23. Compounding Schedules
Annually: once per year
Semiannually: twice per year (period – 6 months)
Quarterly: four times per year (period – 3 months)
Monthly
Daily
In general, when compound interest is paid n times per year, we say that there are n compounding periods per year
Accumulated value 𝐴=𝑃(1+ 𝑟 𝑛 ) 𝑛𝑡
Time, in years, is denoted by t
Continuous compounding – number of compounding periods increases infinitely
Accumulated value 𝐴=𝑃 𝑒 𝑟𝑡

24. Example 7. Choosing between Investments
You decide to invest $8000 for 6 years and there are two account options.
The first pays 7% per year, compounded monthly. The second pays 6.85% per year, compounded continuously.
Which is the better investment?

25. Page 421 # 53
Find the accumulated value of an investment of $10,000 for 5 years at an interest rate of 5.5% if the money is
compounded semiannually
compounded quarterly;
compounded monthly;
compounded continuously.
Use the compound interest formulas 𝐴=𝑃(1+ 𝑟 𝑛 ) 𝑛𝑡 and 𝐴=𝑃 𝑒 𝑟𝑡 . Round answers to the nearest cent.

26. The formula 𝑆=𝐶(1+𝑟 ) 𝑡 models inflation, where C = the value today, r = the annual inflation rate, and S = inflated value t years from now.
Page 422 # 67
If the inflation rate is 6%, how much will a house now worth $465,000 be worth in 10 years?
Page 422 # 68
If the inflation rate is 3%, how much will a house now worth $510,000 be worth in 5 years?

27. Exit Slip
The graphs labeled (a)–(d) in the figure represent 𝑦 =3 𝑥 , 𝑦 =5 𝑥 , 𝑦 =( ) 𝑥 and 𝑦 =( ) 𝑥 but not necessarily in that order. Which is which? Explain.