1. Definition: One-to-one Function
4.1 – Inverse Functions
Definition: One-to-one Function
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2. Are the listed functions one to one ?
4.1 – Inverse Functions
Are the listed functions one to one ?
Function A: { (11,14), (12,14) , (16,7), (18,13) } No
(11,14), (12,14)
Function B: { (3,12), (4,13), (6,14), (8,1) }
Yes
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3. 4.1 – Inverse Functions
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4. 4.1 – Inverse Functions
For each function, use the graph to determine whether the function is one-to-one.
𝑓 𝑥 = 𝑥 2
𝑓 𝑥 = 𝑥 3
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5. 4.1 – Inverse Functions
A function that is increasing on an interval I is a one-to-one function in I.
A function that is decreasing on an interval I is a one-to-one function on I.
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6. Inverse function definition:
4.1 – Inverse Functions
Inverse function definition:
An inverse function is a function that undoes the action of the another function.
A function g is the inverse of a function f if whenever,
𝑦=𝑓(𝑥) then 𝑥=𝑓(𝑦).
In other words, the domain of the original function is the range of the inverse function and the range of the original function is the domain of the inverse function.
This inverse function is unique and is denoted by  𝑓 −1 𝑥  and called “f inverse.”
A function must be one-to-one in order to have an inverse function.
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7. Verifying two functions are inverses.
4.1 – Inverse Functions
Verifying two functions are inverses.
If 𝑓∘𝑔 𝑥 =𝑥 and 𝑔∘𝑓 𝑥 =𝑥, then f and g are inverse functions.
𝑓 𝑥 = 3 𝑥+5 and 𝑔 𝑥 = 3 𝑥 −5
𝑓∘𝑔 𝑥 =𝑓 𝑔 𝑥 =𝑥
𝑔∘𝑓 𝑥 =𝑔 𝑓 𝑥 =𝑥
𝑓 𝑔 𝑥 = 3 3 𝑥 −5+5
𝑔 𝑓 𝑥 = 3 3 𝑥+5 −5
𝑓 𝑔 𝑥 = 3 3 𝑥
𝑔 𝑓 𝑥 = 3 𝑥+5 3 −5
𝑓 𝑔 𝑥 = 3𝑥 3
𝑔 𝑓 𝑥 =𝑥+5−5 =𝑥 =𝑥
𝑓 𝑥 𝑎𝑛𝑑 𝑔 𝑥 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠.
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8. Finding an Equation of an Inverse Function
4.1 – Inverse Functions
Finding an Equation of an Inverse Function
Replace f(x) by y in the equation describing the function.
Interchange x and y. In other words, replace every x by a y and vice versa.
Solve for y.
Replace y by 𝑓 −1 𝑥 .
𝑓 −1 𝑥 = 1 2 𝑥−3
𝑓 𝑥 =2𝑥+3
𝑦=2𝑥+3
𝑥=2𝑦+3
𝑥−3=2𝑦
1 2 𝑥−3 =𝑦
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

9. Check the Inverse Function
4.1 – Inverse Functions
Check the Inverse Function
𝑓 −1 𝑥 = 1 2 𝑥−3
𝑓 𝑥 =2𝑥+3
𝑓 −1 23 = −3
𝑓 10 =2(10)+3
𝑓 10 =23
𝑓 −1 23 =
10, 23
𝑓 −1 23 =10
23, 10
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10. Graphing an Inverse Function
4.1 – Inverse Functions
Graphing an Inverse Function
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11. 4.1 – Inverse Functions
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

12. 4.2 – Exponential Functions
Review: Laws of Exponents
If s, t, a, and b are real numbers where a > 0 and b > 0, then:
𝑎 𝑠 ∙ 𝑎 𝑡 = 𝑎 𝑠+𝑡
( 𝑎 𝑠 ) 𝑡 = 𝑎 𝑠𝑡
(𝑎𝑏) 𝑡 = 𝑎 𝑡 𝑏 𝑡
𝑎 −𝑡 = 1 𝑎 𝑡 = 1 𝑎 𝑡
1 𝑡 =1
𝑎 0 =1
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

13. 4.2 – Exponential Functions
DEFINITION:
An exponential function f is given by
where x is any real number, a > 0, and a ≠ 1. The number a is called the base.
Examples:
Examples of problems involving the use of exponential functions:
growth or decay,
compound interest,
the statistical "bell curve,“
the shape of a hanging cable (or the Gateway Arch in St. Louis),
some problems of probability,
some counting problems,
the study of the distribution of prime numbers.
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14. 4.2 – Exponential Functions
𝑦= 𝑒 𝑥
𝑦= 2 −𝑥
𝑦= 2 𝑥
𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑦=0
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15. 4.2 – Exponential Functions
Exponential Growth Functions
The amount of money spent (in billions of dollars) by Americans on organic food and beverages since 1995 can be approximated by:
𝐴 𝑡 =2.43 𝑒 0.18𝑡 .
The variable t represents the number of year after 1995.
a) How much was spent in 2009?
b) How much was spent in 2006?
𝐴 𝑡 =2.43 𝑒 0.18𝑡
𝑡=2009−1995=14
𝐴 14 =2.43 𝑒 0.18(14)
𝐴 14 =$30.2 𝑏𝑖𝑙𝑙𝑖𝑜𝑛
2012 Pearson Education, Inc. All rights reserved

16. 4.2 – Exponential Functions
The amount of money spent (in billions of dollars) by Americans on organic food and beverages since 1995 can be approximated by:
𝐴 𝑡 =2.43 𝑒 0.18𝑡 .
The variable t represents the number of year after 1995.
a) How much was spent in 2009?
b) How much was spent in 2006?
𝐴 𝑡 =2.43 𝑒 0.18𝑡
𝑡=2006−1995=11
𝐴 11 =2.43 𝑒 0.18(11)
𝐴 11 =$17.6 𝑏𝑖𝑙𝑙𝑖𝑜𝑛
2012 Pearson Education, Inc. All rights reserved

17. 4.2 – Exponential Functions
The amount of money spent (in billions of dollars) by Americans on organic food and beverages since 1995 can be approximated by:
𝐴 𝑡 =2.43 𝑒 0.18𝑡 .
The variable t represents the number of year after 1995.
c) Estimate how much will be spent in 2020? Does the answer seem reasonable?
𝐴 𝑡 =2.43 𝑒 0.18𝑡
𝑡=2020−1995=25
𝐴 25 =2.43 𝑒 0.18(25)
𝐴 25 =$218.7 𝑏𝑖𝑙𝑙𝑖𝑜𝑛
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18. 4.2 – Exponential Functions
Simple Interest Formula
𝐼=𝑃𝑟𝑡
𝐼=𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡
𝑃=𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑙𝑒 𝑖𝑛𝑣𝑒𝑠𝑡𝑒𝑑
𝑟=𝑎𝑛𝑛𝑢𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑒𝑑 𝑎𝑠 𝑎 𝑑𝑒𝑐𝑖𝑚𝑎𝑙
𝑡=𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡𝑖𝑚𝑒 𝑖𝑛 𝑦𝑒𝑎𝑟𝑠
Compound Interest Formula
𝐴=𝑃∙ 1+ 𝑟 𝑛 𝑛∙𝑡
𝐴=𝑟𝑒𝑡𝑢𝑟𝑛 𝑜𝑛 𝑡ℎ𝑒 𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑙𝑒
𝑛=𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑖𝑛 𝑎 𝑦𝑒𝑎𝑟 𝑖𝑛 𝑤ℎ𝑖𝑐ℎ 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑖𝑠 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑
Continuous Compounding Interest Formula
𝐴=𝑃 𝑒 𝑟∙𝑡

19. Example - Simple Interest Examples - Compound Interest
4.2 – Exponential Functions
Example - Simple Interest
𝐼=𝑃𝑟𝑡
What is the future value of a $34,100 principle invested at 4% for 3 years
𝐼= (.04)(3)
𝐹𝑢𝑡𝑢𝑟𝑒 𝑉𝑎𝑙𝑢𝑒=
𝐼=$
𝐹𝑢𝑡𝑢𝑟𝑒 𝑉𝑎𝑙𝑢𝑒=$38,192.00
Examples - Compound Interest
𝐴=𝑃∙ 1+ 𝑟 𝑛 𝑛∙𝑡
The amount of $12,700 is invested at 8.8% compounded semiannually for 1 year. What is the future value?
𝐴=12700∙ ∙1
𝐴=$13,842.19
$21,000 is invested at 13.6% compounded quarterly for 4 years. What is the return value?
𝐴=21000∙ ∙4
𝐴=$35,854.85

20. Examples - Compound Interest Example - Continuous Compounding Interest
4.2 – Exponential Functions
Examples - Compound Interest
𝐴=𝑃∙ 1+ 𝑟 𝑛 𝑛∙𝑡
How much money will you have if you invest $4000 in a bank for sixty years at an annual interest rate of 9%, compounded monthly?
𝐴=4000∙ ∙60
𝐴=$867,959.49
Example - Continuous Compounding Interest
𝐴=𝑃 𝑒 𝑟∙𝑡
If you invest $500 at an annual interest rate of 10% compounded continuously, calculate the final amount you will have in the account after five years.
𝐴=500 𝑒 0.10∙5
𝐴=$824.36