Definition: One-to-one Function 4.1 – Inverse Functions Definition: One-to-one Function Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

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 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

4.1 – Inverse Functions Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

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 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

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. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

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. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

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. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

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.

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 = 1 2 23−3 𝑓 10 =2(10)+3 𝑓 10 =23 𝑓 −1 23 = 1 2 20 10, 23 𝑓 −1 23 =10 23, 10 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Graphing an Inverse Function 4.1 – Inverse Functions Graphing an Inverse Function Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

4.1 – Inverse Functions Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

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.

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. 2012 Pearson Education, Inc. All rights reserved

4.2 – Exponential Functions 𝑦= 𝑒 𝑥 𝑦= 2 −𝑥 𝑦= 2 𝑥 𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝐴𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒 @ 𝑦=0 2012 Pearson Education, Inc. All rights reserved

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

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

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 2012 Pearson Education, Inc. All rights reserved

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

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= 34100 (.04)(3) 𝐹𝑢𝑡𝑢𝑟𝑒 𝑉𝑎𝑙𝑢𝑒=34100+4092.00 𝐼=$4092.00 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+ 0.088 2 2∙1 𝐴=$13,842.19 $21,000 is invested at 13.6% compounded quarterly for 4 years. What is the return value? 𝐴=21000∙ 1+ 0.136 4 4∙4 𝐴=$35,854.85

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∙ 1+ 0.09 12 12∙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