4.1 – inverse functions copyright © 2012 pearson education, inc. publishing as prentice hall....

4.1 – Inverse Functions

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

Definition: One-to-one Function



Are the listed functions one to one ?

Function B: { (3,12), (4,13), (6,14), (8,1) }

Function A: { (11,14), (12,14) , (16,7), (18,13) }

No (11,14), (12,14)

Yes



For each function, use the graph to determine whether the function is one-to-one.

𝑓 (𝑥 )=𝑥2 𝑓 (𝑥 )=𝑥3



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.



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,

then .

This inverse function is unique and is denoted by and called “f inverse.”

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.

A function must be one-to-one in order to have an inverse function.



Verifying two functions are inverses.

If and then f and g are inverse functions.

𝑓 (𝑥 )= 3𝑥+5

∧𝑔 (𝑥 )=3𝑥

−5

𝑓 (𝑔 (𝑥 ) )= 33𝑥

− 5+5

𝑓 (𝑔 (𝑥 ) )= 33𝑥

𝑓 (𝑔 (𝑥 ) )=3 𝑥3 ¿ 𝑥

𝑔 ( 𝑓 (𝑥 ) )= 33

𝑥+5

−5

𝑔 ( 𝑓 (𝑥 ) )=3 (𝑥+5 )3

− 5

𝑔 ( 𝑓 (𝑥 ) )=𝑥+5−5¿ 𝑥

𝑓 (𝑥 )𝑎𝑛𝑑𝑔 (𝑥 )𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠 .



Finding an Equation of an Inverse Function

𝑓 (𝑥 )=2𝑥+3

1.Replace f(x) by y in the equation describing the function.

2.Interchange x and y. In other words, replace every x by a y and vice versa.

3.Solve for y.

4.Replace y by .

𝑦=2 𝑥+3𝑥=2 𝑦+3𝑥−3=2 𝑦12

(𝑥−3 )=𝑦

𝑓 −1 (𝑥 )=12

(𝑥−3 )



Check the Inverse Function

𝑓 (𝑥 )=2𝑥+3 𝑓 −1 (𝑥 )=12

(𝑥−3 )

𝑓 (10 )=2(10)+3

𝑓 (10 )=23

(10 , 23 )

𝑓 −1 (23 )=12

(23 −3 )

𝑓 −1 (23 )=12

(20 )

𝑓 −1 (23 )=10

(23 , 10 )



Graphing an Inverse Function


4.2 – Exponential Functions

Review: Laws of Exponents

If s, t, a, and b are real numbers where a > 0 and b > 0, then:

𝑎𝑠 ∙𝑎𝑡=𝑎𝑠+𝑡 (𝑎𝑠)𝑡=𝑎𝑠𝑡 (𝑎𝑏)𝑡=𝑎𝑡𝑏𝑡

1𝑡=1 𝑎0=1 𝑎−𝑡= 1𝑎𝑡=( 1

𝑎 )𝑡

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DEFINITION:An exponential function f is given bywhere x is any real number, a > 0, and a ≠ 1. The number a is called the base.

f (x) ax ,

Examples: 12 , , 0.4

2

xxxf x f x f x

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.



𝑦=2𝑥𝑦=2−𝑥 𝑦=𝑒𝑥


𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝐴𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒@ 𝑦=0


𝑡=2009 −1995=14

The amount of money spent (in billions of dollars) by Americans on organic food and beverages since 1995 can be approximated by:

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 )=2.43𝑒0.18 𝑡

𝐴 (14 )=2.43𝑒0.18(14)

𝐴 (14 )=$ 30.2𝑏𝑖𝑙𝑙𝑖𝑜𝑛


Exponential Growth Functions


𝑡=2006 − 1995=11


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 )=2.43𝑒0.18 𝑡

𝐴 (11 )=2.43 𝑒0.18 (11)

𝐴 (11 )=$ 17.6𝑏𝑖𝑙𝑙𝑖𝑜𝑛



𝑡=2020 −1995=25


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 )=2.43𝑒0.18 𝑡

𝐴 (25 )=2.43𝑒0.18(25)

𝐴 (25 )=$ 218.7𝑏𝑖𝑙𝑙𝑖𝑜𝑛


𝐼=𝑃𝑟𝑡𝐼=𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡𝑃=𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑙𝑒 𝑖𝑛𝑣𝑒𝑠𝑡𝑒𝑑𝑟=𝑎𝑛𝑛𝑢𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡𝑟𝑎𝑡𝑒𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑒𝑑𝑎𝑠𝑎𝑑𝑒𝑐𝑖𝑚𝑎𝑙

Simple Interest Formula

Compound Interest Formula

𝐴=𝑃 ∙(1+ 𝑟𝑛 )

𝑛 ∙𝑡

Continuous Compounding Interest Formula

𝐴=𝑃 𝑒𝑟 ∙ 𝑡

𝐴=𝑟𝑒𝑡𝑢𝑟𝑛𝑜𝑛 h𝑡 𝑒𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑙𝑒


𝐼=𝑃𝑟𝑡Example - Simple Interest

Examples - Compound Interest

𝐴=𝑃 ∙(1+ 𝑟𝑛 )

𝑛 ∙𝑡

What is the future value of a $34,100 principle invested at 4% for 3 years

$21,000 is invested at 13.6% compounded quarterly for 4 years. What is the return value?

The amount of $12,700 is invested at 8.8% compounded semiannually for 1 year. What is the future value?

𝐼=(34100 )(.04)(3)𝐼=$ 4092.00

𝐹𝑢𝑡𝑢𝑟𝑒𝑉𝑎𝑙𝑢𝑒=34100+4092.00𝐹𝑢𝑡𝑢𝑟𝑒𝑉𝑎𝑙𝑢𝑒=$ 38,192.00

𝐴=12700 ∙(1+ 0.0882 )

2∙ 1

𝐴=$ 13,842.19

𝐴=21000∙ (1+ 0.1364 )

4 ∙ 4

𝐴=$ 35,854.85


Examples - Compound Interest

𝐴=𝑃 ∙(1+ 𝑟𝑛 )

𝑛 ∙𝑡

Example - Continuous Compounding Interest

𝐴=𝑃 𝑒𝑟 ∙ 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.

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+ 0.0912 )

12∙ 60

𝐴=$ 867,959.49

𝐴=500𝑒0.10 ∙ 5 𝐴=$ 824.36


4.1 – inverse functions copyright © 2012 pearson education, inc. publishing as prentice hall....

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