Graphs of One-to-One Functions

In the following graphs of one-to-one functions, draw a horizontalline through more than one point on the graph if possible.

Write a rule for determining a one-to-one function from a graph of afunction by drawing a horizontal line through points on the graph.

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1. 2.

Chapter 14Discovery 1

Graphs of One-on-One Functions

Check your rule on the following graph of a function, which is notone-to-one:

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3.

Chapter 14Discovery 1

Equivalent Exponential Functions

1. Graph each exponential function on a decimal window.

. ( ) 2 . ( ) 2

1 1. ( ) . ( )

2 2

a b

c d

x x

x x

f x f x

f x f x

2. Match the graphs of the functions.

Write a rule for determining equivalent functions.

Chapter 14Discovery 2

Effect of the Base a on an Exponential Graph

1. Sketch the graphs of the given exponential functions of the formf(x) = ax, where a > 0, on the same coordinate plane. Use the decimalwindow.

. ( ) 2 . ( ) 5 . ( ) 10

1 1 1. ( ) . ( ) . ( )

2 5 10

a b c

d e f

x x x

x x x

f x f x f x

f x f x f x

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Chapter 14Discovery 3

Effect of the Base a on an Exponential Graph

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Use your graphs to complete the following exercises:2. Determine the domain of each function.3. Determine the range for each function.4. Are all the functions one-to-one?5. Determine the y-intercept of each function.6. Determine the x-intercept of each function.

Choose the correct answer:7. In exercises 1a - 1c, a > 1. The function is increasing/decreasing. The larger the value of a, the steeper/shallower the graph.8. In exercises 1d -1f, 0 < a < 1. The function is increasing/decreasing.The smaller the value of a (as a approaches 0), the steeper/shallowerthe graph.

Chapter 14Discovery 3

Properties of Logarithms

Determine the following logarithms in the form logax:

3 5

3 5

2 2

3 5

3 5

3 5

. . log 1 . log 1

. . log 3 . log 5

. . log 3 . log 5

. . log 1 . log ( 1)

. . log 0 . log 0

1 a b

2 a b

3 a b

4 a b

5 a b

6. In exercise 1, x = 1. The logarithms are ____.7. In exercise 2, x = a, the base. The logarithms are ____.8. In exercise 3, x = a2, the base squared. The logarithms are ____.9. In exercise 4, x = -1. The logarithms are ____.10. In exercise 5, x = 0. The logarithms are ____.

Chapter 14Discovery 4

Product, Quotient, and Power Rulesof Logarithms

Approximate each expression, and compare the results obtained in theleft column with the corresponding results in the right column.

1. Product rule. log(3 4) . log3 log 4

. ln (2 5) . ln 2 ln5

a b

c d

2. Quotient rule3

. log . log3 log 44

2. ln . ln 2 ln5

5

a b

c d

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Chapter 14Discovery 5

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Product, Quotient, and Power Rulesof Logarithms

3. Power rule4

5

. log3 . 4log3

. ln 2 . 5 In 2

a b

c d

Write the following rules of logarithms:4. Product Rule:

5. Quotient Rule:

6. Power Rule:

Chapter 14Discovery 5

Approximate each expression, and compare the results obtained in theleft column with the corresponding results in the right column.

Properties of Exponential and Logarithms Equations

Complete the following statements with “true” or “false”:

1. a. If 2 = 2 is true, the 52 = 52 is ____. b. If 2 = 2 is true, then 122 = 122 is ____. c. If 2 = 2 is true, then e2 = e2 is ____.Write a rule for determining a true equation using exponentials.

2. a. If 2 = 2 is true, then log52 = log52 is ____. b. If 2 = 2 is true, then log122 = log122 is ____. c. If 2 = 2 is true, then ln 2 = ln 2 is ____.Write a rule for determining a true equation using logarithms.

Chapter 14Discovery 6