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Exponential and Logarithmic Functions Exponential Functions and Their Applications Section 4.1 Copyright © 2013, 2009, 2005, 2002 Pearson Education, Inc.

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Page 1: Exponential and Logarithmic Functions · Definition: Exponential Function An exponential function with base a is a function of the form f(x) = ax, where a and x are real numbers such

Exponential and Logarithmic Functions

Exponential Functions and Their Applications

Section 4.1

Copyright © 2013, 2009, 2005, 2002 Pearson Education, Inc.

Page 2: Exponential and Logarithmic Functions · Definition: Exponential Function An exponential function with base a is a function of the form f(x) = ax, where a and x are real numbers such

Definition: Exponential FunctionAn exponential function with base a is a function of the form

f(x) = ax,where a and x are real numbers such that a > 0 and a ≠ 1.

Domain of an Exponential FunctionThe domain of f(x) = ax for a > 0 and a ≠ 1 is the set of all real numbers.

The Definition and Domain of Exponential Functions

Page 3: Exponential and Logarithmic Functions · Definition: Exponential Function An exponential function with base a is a function of the form f(x) = ax, where a and x are real numbers such

Exponential Functions

Exponential Functions are neither odd or even

Page 4: Exponential and Logarithmic Functions · Definition: Exponential Function An exponential function with base a is a function of the form f(x) = ax, where a and x are real numbers such

Solution x f(x) = 2x

–2 1/4

–1 1/2

0 1

1 2

2 4

3 8

Example 1 Graphing an exponential function (a >1)

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Page 5: Exponential and Logarithmic Functions · Definition: Exponential Function An exponential function with base a is a function of the form f(x) = ax, where a and x are real numbers such

Solution

Example 1 Graphing an exponential function (a >1)

Domain: (–∞, ∞)Range: (0, ∞)

Copyright © 2013, 2009, 2005, 2002 Pearson Education, Inc.

Page 6: Exponential and Logarithmic Functions · Definition: Exponential Function An exponential function with base a is a function of the form f(x) = ax, where a and x are real numbers such

Properties of Exponential FunctionsThe exponential function f(x) = ax has the following properties:

1. The function f is increasing for a > 1 and decreasing for 0 < a < 1.

2. The y-intercept of the graph of f is (0, 1).

3. The graph has the x-axis as a horizontal asymptote.

4. The domain of f is (–∞, ∞), and the range of f is (0, ∞).

5. The function f is one-to-one.

Graphing Exponential Functions

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Page 7: Exponential and Logarithmic Functions · Definition: Exponential Function An exponential function with base a is a function of the form f(x) = ax, where a and x are real numbers such

Solution x

–2 4

–1 2

0 1

1 1/2

2 1/4

3 1/8

Example 2 Graphing an exponential function (0 < a < 1)

Copyright © 2013, 2009, 2005, 2002 Pearson Education, Inc.

Page 8: Exponential and Logarithmic Functions · Definition: Exponential Function An exponential function with base a is a function of the form f(x) = ax, where a and x are real numbers such

Solution

Example 2 continuedGraphing an exponential function (0 < a < 1)

Copyright © 2013, 2009, 2005, 2002 Pearson Education, Inc.

Page 9: Exponential and Logarithmic Functions · Definition: Exponential Function An exponential function with base a is a function of the form f(x) = ax, where a and x are real numbers such

Any function of the form g(x) = b × ax – h + k is a member of the exponential family of functions.

The graph of f moves to the left if h < 0 or to the right if h > 0.

The graph of f moves upward if k > 0 or downward if k < 0. (k – horizontal asymptote)

The graph of f is stretched if b > 1 and shrunk if 0 < b < 1.

The graph of f is reflected in the x-axis if b is negative.

The Exponential Family of Functions

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Page 10: Exponential and Logarithmic Functions · Definition: Exponential Function An exponential function with base a is a function of the form f(x) = ax, where a and x are real numbers such

Example 4: Which equation represents the graph?

Copyright © 2013, 2009, 2005, 2002 Pearson Education, Inc.

g(x) = b × ax – h + k

𝑏. 𝑓 𝑥 = 15(

)𝑓. 𝑓 𝑥 = 1

2()

𝑔. 𝑓 𝑥 = 14(

)

𝑎. 𝑓 𝑥 = 13(

)

𝑐. 𝑓 𝑥 = 5)

𝑑.𝑓 𝑥 = 2)

𝑒. 𝑓 𝑥 = 3)

Page 11: Exponential and Logarithmic Functions · Definition: Exponential Function An exponential function with base a is a function of the form f(x) = ax, where a and x are real numbers such

Example 5: Which equation represents the graph?

Copyright © 2013, 2009, 2005, 2002 Pearson Education, Inc.

g(x) = b × ax – h + k

𝑏. 𝑓 𝑥 = 15(

)𝑓. 𝑓 𝑥 = 1

2()

𝑔. 𝑓 𝑥 = 14(

)

𝑎. 𝑓 𝑥 = 13(

)

𝑐. 𝑓 𝑥 = 5)

𝑑.𝑓 𝑥 = 2)

𝑒. 𝑓 𝑥 = 3)

Page 12: Exponential and Logarithmic Functions · Definition: Exponential Function An exponential function with base a is a function of the form f(x) = ax, where a and x are real numbers such

Example 6: Which equation represents the graph?

Copyright © 2013, 2009, 2005, 2002 Pearson Education, Inc.

g(x) = b × ax – h + k

𝑎. 𝑓 𝑥 = − 1 3( 3 3) + 2

𝑏. 𝑓 𝑥 = −1 3( 3 3) − 1

𝑐. 𝑓 𝑥 = 3 3 1 3()− 1

𝑒. 𝑓 𝑥 = − 1 3( 3 1 3()− 1

𝑓. 𝑓 𝑥 = 3 3 3) + 2

𝑔. 𝑓 𝑥 = 13( 3 3) − 1

𝑑. 𝑓 𝑥 = −3 3 1 3()+ 1 ℎ. 𝑓 𝑥 = 1

3( 3 3) + 2

Page 13: Exponential and Logarithmic Functions · Definition: Exponential Function An exponential function with base a is a function of the form f(x) = ax, where a and x are real numbers such

One-to-One Property of Exponential FunctionsFor a > 0 and a ≠ 1,

., 2121 xxaa xx == then if

Exponential Equations

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Page 14: Exponential and Logarithmic Functions · Definition: Exponential Function An exponential function with base a is a function of the form f(x) = ax, where a and x are real numbers such

Solution

1

1a. 4 =414 = 44

1

x

x

x

−=

= −

Example 7 Solving exponential equations

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Page 15: Exponential and Logarithmic Functions · Definition: Exponential Function An exponential function with base a is a function of the form f(x) = ax, where a and x are real numbers such

Compound Interest FormulaIf a principal P is invested for t years at an annual rate rcompounded n times per year, then the amount A, or ending balance, is given by

Continuous Compounding FormulaIf a principal P is invested for t years at an annual rate rcompounded continuously, then the amount A, or ending balance, is given by A = P · ert.

The Compound Interest Model and Continuous Compounding

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Page 16: Exponential and Logarithmic Functions · Definition: Exponential Function An exponential function with base a is a function of the form f(x) = ax, where a and x are real numbers such

Solution

Example 8Interest compounded continuoslyFind the amount when a principal of $5600 is invested at 6 1/4% annual rate compounded continuously for 5 years and 9 months.

Convert 5 years and 9 months to 5.75 years. Use r = 0.0625, t = 5.75, and P = $5600 in the continuous compounding formula:

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