Exponential and Logarithmic Functions · Definition: Exponential Function An exponential function...
Transcript of Exponential and Logarithmic Functions · Definition: Exponential Function An exponential function...
Exponential and Logarithmic Functions
Exponential Functions and Their Applications
Section 4.1
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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
Exponential Functions
Exponential Functions are neither odd or even
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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Solution
Example 1 Graphing an exponential function (a >1)
Domain: (–∞, ∞)Range: (0, ∞)
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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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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)
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Solution
Example 2 continuedGraphing an exponential function (0 < a < 1)
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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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Example 4: Which equation represents the graph?
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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)
Example 5: Which equation represents the graph?
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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)
Example 6: Which equation represents the graph?
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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
One-to-One Property of Exponential FunctionsFor a > 0 and a ≠ 1,
., 2121 xxaa xx == then if
Exponential Equations
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Solution
1
1a. 4 =414 = 44
1
x
x
x
−=
= −
Example 7 Solving exponential equations
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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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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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