Section 6.3 – Exponential Functions Laws of Exponents If s, t, a, and b are real numbers where a > 0 and b > 0, then: ∙ = + ( ) = ( ) = 1 = 1 0 =1 − = 1 = ( 1 ) Definiti on: An Exponential Function is in the form, “a” is a positive real number and does not equal 1 “C” is a real number and does not equal 0 The domain of f(x) is the set of all real numbers “a” is the base and is the Growth Factor “C” is the Initial Value because ( + 1 ) ( ) = → + 1 = 1 =
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Section 6.3 – Exponential Functions Laws of Exponents If s, t, a, and b are real numbers where a > 0 and b > 0, then: Definition: “a” is a positive real.
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Section 6.3 – Exponential FunctionsLaws 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
Definition:
An Exponential Function is in the form,
“a” is a positive real number and does not equal 1
“C” is a real number and does not equal 0
The domain of f(x) is the set of all real numbers
“a” is the base and is the Growth Factor
“C” is the Initial Value because
𝑓 (𝑥+1)𝑓 (𝑥 )
=𝑎→𝐶𝑎𝑥+1
𝐶𝑎𝑥 =𝐶𝑎𝑥𝑎1
𝐶𝑎𝑥 =𝑎
Section 6.3 – Exponential FunctionsExamples
𝑓 (0 )=1 , h𝑡 𝑒𝑟𝑒𝑓𝑜𝑟𝑒𝐶=1
𝑥 𝑓 (𝑥) 𝑓 (𝑥+1)𝑓 (𝑥 )
=𝑎
−1
0
1
2
3
23
132
94
278
12/3
=32
3/21
=32
9/ 43 /2
=32
27/ 89/4
=32
𝑎=32
𝑓 (𝑥 )=𝐶𝑎𝑥
𝑓 (𝑥 )=1( 32 )
𝑥
¿ ( 32 )
𝑥
Section 6.3 – Exponential FunctionsExamples
𝑓 (0 )=1/ 4 , h𝑡 𝑒𝑟𝑒𝑓𝑜𝑟𝑒𝐶=1/4
𝑥 𝑓 (𝑥) 𝑓 (𝑥+1)𝑓 (𝑥 )
=𝑎
−1
0
1
2
3
12
14
18
116
132
1/41/2
=12
1/81/4
=12
1/161/8
=12
1/321/16
=12
𝑎=12
𝑓 (𝑥 )=𝐶𝑎𝑥
𝑓 (𝑥 )=14 ( 3
2 )𝑥
Section 6.3 – Exponential FunctionsProperties of the Exponential Function,
The domain is the set of all real numbers.
The range is the set of all positive real numbers.
The y-intercept is 1; x-intercepts do not exist.
The x-axis (y = 0) is a horizontal asymptote, as x.If a > 1, the f(x) is increasing function.
The graph contains the points (0, 1), (1, a), and (-1, 1/a).
The graph is smooth and continuous.
Section 6.3 – Exponential Functions
𝑓 (𝑥 )=𝑎𝑥 ,𝑎>1The graph of the exponential function is shown below.
𝑦= 𝑓 (𝑥 )=𝑎𝑥
a
Section 6.3 – Exponential FunctionsProperties of the Exponential Function,
The domain is the set of all real numbers.
The range is the set of all positive real numbers.
The y-intercept is 1; x-intercepts do not exist.
The x-axis (y = 0) is a horizontal asymptote, as x.If 0 < a < 1, then f(x) is a decreasing function.
The graph contains the points (0, 1), (1, a), and (-1, 1/a).
The graph is smooth and continuous.
Section 6.3 – Exponential Functions
𝑓 (𝑥 )=𝑎𝑥 ,0<𝑎<1The graph of the exponential function is shown below.
𝑓 (𝑥 )=𝑎𝑥 ,0<𝑎<1
Section 6.3 – Exponential FunctionsEuler’s Constant – e
The value of the following expression approaches e,
(1+ 1𝑛 )
𝑛
as n approaches .
Using calculus notation,
Growth and decay
Compound interest
Differential and Integral calculus with exponential functions