section 9.3 we have previously worked with exponential expressions, where the exponent was a...
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Section 9.3
• We have previously worked with exponential expressions, where the exponent was a rational number
• The expression bx can actually be defined for all real numbers, x, including irrational numbers.
• However, the proof of this would have to wait until a higher level math course.
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A function of the form f(x) = bx is called an exponential function if b > 0, b is not 1, and x is a real number.
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We can graph exponential functions of the form f(x) = 3x, g(x) = 5x or h(x) = (½)x by substituting in values for x, and finding the corresponding function values to get ordered pairs.
We would find all graphs satisfy the following properties:
• 1-to-1 function
• y-intercept (0, 1)
• no x-intercept
• domain is (-, )
• range is (0, )
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We would find a pattern in the graphs of all the exponential functions of the type bx, where b > 1.
x
y
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We would find a pattern in the graphs of all the exponential functions of the type bx, where 0 < b < 1.
x
y
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We would find a pattern in the graphs of all the exponential functions of the type bx-h, where b > 1.
x
y
(h, 1)
The graph has the same shape as the graph for bx, except it is shifted to the right h units.
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We would find a pattern in the graphs of all the exponential functions of the type bx+h, where b > 1.
x
y
(-h, 1)
The graph has the same shape as the graph for bx, except it is shifted to the left h units.
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Since an exponential function is a 1-to-1 function,
if b > 0, and b 1, then bx = by is equivalent to x = y.
Example
Solve 6x = 36
6x = 62
x = 2
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Example
Solve 92x+1 = 81
92x+1 = 92
2x + 1 = 2
2x = 1
x = ½
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Example
Solve x2327
1
3-3 = 32x
-3 = 2x
2
3x
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Example
Solve 43x-6 = 322x
(22)3x-6 = (25)2x
(22)3x-6 = 210x
26x-12 = 210x
6x-12 = 10x
-12 = 4x
x = -3
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• Many applications use exponential functions of various types.
• Compound interest formulas are exponential functions used to determine the amount of money accumulated or borrowed.
• Exponential functions with negative exponents can be used to describe situations of decay, while those with positive exponents can be used to describe situations of growth.
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Example
Find the total amount invested in a savings account if $5000 was invested and earned 6% compounded monthly for 18 years. Round your answer to two decimal places.
The formula that is used for calculating compound interest isnt
n
rPA
1
where P is the initial principal invested, r is the interest rate, n is the number of times interest is compounded each year, t is the time of the investment (in years) and A is the amount of money in the account.
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Example (cont.)
1812
12
06.0150001
nt
n
rPA
216005.015000
216005.15000
14683.83$
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Example
An accidental spill of 100 grams of radioactive material in a local stream has led to the presence of radioactive debris decaying at a rate of 5% each day. Find how much debris still remains after 30 days.
The formula that would be used for this problem is
rtAy )7.2(
where A is the amount of radioactive material to start, r is the rate of decay, t is the number of days and y is the amount of radioactive material after the time period.
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Example (cont.)
3005.0)7.2(100)7.2( rtAy5.1)7.2(100 (exact answer)
22.54 grams