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**Unformatted text preview: **c Kendra Kilmer January 4, 2012 Section 1.5 - Exponential Functions
Deﬁnition: The exponential functions are the functions of the form f (x) = ax , where the base a is a positive
constant with a = 1. Properties of the Graphs of f (x) = ax
1. Domain is the set of all real numbers.
2. Range is the set of all positive real numbers.
3. All graphs pass through the point (0, 1).
4. The graph is continuous (no holes or jumps).
5. The x axis is a horizontal asymptote (but only in one direction).
6. If b > 1, the graph is increasing (exponential growth).
7. If 0 < b < 1, the graph is decreasing (exponential decay).
Example 1: Graph f (x) = 2x , g(x) = 5x , h(x) = 1
2 x , and k(x) = 17 1
5 x . c Kendra Kilmer January 4, 2012 Deﬁnition: The most common base is the number e. Properties of Exponents: If a and b are positive numbers and x and y are any real numbers, then
1. ax+y = ax ay
ax
2. ax−y = y
a
x )y = axy
3. (a
4. (ab)x = ax bx
5. ax = ay if and only if x = y
6. For x = 0, ax = bx if and only if a = b
Example 2: Simplify ex+4
e4−x Example 3: Solve each equation for x:
a) 9x−1 = 31+x b) x2 ex − 5xex = 0 18 c Kendra Kilmer January 4, 2012 Example 4: The population of a particular city doubles every 5 years. If 30,000 people currently live in the city,
a) what will the population be in 15 years? b) what will the population be in t years? c) estimate the size of the population in 27 years. Example 5: The half-life of a particular substance is 8 hours. If a sample of this substance initially has a mass of
100 grams,
a) what will the mass be in 16 hours? b) what will the mass be in t hours? c) estimate the mass in 45 hours. Section 1.5 Highly Suggested Homework Problems: 1, 3, 9, 11, 15, 17, 21, 29, 31, 33 19 ...

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