Math 150 – Fall 2015
Section 5C
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Section 5C – Exponential Functions
3
Example 1. What is 16 2 ?
√
Question. What is 2
2
?
Theorem. To evaluate ab , when b is irrational (so b is not a fraction of integers), we
p
approximate it by choosing rational numbers pq that are close to b and evaluating a q .
We can find rational numbers pq as close to b as we want. As we let our fraction pq
p
approach b, the numbers a q are also approaching a number, we call this number ab .
√
Example 2. Compute 2x for a sequence of x’s getting close to 2.
Properties of Exponents: For any positive real number a, and any real numbers x
and y, the following properties hold:
1. ax ay = ax+y
2.
ax
ay
= ax−y
3. a−x =
1
ax
4. a0 = 1 for a 6= 0
x
5. (ab) = ax bx
x
x
6. ab = abx
y
7. (ax ) = axy
Definition of Exponential Functions
Definition. The function f (x) = ax , with 0 < a and x any real number, is called the
exponential function with base a
x
Example 3. Fill in the table below for the functions f (x) = 2x and f (x) = 12 . Use
the points to graph the function.
x
-3
1 x
2
2x
-2
-1
0
1
2
3
.
Math 150 – Fall 2015
Section 5C
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Theorem. Consider the exponential function, f (x) = ax .
1. If a > 1, then the graph of ax looks like the graph of 2x . The higher the value of
a the steeper the graph will be.
2. If 0 < a < 1, then the graph of ax looks like the graph of
value of a, the steeper the graph will be.
a x
.
2
The smaller the
3. If a = 1, then the graph of ax is the horizontal line y = 1.
Properties for f (x) = ax : You need to know these properties!
1. The domain of f (x) = ax is all real numbers.
2. The range of f (x) = ax , if a 6= 1, is all positive real numbers. If a = 1, then the
range is the set consisting of the number 1 only. Note: ax can never equal 0.
3. If a > 1, then as x → ∞, f (x) = ax → ∞, and as x → −∞, f (x) = ax → 0.
4. If 0 < a < 1, then as x → ∞, f (x) = ax → 0, and as x → −∞, f (x) = ax → ∞.
5. If 1 < a and x < y, then ax < ay . Therefore, f (x) = ax is an increasing function
if a > 1.
6. If 0 < a < 1 and x < y, then ax > ay . Therefore, f (x) = ax is an decreasing
function if 0 < a < 1.
Math 150 – Fall 2015
Section 5C
Example 4. Which number is bigger:
1 6
3
or
3 of 5
1 5
3 ?
Example 5. What does the plot of 2 · 2x look like compared to the plot of 2x ?
Consider the graphs below of y = 1.5x , y = 2x , y = 2.5x , y = 3x , and y = 3.5x for
−1 ≤ x ≤ 1.
Also, compare the graphs of y = a0.25 , y = a0.5 , and y = a.75 .
Theorem. In general, we have the following:
If a < b, and x > 0, then ax < bx .
If a < b, and x < 0, then ax > bx .
Example 6. Which number is larger:
√
√
(a) 2− 2 or 3− 2
π
π
(b) 23 or 73
(c)
√
5 − 7
2
or
√
13 − 7
11
Math 150 – Fall 2015
Section 5C
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The Natural Exponential Function
Definition. The natural
exponential function is the function f (x) = ex , where e
1 n
is the limit of 1 + n as n goes to infinity.
Note. The value of e can be approximated by substituting larger and larger numbers
n
for n in the expression 1 + n1 .
A closer approximation is
e ≈ 2.71828182845905
Then 2 < e < 3 so the graph of ex is between the graphs of 2x and 3x .
Example 7. Place the following numbers in increasing order: 2.85 , 2.33 , e4 , 35 , e5 ,
2.54 .
Theorem. To evaluate ex , we can use the following limit
x n
ex = lim 1 +
n→∞
n
x
Example 8. Use the formula ex = limn→∞ 1 + nx to approximate e3 (You need a
calculator)
n n
1 + n3
1
10
100
1000
10,000
100,000,000
Math 150 – Fall 2015
Section 5C
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Example 9. For the following functions, find the domain, range, x-intercepts, and
y-intercepts. Also, describe the end behavior of the functions and the intervals where
the function is increasing and decreasing.
(a) f (x) = −3ex−1 − 2
(b) g(x) = 2
1 x+1
3
− 18
(c) h(x) = −2(4)−x−2 + 4