MATH 150
Quiz Key #7
11/3-5/2015
(1) For each pair, choose the larger of the two numbers.
Solution:
x
3 4
is a decreasing function so 15 > 15 .
−e
x
−2e
(b) f ( x ) = 23 is a decreasing function. 49
= 23
. Since f ( x ) is
−e
−
2e
−
e
−
e
decreasing, 23
> 32
. Therefore, 49
> 23
.
x
π
(c) π > 3 > e. Therefore, e > 1. Hence, f ( x ) = πe
is an increasing
π 7
π 6
function. Therefore, e > e .
(a) f ( x ) =
1
5
√
√
(d) 4 < 5. Thus, 4−1 > 5−1 . Therefore, 4− 7 > 5− 7 .
(e) Skip.
√
√
√
√
(f) Note that 5 2 < 5 2+1 . Therefore, −5 2 > −5 2+1 .
(2) For the function f ( x ) = −3e x+2 − 1, find the domain, range, intercepts,
asymptotes, and the intervals where f is increasing or decreasing.
Solution:
DOMAIN: (−∞, ∞)
RANGE: (−∞, −1)
x-INTERCEPTS: none
y-INTERCEPTS: 0, −3e2 − 1
INCREASING: never
DECREASING: (−∞, ∞)
ASYMPTOTES: Horizontal: y = −1. No Vertical Asymptotes.
(3) Determine the end behavior of the function g ( x ) = 3− x .
Solution: Note that g ( x ) is y = 3x reflected about the y-axis. Therefore,
typical end behavior is reversed. Hence, as x → ∞, g ( x ) → 0, and as x →
−∞, g ( x ) → ∞.
(4) If a2 = 3 and b−2 = 5, what does a3 b
2
equal?
Solution:
3
a b
2
3
a2
33
27
= a b = −2 =
= .
5
5
b
6 2
1
(5) Plot the graph of the function f ( x ) = 2
on the graphs and label any asymptotes.
x
1
2
+ 1. Label at least two points
0
Solution: First we choose two points. Note that f (0) = 2 12 + 1 = 2 + 1 =
1
3 and f (1) = 2 12 + 1 = 1 + 1 = 2. Therefore, two points on the graph
are (0, 3) and (1, 2) . [This also shows that the y-intercept is (0, 3) .] This is the
x
shifted up 1. Therefore, the horizontal asymptote of our
graph of y = 2 12
graph is y = 1. The graph should be a decreasing exponential curve. It has
domain (−∞, ∞) and range (1, ∞) . Therefore, it never intersects the x-axis,
and so it has no x-intercepts.
(6) Answer the following questions:
√
100 = 12 .
(b) log7 17 = log7 7−1 = −1.
√ 2
7
= 2.
(c) log√7 7 = log√7
(d) log2 64 = log2 26 = 6.
(e) log11 1 = log11 110 = 0.
(f) log11 11 = 1.
(a) log100 10=log100
(7) Write the statement “the logarithm of x to the base 4 is 3” as an equation
with a logarithm. For this equation, what is the value of x?
Equation: log4 x = 3.
Now to solve this equation, we rewrite in exponential form: x = 43 = 64.
2