Name: Solutions
Due Date: Friday, December 9th in Class
Extra Credit
Directions: For problems 1-10 put your solutions in the space provided on the front of this assignment.
For problem 11 work out your problem in the space provided. Show all work when necessary to receive
full credit.
Student Solutions
1
2
3
4
5
6
b
a
c
d
d
c
7
8
9
10
c
a
d
d
Scoring (For Instructor use only)
11
Total
10
1
1.
Which of the following rational functions represents the graph above:
x2 + 3x + 2
x3 − 7x + 6
x2 − 3x + 2
(b) f(x) = 3
x + 2x2 − 5x − 6
x2 + x − 2
(c) f(x) = 3
x + 2x2 − 5x − 6
x2 − x − 2
(d) f(x) = 3
x − 7x + 6
(a) f(x) =
2. Which of the following graphs represents the function
f(x) =
x3 + 2x2 − 5x − 6
x2 − 3x + 2
Remark: (the holes are too small to be seen on the graphs but are understood to be included on the graphs shown)
(a)
(b)
2
(c)
(d)
4x2 − 1
, which of the following intervals represents the solution set to the inequality f(x) ≤ 0?
x
[
) (
]
1
1
(a) − , 0 ∪ 0,
2
2
(
]
1
(b) −∞, − 2 ∪ (0, −∞]
(
] (
]
(c) −∞, − 12 ∪ 0, 12
[
) [
)
(d) − 12 , 0 ∪ 12 , −∞
√
4. If f(x) = 8 − x and g(x) = x2 − 8, which of the following intervals represents the domain of (f ◦ g)(x)?
3. If f(x) =
(a) (−∞, −4] ∪ [4, ∞)
(b) (−∞, 4]
(c) [−4, 0] ∪ [4, ∞)
(d) [−4, 4]
(e) (−∞, 0] ∪ [4, ∞)
(f) (−∞, −4] ∪ [0, ∞)
5. Does the following graph represent a function of y, why or why not?
(a) Yes it passes the vertical line test.
(b) No it fails the vertical line test.
(c) Yes it passes the horizontal line test.
(d) No it fails the horizontal line test.
3
6. Define the functions f(x) and g(x) by their graphs below
Use the following graph bank to choose the correct order of graphs that represent the ordered list
(fg)(x), (f ◦ g)(x), (f + g)(x)
(a) A,B,C
(b) B,D,A
(c) B,C,A
(d) A,C,B
(e) C,A,D
(f) B,D,A
(g) B,C,A
4
7. Define f(x) = 2x3 − 2. Which of the following represents the difference quotient
f(x + h) − f(x)
?
h
2h3 − 2 − x
h
2(x + h)3 − 2 − x
(b)
h
2(x + h)3 − 2x3
(c)
h
2(x + h)3 − 2(x + h)3 − 4
(d)
h
(a)
8. The number of bacteria in a culture is modeled by the function n(t) = 10ert , where t is in hours. If there are 160
bacteria after 2 hours what is the rate r.
(a) ln(4)
(b) ln(8)
(c) ln(16)
(d) ln(160)
9. Define f(x) = 3x−1 + 2. Which of the following represents a graph of f(x) and its inverse.
10. Which of the following represents a fourth degree polynomial, p(x), with zeros −1, 6, −2i.
(a) p(x) = (x + 1)(x − 6)2 (x + 2i)
(b) p(x) = (x + 1)(x − 6)(x2 − 4)
(c) p(x) = (x + 1)2 (x − 6)(x − 2i)(x + 2i)
(d) p(x) = (x + 1)(x − 6)(x2 + 4)
5
11. Solve for x. Identify any extraneous solutions, write NONE if there are not any.
(a) ln(x − 5) − ln(x + 3) = ln(x) − ln(x − 2)
Solution. Observe that the solution space for this equation is (5, ∞) ∩ (−3, ∞) ∩ (0, ∞) ∩ (2, ∞) = (5, ∞),
which is the intersection of the domains of each logarithm in the equation. Thus we can only except solutions
in (5, ∞). Solving for x we get
ln(x − 5) − ln(x + 3) = ln(x) − ln(x − 2)
(
)
(
)
x−5
x
= ln
⇒ ln
x+3
x−2
x−5
x
⇒
=
x+3
x−2
since logs are 1-1 functions,
x−5
x
−
=0
x+3 x−2
(x − 5)(x − 2) − x(x + 3)
⇒
=0
(x + 3)(x − 2)
⇒ (x − 5)(x − 2) − x(x + 3) = 0
⇒
⇒ x2 − 7x + 10 − x2 − 3x = 0
⇒ 10 − 10x = 0
⇒x=1
Since 1 ̸∈ (5, ∞) we reject the solution as extraneous and say there are no valid solutions.
(b) log3 (x) + log3 (x + 8) = 2
Solution. Observe that the solution space is (0, ∞)∩(−8, ∞) = (0, ∞), which is the intersection of the domains
of the logarithms in the equation. Solving for x we get
log3 (x) + log3 (x + 8) = 2
⇒ log3 (x2 + 8x) = 2
⇒ 32 = x2 + 8x
⇒ (x + 9)(x − 1) = 0
⇒ x = 1, −9
Since 1 is in our solution space we accept it as valid, but -9 we reject as extraneous.
(c) 34x − 7 = 8
Solution. Solving for x we get
34x − 7 = 8
⇒ 34x = 15
( )
⇒ log3 34x = log3 (15)
⇒ 4x = log3 (15)
⇒x
6
log3 (15)
4