Practice Final Exam
Name:
The answer sheets seen on previous practice exams are omitted in an effort to save paper.
Discrete Math
1. Write
4
X
(2i2 + 3i) as an integer in standard form.
i=1
2. What does the following equal: 1 + 27 +
4
49
+
8
343
+ · · ·?
3. What is the 129th term in the sequence 8, 15, 22, 29, . . .?
4. What is the 43rd term in the sequence 2, 34 , 89 , . . .?
5. What is the sum of the first 70 terms of the sequence 39, 34, 29, 24, . . . as an iteger
in standard form?
6. An ice-cream shop has 28 flavors. You need to select 12 flavors. In how many ways
can you select these flavors?
7. A office supply store has 45 employees. A departmental team needs to be selected
with a head manager, assistant manager, stock person, and greeter. In how many
ways can you select these positions?
8. The world championship of boxing has 64 competitors. How many possible ways are
there to rank the competitors from first to last?
9. You need to stop at three places during a road trip. At the first stop, there are 3 gas
stations, at the second stop there are 15 gas stations, and at the third stop there are
4 gas stations. How many different gas stations can you stop at if you stop at one
station at each place?
10. What is
8
5
? Your answer should be a natural number in standard form.
Algebra
52
9
11. Write
as a rational number in standard form.
25
40
2
12. Write 8−98 (82 ) (830 ) 3 as a rational number in standard form.
13. If a 6= 0, then what is a0 as a rational number in standard form?
14. If a > 0, then what is loga (1) as a rational number in standard form?
15. Write log4
q 3
1
16
as a rational number in standard form.
√
2
x2 − 4x
√
16. Solve for x when
= 6.
2
9x
17. Solve for x when 4e3x+8 = 8e−8x+13 .
4
18. Solve for x when 2 + loge (x ) = 7 − loge
1
x5
19. If f (x) = x3 + 2 and g(x) = −x + 8, then what is f ◦ g(x)?
20. Find the inverse of f (x) = 4(x − 1)3 + 12. You can check your answer by seeing if
f −1 ◦ f (x) = x.
21. What is the implied domain of f (x) =
x3 − 4x + x − 10
?
x2 − 9
r
22. What is the implied domain of g(x) =
23. Find
3
6x −
2 √
− 2 5x − 19?
3
5x4 − x3 + 2x2 − 3x + 1
x2 + 8
24. Complete the square: Write −4x2 + 16x − 13 in the form α(x + β)2 + δ where
α, β, δ ∈ R.
25. How man roots does −x2 + 14x − 49 have?
26. Find the roots of 2x2 + 3x − 5. (Write both roots.)
27. Completely factor 6x3 + 19x2 + 2x − 3 (Hint: −3 is a root.)
28. |x − y| is the distance between which two numbers?
29. Solve for x when | − 31 x − 5| < 14.
Linear Algebra
30. What is the determinant of the matrix below?
3 4
−1 8
31. Find the product
2 −5
8
0
1 2
−1 3
32. What is the inverse of the matrix below?
3 0
−1 1
33. Write the system of three linear equations in three variables below as a matrix equation.
y + 3z = 10
x+ y
= 6
−3x + 3y + 2z = 13
34. Use that

−1 
1/4 −2 7/2
4 2 −6
 0 2
1 −1 
1  = 0
0 −1
2
0 1
1

to find x, y, z ∈ R if
4x + 2y − 6z = −4
2y + z = 4
y+ z= 6
Graphs
35. Graph the following ten common functions: 8, x, x2 , x3 ,
√
2
x,
√
3
x, 1/x, 1/x2 , ex , loge (x).
36. Graph f : [−5, 3) → R where f (x) = −x + 2.
37. Graph g : {1, 2, 3, 4} → R where g(x) = x2 − 6
38. Graph e−x−4 and label its x− and y− intercepts (if there are any).
39. Graph −2(x − 4)2 − 1 and label its vertex.
40. Graph
√
3
x + 4 and label its x− and y− intercepts.
41. Graph p(x). Label all x−intercepts.
p(x) = (x + 1)(x − 2)(x − 6)(x2 + 1)
42. Graph r(x). Label all x−intercepts and all vertical asymptotes.
r(x) =
−3(x + 1)(x + 1)
(x + 4)(x − 1)(x − 4)
43. Graph
f (x) =
44. Graph
√
3
f (x) =
x2 − 3
3
x
−x + 4
if x 6= 1
if x = 1
if x ∈ (−∞, 1);
if x ∈ [1, 5]