Math 110 Practice Final Exam

1. Evaluate using the Piecewise Function:

𝑓(𝑥) = {−3𝑥2 + 4𝑥 − 7 𝑥 ≤ −5

5 −5 < 𝑥 ≤ 3|2𝑥 − 1| 𝑥 > 3

a. 𝑓(2) = −5 b. 𝑓(−5) = 5

c. 𝑓(−6) = 13 d. 𝑓(8) = 15

2. Given the Piecewise Function, Identify an open point of the graph.

𝑓(𝑥) = {2𝑥 − 12 𝑥 < −2

𝑥 + 4 −2 ≤ 𝑥 < 2−6𝑥 + 3 𝑥 ≥ 2

a. (−2, −2) b. (2, −9)

c. (−2, 16) d. (2, 6)

3. Find the new function given the set of equations: (𝑔 + 2𝑓)(𝑥)

𝑓(𝑥) = −𝑥 + 1 𝑔(𝑥) = 𝑥2 + 1 ℎ(𝑥) = (𝑥 + 1)2

a. 𝑥2 + 2 b. 𝑥2 − 𝑥 + 2

c. 𝑥2 − 2𝑥 + 3 d. 𝑥 + 2

4. Find the Difference Quotient of the function: 𝐷𝑄 =𝑓(𝑥+ℎ)−𝑓(𝑥)

ℎ

𝑓(𝑥) = −5𝑥2 + 2

a. −10ℎ b. −10𝑥 − 5ℎ

c. −5𝑥 d. −10𝑥ℎ − 5ℎ2

5. Use the given Table of Values to identify the correct Evaluation.

𝑥 𝑓(𝑥) 𝑔(𝑥) 1 3 8 3 1 9 8 9 1 9 8 3

a. (𝑓 ∘ 𝑔)(8) = 1 b. (𝑓 ∘ 𝑔)(9) = 3

c. (𝑔 ∘ 𝑓)(3) = 8 d. (𝑔 ∘ 𝑓)(9) = 9

6. Find the Composite function given the set of equations: (ℎ ∘ 𝑓)(𝑥)

𝑓(𝑥) = 3𝑥 − 4 𝑔(𝑥) = 2𝑥2 − 5 ℎ(𝑥) = 3(𝑥 − 1)2

a. 27𝑥2 − 75 b. 27𝑥2 − 90𝑥 + 75

c. 18𝑥2 − 17 d. 18𝑥2 − 27𝑥 + 45

7. Given the Graph, identify the correct evaluation:

a. (𝑓 ∘ 𝑔)(4) = 4 b. (𝑔 ∘ 𝑓)(4) = 4

c. (𝑓 ∘ 𝑓)(4) = 4 d. (𝑔 ∘ 𝑔)(4) = 4

8. Given the table of data, determine if the table can represent a One-to-One Function.

𝑥 (Semester)

𝑦 (Percent Passing)

F2012 48

S2013 55

F2013 52

S2014 46

F2014 51

a. No, 𝑥 repeats b. Yes, 𝑥 & 𝑦 are unique

c. No, 𝑦 repeats d. Yes, 𝑥 & 𝑦 both repeat

9. Consider the One-to-One function, Identify the correct statement about it’s inverse.

𝐹(𝑥) = {(2,1), (3,2), (4,3), (5,4), (1,7)}

a. 𝑓−1(2) = 3 b. 𝑓−1(1) = 0

c. 𝑓−1(4) = 7 d. 𝑓−1(2) = 1

10. Consider the One-to-One graph of 𝑓(𝑥), Identify the correct evaluation for its inverse.

a. 𝑓−1(−1) = 2 b. 𝑓−1(0) = −1

c. 𝑓−1(1) = 0 d. 𝑓−1(−3) = 4

11. Solve the Rational Equation: 4

𝑥−1+

7

𝑥+2=

3𝑥−7

(𝑥−1)(𝑥+2)

a. 𝑥 = 2 b. 𝑥 = −1

c. 𝑥 = 1 d. 𝑁𝑜 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛

12. Find the Vertical Asymptotes: 𝑓(𝑥) =(𝑥+6)(𝑥−2)

(𝑥−3)(𝑥+4)

a. −6 𝑎𝑛𝑑 2 b. 6 𝑎𝑛𝑑 − 2

c. −3 𝑎𝑛𝑑 4 d. 3 𝑎𝑛𝑑 − 4

13. Identify the Domain: 𝑓(𝑥) =(𝑥−2)(𝑥+2)

(𝑥−2)(𝑥−1)

a. (−∞, −2) ∪ (−2, −1) ∪ (−1, ∞) b. (−∞, 1) ∪ (1, 2) ∪ (2, ∞)

c. (−∞, −1) ∪ (−1, ∞) d. (−∞, −2) ∪ (−1, ∞)

14. Which Rational Function has 2 Holes and 1 Vertical Asymptote?

a. 𝑓(𝑥) =(𝑥+3)(𝑥−1)(𝑥+1)

(𝑥−1)(𝑥+3)(𝑥+2) b. 𝑓(𝑥) =

(𝑥+2)(𝑥−7)(𝑥+3)

(𝑥+2)(𝑥−5)(𝑥−1)

c. 𝑓(𝑥) =(𝑥−2)(𝑥+1)(𝑥+5)

(𝑥−5)(𝑥+2) d. 𝑓(𝑥) =

(𝑥+5)(𝑥−3)(𝑥+1)

(𝑥−3)(𝑥+1)

15. Find the x-intercept values: 𝑓(𝑥) =𝑥2−5𝑥+6

𝑥2−4𝑥−5

a. −3 𝑎𝑛𝑑 − 2 b. −5 𝑎𝑛𝑑 1

c. −1 𝑎𝑛𝑑 5 d. 2 𝑎𝑛𝑑 3

16. Find the y-intercept value: 𝑓(𝑥) =𝑥2+4

𝑥−2

a. −2 b. 2

c. −1 d. 1

2

17. Find the equation of the Horizontal Asymptote: 𝑓(𝑥) =3𝑥2+4𝑥−9

2𝑥3−2𝑥2+5𝑥−1

a. 𝑦 =3

2 b. 𝑦 = 1

c. 𝑦 = 0 d. 𝑁𝑜 𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝐴𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒

18. Find the equation of the Oblique Asymptote: 𝑓(𝑥) =−3𝑥3+𝑥2+2𝑥−6

𝑥2−2𝑥+1

a. 𝑦 = −3𝑥 − 5 b. 𝑦 = −3𝑥 + 7

c. 𝑦 = 3𝑥 − 7 d. 𝑁𝑜 𝑂𝑏𝑙𝑖𝑞𝑢𝑒 𝐴𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒

19. Identify the equation based on the graph:

a. 𝑓(𝑥) =(𝑥+1)(𝑥−7)(𝑥+2)

(𝑥+1)(𝑥−3) b. 𝑓(𝑥) =

(𝑥−1)(𝑥+7)(𝑥+2)

(𝑥+2)(𝑥−3)

c. 𝑓(𝑥) =(𝑥−1)(𝑥−7)(𝑥−2)

(𝑥−2)(𝑥+3) d. 𝑓(𝑥) =

(𝑥+1)(𝑥−7)(𝑥+2)

(𝑥+2)(𝑥−3)

20. Solve: 𝑥 + 3 = √14𝑥 − 6

a. −3 𝑎𝑛𝑑 − 5 b. 3 𝑎𝑛𝑑 5

c. 3 d. 𝑁𝑜 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛

21. Solve: |2𝑥 − 9| = 7

a. 1 & 8 b. 8 & − 8

c. 1 & − 1 d. −1 & − 8

22. Solve:

4|2𝑥 − 9| + 7 = 3

a. 4 & 5 b. 4 & − 5

c. −4 & − 5 d. 𝑁𝑜 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛

23. George has $5000 to invest in an account which compounds weekly at 3.8%. If he needs

$7250, how long will he need to leave his money invested? 𝐴 = 𝑃 (1 +𝑟

𝑛)

𝑛𝑡 𝑜𝑟 𝐴 = 𝑃𝑒𝑟𝑡

a. 508.6 𝑦𝑒𝑎𝑟𝑠 b. 9.7 𝑦𝑒𝑎𝑟𝑠

c. 39.3 𝑦𝑒𝑎𝑟𝑠 d. 17.5 𝑦𝑒𝑎𝑟𝑠

24. The bank currently offers monthly interest at 5.4%, and George has $1200 to invest. How much

money will be in his account at the end of 6 years? 𝐴 = 𝑃 (1 +𝑟

𝑛)

𝑛𝑡 𝑜𝑟 𝐴 = 𝑃𝑒𝑟𝑡

a. $1658 b. $1578

c. $1868 d. $1388

25. Solve:

log2(3𝑥 + 25) + 4 = 8

a. 3 b. 0

c. −3 d. 𝑁𝑜 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛

26. Solve:

log(𝑥 − 7) + log(2𝑥 + 3) = log (19)

a. −3

2 𝑎𝑛𝑑 7 b. −

5

2 𝑎𝑛𝑑 8

c. 7 d. 8

27. Given the Hydronium concentration, find the correct 𝑝𝐻 of the substance.

𝑝𝐻 = − log[𝐻3𝑂+]

𝐺𝑖𝑣𝑒𝑛 [𝐻3𝑂+] = 3.7 × 10−4.83

a. 3.74 b. 4.83

c. 4.26 d. 5.68

28. A sound has a decibel level of 132. If it takes 5 machines to produce this loudness, find the

intensity of 1 machine.

𝐷 = 10 ∙ log (5 ∙ 𝐼

10−12)

𝑁𝑜𝑡𝑖𝑐𝑒: 5 times the intensity is because we needed 5 machines to produce the loudness.

a. 3.1698 b. 15.8489

c. 1.58 × 1013 d. 12.6990

29. An organism in a polluted water sample is recognized as active. Measurement #1 showed 187

organisms as active. 3 hours later, Measurement #2 showed 123 organisms as active. Find the

rate of decay for this population in this polluted environment.

𝑦 = 𝑏𝑒𝑟𝑡

a. −0.2193 b. −0.5207

c. −0.1396 d. −0.6578

30. We want a colony to double in 36 months. What rate of growth should we hope to have?

𝑦 = 𝑏𝑒𝑟𝑡

a. 0.1980 b. 0.1792

c. 0.0167 d. 0.0193

31. Identify the graph which shows the correct intersection of the lines.

{3𝑥 − 2𝑦 = 52𝑥 + 3𝑦 = 7

a.

b.

c.

d.

32. Find the point of intersection for the given system of linear equations.

{3𝑥 − 4𝑦 = 1

2𝑥 + 3𝑦 = 12

a. (−1, −1) b. (5, −1)

c. (6, 0) d. (3, 2)

33. Identify the system of nonlinear equations which creates the graph.

a. {3𝑥2 + 2𝑦2 = 5

𝑥2 − 3𝑦2 = 3 b. {

3𝑥2 − 2𝑦2 = 5

𝑥2 + 3𝑦2 = 3

c. {3𝑥2 − 2𝑦 = 5

𝑥 + 3𝑦2 = 3 d. {

3𝑥2 − 2𝑦2 = −5

𝑥2 − 3𝑦2 = 3

34. Solve the nonlinear system of equations. Find all real points of intersection.

{𝑥2 + 𝑦2 = 12

−𝑥2 + 𝑦 = 0

a. (√3, 3), (−√3, 3) b. (3, −4), (−3, −4)

c. (3, √3), (3, −√3), (−4, 2), (4, 2) d. (√3, 3), (−√3, 3), (2, −4), (−2, 4)

35. Write the Augmented matrix as a system of equations.

[3 1 −2 ⋮ 71 −1 −3 ⋮ 52 2 0 ⋮ −3

]

a. {3𝑥2 + 𝑥 − 2 = 7𝑥2 − 𝑥 − 3 = 52𝑥2 + 2𝑥 = −3

b. {

3𝑥 + 𝑦 − 2𝑧 = 7𝑥 − 𝑦 − 3𝑧 = 5

2𝑥 + 2𝑦 + 𝑧 = −3

c. {

3𝑥 + 𝑦 − 2𝑧 = 7𝑤𝑥 − 𝑦 − 3𝑧 = 5𝑤2𝑥 + 2𝑦 = −3𝑤

d. {

3𝑥 + 𝑦 − 2𝑧 = 7𝑥 − 𝑦 − 3𝑧 = 52𝑥 + 2𝑦 = −3

36. Solve the system of equations.

{

8𝑥 + 2𝑦 − 5𝑧 = −13𝑥 + 2𝑦 + 3𝑧 = −4−5𝑥 + 4𝑦 + 𝑧 = 8

a. (1, 1, −1) b. (−1, 1, 1)

c. (−1, 1, −1) d. (1, −1, 1)

37. Identify the type of system:

{4𝑥 + 1𝑦 = 5

8𝑥 + 2𝑦 = 10

a. Dependent b. Inverse

c. Consistent d. Inconsistent

38. Given the reduced matrix, write the solution:

[1 0 1 ⋮ 10 1 1 ⋮ 00 0 0 ⋮ 4

]

a. (1 − 𝑧, −𝑧, 𝑧) b. (1, 0, 4)

c. (1 − 𝑧, −𝑧, 4) d. 𝑁𝑜 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛

39. Given the reduced matrix, write the solution:

[1 0 3 40 1 −1 00 0 0 0

]

a. (−3𝑧 + 4, 𝑧, 𝑧) b. (4

3, 1, 𝑧)

c. (3𝑧, −1𝑧, 4𝑧) d. (𝑧, 𝑧, 𝑧)

40. The table shown represents 3 people’s trips to the concession stand at a baseball game. Use the

information to determine the price of one hotdog.

Hotdogs Cokes Pretzels Total spent

Sally 4 6 2 $26.80

Bink 6 3 1 $22.80

Roana 2 4 8 $28.80

a. $1.88 b. $2.28

c. $2.15 d. $2.35

41. A volleyball team paid $5 for a pair of socks and $17 for a pair of shorts. Last year, the total bill

was $315. This year, they purchased the same number of socks and shorts, but the total was

$342. Socks are now $6 a pair and shorts are $18 a pair. How many pairs of shorts did the team

buy? (Hint: try to set up a table like the one above)

a. 12 b. 15

c. 21 d. 8

42. Identify the system of inequalities that creates the feasible region shown:

a. {

𝑥 + 3𝑦 ≤ 198𝑥 + 𝑦 ≤ 73

−𝑥 + 7𝑦 ≤ −16𝑥 ≥ 2

b. {

−𝑥 − 3𝑦 ≤ 198𝑥 + 𝑦 ≥ 73

𝑥 − 7𝑦 ≤ −16𝑥 ≥ 2

c. {

−𝑥 + 3𝑦 ≤ 198𝑥 + 𝑦 ≤ 73

−𝑥 − 7𝑦 ≤ −16𝑥 ≥ 2

d. {

−𝑥 + 3𝑦 ≤ 198𝑥 + 𝑦 ≥ 73

−𝑥 − 7𝑦 ≥ −16𝑥 ≥ 2

43. Use the optimizing function to Minimize the feasible region created by the system:

{𝑦 ≤ 5𝑥 ≥ 2

−𝑥 + 𝑦 ≥ −1

𝑧 = 8𝑥 − 2𝑦

a. 22 b. 38

c. 14 d. 6

44. The matrix simplification is not possible because _____.

[1 5 42 9 7

] + [8 91 30 2

]

a. Adding inverse matrices is wrong and causes mass destruction of sanity.

b. Addition is only possible when both matrices are perfectly square.

c. The number of rows of the first matrix do not match the number columns of the second matrix.

d. Both matrices must be exactly the same size in order to add them together.

45. Simplify the Matrix Expression:

2 [3 41 −1

] [32

]

a. [6 122 −2

] b. [342

]

c. [24 12] d. Not Compatible

46. The inverse of this matrix is impossible to compute because ______.

[3 −42 6

−1 5]

a. All entries must be positive values. b. The matrix must be square.

c. This matrix is too big. d. All entries must be 0 𝑜𝑟 1 𝑜𝑟 − 1.

47. Compute the inverse of the matrix:

𝐴 = [1 0 00 −1 0

−1 0 1]

a. [1 0 00 −1 01 0 1

] b. [−1 0 00 1 01 0 −1

]

c. [1 0 00 −1 01 0 −1

] d. [1 0 00 1 0

−1 0 −1]

48. Given the matrix equation, solve for 𝑋: 𝐴𝑥 = 𝑏

a. 𝑥 = 𝑏/𝐴 b. 𝑥 = 𝐴−1𝑏

c. 𝑥 = 𝑏−1𝐴 d. 𝑥 = 𝑏𝐴−1

49. Given the matrix equation, solve for 𝑋:

𝐴 = 𝑋𝐶 when 𝐴 = [5 72 3

] 𝑎𝑛𝑑 𝐶 = [2 31 1

]

a. [2 −10 1

] b. [2 11 0

] c. [1 21 1

] d. [−1 21 −1

]

50. Write the system of equations as a Matrix Equation:

{5𝑥 − 3𝑦 + 7𝑧 = 12

𝑥 − 𝑦 = 2𝑦 + 8𝑧 = 9

a. [5 −3 7 121 −1 0 20 1 8 9

] b. [5 −3 71 −1 00 1 8

] [𝑥𝑦𝑧

] = [1229

]

c. [𝑥𝑦𝑧

] = [5 −3 71 −1 00 1 8

] [1229

] d. [𝑥𝑦𝑧

] = [5 −3 71 −1 00 1 8

]

−1

[1229

]

Answer Key:

1 D

2 D

3 C

4 B

5 C

6 B

7 C

8 B

9 A

10 D

11 B

12 D

13 B

14 A

15 D

16 A

17 C

18 A

19 D

20 B

21 A

22 D

23 B

24 A

25 C

26 D

27 C

28 A

29 C

30 D

31 C

32 D

33 B

34 A

35 D

36 C

37 A

38 D

39 A

40 D

41 B

42 C

43 D

44 D

45 B

46 B

47 A

48 B

49 B

50 B