110 Practice Final Fall 2023

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The University of Tennessee, Knoxville *

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MATH-113

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Mathematics

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Jan 9, 2024

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pdf

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110 Practice Final Fall 2023 Name: Date: Instructor: This is 50 question multiple choice exam. You have exactly 2 hours to finish the exam. You must use a number two pencil. Bubble in your name and identification number before you start the exam on the Akindi Sheet. Bubble in your answers on positions 1-50. Fill in each circle completely and do not make marks anywhere else on the Akindi Sheet. It is recommended that you circle your answers on your exam as well. 1
1. Identify the equation of the graphed piecewise defined function: − 2 − 1 1 2 3 4 − 12 − 10 − 8 − 6 − 4 − 2 2 4 6 8 10 12 (A) f ( x ) = x − 9 − 1 ≤ x < 0 3 0 ≤ x ≤ 1 − x + 9 1 < x < 3 (B) f ( x ) = 3 − 1 ≤ x < 0 x − 9 0 ≤ x ≤ 1 − x + 9 1 < x < 3 (C) f ( x ) = x − 9 − 1 < x ≤ 0 3 0 ≤ x ≤ 1 − x + 9 1 < x < 3 (D) f ( x ) = 3 − 1 < x ≤ 0 x − 9 0 ≤ x ≤ 1 − x + 9 1 < x < 3 2
2. Given the Piecewise Function, identify the correct evaluation. f ( x ) = − 3 √ − x x < − 4 x − 2 − 4 ≤ x < 4 − | x − 3 | x ≥ 4 (A) f (5) = 2 (B) f (4) = − 1 (C) f (4) = 0 (D) f (5) = 3 3. Find the new function given the set of equations: ( f − g )( x ) . f ( x ) = x 2 − 4 x + 4 g ( x ) = 3 x 2 + 2 (A) 4 x 2 − 4 x + 6 (B) 4 x 2 − 4 x + 2 (C) − 2 x 2 − 4 x + 6 (D) − 2 x 2 − 4 x + 2 3
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4. Evaluate using the given set of equations: ( f · g )( − 1) f ( x ) = x 2 − 6 x + 9 g ( x ) = 4 x + 2 (A) 7 (B) − 32 (C) − 96 (D) 14 5. Evaluate using the graph of f ( x ) and g ( x ) : ( f ÷ g )(5) 1 2 3 4 5 6 1 2 3 4 5 6 f ( x ) g ( x ) (A) − 2 5 (B) 13 5 (C) 3 5 (D) − 12 5 4
6. Find the composite function given the set of equations: ( f ◦ g )( x ) f ( x ) = x 2 + 5 x g ( x ) = 3 x 2 − 7 (A) 3 x 4 + 30 x 3 + 75 x 2 − 7 (B) 27 x 4 − 126 x 2 + 140 (C) x 4 + 10 x 3 + 30 x 2 + 25 x (D) 9 x 4 − 27 x 2 + 14 7. Given the Graph, identify the correct evaluation. 1 2 3 4 5 6 1 2 3 4 5 6 f ( x ) g ( x ) (A) ( g ◦ f )(3) = 2 (B) ( f ◦ g )(3) = 4 (C) ( g ◦ f )(3) = 5 (D) ( f ◦ g )(3) = 1 5
8. Given the table of data, determine if the table can represent a One-to-one function. age of weight of student backpack 16 19 13 18 15 30 17 18 14 28 (A) Neither one-to-one nor a function (B) Not a function (C) Not one-to-one (D) One-to-one function 9. Use the Horizontal Line Test to determine if the given graph is a One-to-One function. − 1 1 2 3 4 − 9 − 8 − 7 − 6 − 5 − 4 − 3 − 2 − 1 1 2 (A) Not a function (B) One-to-one function (C) Not one-to-one (D) Neither one-to-one nor a function 6
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10. Consider the One-to-One function, Identify the correct statement about its inverse. f ( x ) = { ( − 8 , 2) , (6 , 5) , ( − 1 , − 2) , ( − 7 , 0) , (8 , − 4) } (A) f − 1 ( − 4) = 8 (B) f − 1 ( − 8) = 2 (C) f − 1 ( − 7) = 0 (D) f − 1 ( − 1) = − 2 11. The graph shows the height (in hundreds of feet) as a function of time (in seconds) of a ball falling off a cliff. At what time is the height 6 hundred feet? − 1 1 2 3 4 5 6 − 3 − 2 − 1 1 2 3 4 5 6 7 8 9 10 Time (seconds) Height (hundreds of feet) (A) 2 seconds (B) 4 seconds (C) 400 seconds (D) 200 seconds 7
12. Find the inverse of the given one-to-one function: f ( x ) = 8 x + 5 (A) x 8 − 1 5 (B) x 8 + 1 5 (C) x 8 − 5 8 (D) x 8 + 5 8 13. Solve the Rational Equation: 3 x + 3 + 4 x − 2 = 5 x − 2 ( x − 2) ( x + 3) (A) {− 4 } (B) { 4 } (C) {− 4 , 4 } (D) ∅ 8
14. Find the x -intercept values: f ( x ) = ( x − 3) ( x − 1) ( x − 8) ( x + 6) (A) {− 6 , 8 } (B) {− 8 , 6 } (C) {− 3 , − 1 } (D) { 1 , 3 } 15. Identify the Domain: f ( x ) = ( x − 5) ( x + 9) ( x − 3) ( x − 2) (A) ( −∞ , − 9) ∪ ( − 9 , 5) ∪ (5 , ∞ ) (B) ( −∞ , − 3) ∪ ( − 3 , − 2) ∪ ( − 2 , ∞ ) (C) ( −∞ , − 5) ∪ ( − 5 , 9) ∪ (9 , ∞ ) (D) ( −∞ , 2) ∪ (2 , 3) ∪ (3 , ∞ ) 9
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16. Which Rational Function has 1 hole and 2 Vertical Asymptotes? (A) f ( x ) = ( x − 6) ( x − 1) ( x + 8) ( x + 3) ( x + 5) (B) f ( x ) = ( x − 6) ( x − 1) ( x + 8) ( x − 2) ( x + 3) (C) f ( x ) = ( x − 6) ( x − 1) ( x + 8) ( x − 2) ( x + 3) ( x + 8) (D) f ( x ) = ( x − 6) ( x − 1) ( x + 8) ( x − 2) ( x + 3) ( x + 5) 17. Find the y -intercept value: f ( x ) = 7 x 2 − 5 x + 2 6 x 5 + 8 x 3 − 6 x + 8 (A) 1 4 (B) 4 (C) 7 6 (D) 0 10
18. Find the equation of the Oblique Asymptote: f ( x ) = 5 x 3 + 2 x 2 − 6 x 2 + 2 x − 4 (A) y = 5 x − 1 (B) ∅ (C) y = 5 x − 9 (D) y = 5 x − 8 19. Identify the characteristics of the given funciton: f ( x ) = ( x − 6) ( x − 3) ( x + 8) ( x − 8) ( x − 6) (A) 0 hole(s) 2 vertical asymptote(s) 3 x -intercept(s) (B) 0 hole(s) 3 vertical asymptote(s) 2 x -intercept(s) (C) 2 hole(s) 0 vertical asymptote(s) 1 x -intercept(s) (D) 1 hole(s) 1 vertical asymptote(s) 2 x -intercept(s) 11
20. Find the equation of the Horizontal Asymptote: 4 x 2 + 20 x + 24 3 x 3 + 45 x 2 + 177 x + 135 (A) y = 0 (B) y = 4 3 (C) ∅ (D) y = 1 21. Simplify the radical expression: √ 48 (A) 4 √ 12 (B) 12 √ 12 (C) 4 √ 3 (D) 12 √ 3 12
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22. Simplify the radical expression: 3 √ 135 (A) 3 3 √ 5 (B) 15 3 √ 5 (C) 3 3 √ 15 (D) 15 3 √ 15 23. Solve: √ x − 4 = 5 (A) 29 (B) 9 (C) 36 (D) 65 13
24. Solve: 2 x + 1 = √ 2 x + 13 (A) ∅ (B) {− 2 } (C) 3 2 (D) − 2 , 3 2 25. Solve: 9 | 5 x − 2 | − 4 = 10 (A) − 32 45 , 4 45 (B) 4 45 , 32 45 (C) ∅ (D) − 4 45 , 32 45 14
26. Solve: − | x + 8 | < − 2 (A) ∅ (B) R (C) ( −∞ , − 10) ∪ ( − 6 , ∞ ) (D) ( − 10 , − 6) 27. Suppose $5000 is invested in a bank earning 4% interest compounded monthly. How much money will be in the account after 5 years? (Hint: The two potentially relevant formulas are: A = P (1 + r n ) nt and A = Pe rt ) (A) $6410 . 23 (B) $6104 . 98 (C) $6715 . 48 (D) $5189 . 24 15
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28. Suppose $3000 is invested in a bank earning 5% interest compounded continuously. How much money will be in the account after 15 years? (Hint: The two potentially relevant formulas are: A = P (1 + r n ) nt and A = Pe rt ) (A) $6351 . 00 (B) $6668 . 55 (C) $6986 . 10 (D) $5715 . 90 29. Solve: 2 x − 5 = 3 4 x − 9 (A) ∅ (B) 1 . 698 (C) 1 . 735 (D) 1 . 772 16
30. Solve: 2 x − 2 = 5 9 x − 5 (A) 0 . 557 (B) 0 . 483 (C) 0 . 446 (D) ∅ 31. f ( x ) = log( x ) . Find f (11 . 8) . Round to four decimal places. (A) 1 . 0719 (B) 1 . 0934 (C) 1 . 0492 (D) 1 . 0607 17
32. Solve the question below. Eliminate any extraneous solutions. log 6 ( x + 4) + log 6 ( x + 3) = 1 (A) { 6 } (B) { 2 } (C) {− 1 } (D) {− 6 } 33. Suppose an amount of plutonium-241 is modeled by the formula: y ( t ) = 600 e − 0 . 053 t , where t is measured in years. What is the half-life of plutonium-241? Round your answer to the nearest year. (A) 15 years (B) 9 years (C) 13 years (D) 19 years 18
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34. Solve: log(5 x + 38) = log( x + 26) (A) − 3 (B) 0 (C) − 2 (D) − 1 35. Solve: log 4 ( x − 1) − log 4 ( x − 5) = log 4 (2 x − 2) (A) ∅ (B) 11 2 (C) { 1 } (D) 1 , 11 2 19
36. The following table describes the number of cellphones per year (in millions) from 1988 to 1993, where x is the number of years since 1987. Use Exponential Re- gression to determine the exponential function which best models the number of cellphones (in millions) as a function of time. Time 1 2 3 4 5 Cellphones 1 . 6 2 . 9 4 . 6 6 . 4 8 . 9 (A) y = 1 . 03 · 1 . 46 x (B) y = 1 . 17 · 1 . 53 x (C) y = 1 . 1 · 1 . 6 x (D) y = 1 . 24 · 1 . 74 x 37. Identify the type of linear system: 6 x + 8 y = 4 12 x + 16 y = 8 (A) Consistent (B) Dependent (C) Inconsisent (D) Inverse 20
38. Solve the system of linear equations using Gauss-Jordan elimination via the rref command. x + y = − 9 4 x − 8 y = 24 (A) ( − y − 9 , y ) (B) ( − 4 , − 5) (C) ∅ (D) (4 , 5) 39. Solve the system of linear equations using Gauss-Jordan elimination via the rref command. x + 3 y = 14 − 2 x − 6 y = − 36 (A) ( − 3 y + 14 , y ) (B) ∅ (C) (2 , 4) (D) ( − 2 , − 4) 21
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40. Solve the nonlinear system of equations by graphing. Round to three decimal places. x 2 + y 2 = 4 y = ln( x + 1) (A) (-0.837,1.004) (B) (-0.837,-1.816) (C) (1.73,1.004) and (-0.837,-1.816) (D) (1.73,-1.816) and (-0.837,1.004) 41. Write the system of linear equations as an Augmented Matrix 9 x + 5 y − z = − 7 − 6 x − 2 y + 7 z = − 5 − 3 x − 2 y + 9 z = 5 (A) 9 5 − 1 − 7 − 6 − 2 7 − 5 − 3 − 3 9 5 (B) 9 5 − 1 − 7 − 5 − 2 7 − 5 − 3 − 2 9 5 (C) 9 5 − 1 − 7 − 6 − 2 7 − 5 − 3 − 2 9 5 (D) 9 6 − 1 − 7 − 6 − 2 7 − 5 − 3 − 2 9 5 22
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42. Solve the system of equations. 5 x − 3 y + 4 z = 8 − 5 x + 4 y − z = 7 4 x − 4 y − 2 z = − 4 (A) (99 , 117 , − 34) (B) ( − 13 z 5 + 53 5 , − 3 z + 15 , z ) (C) (34 , 42 , − 9) (D) ∅ 43. Given the reduced matrix, write the solution. 1 0 1 − 9 0 1 6 − 8 0 0 0 − 7 (A) ( − z − 9 , − 6 z − 8 , − 7) (B) ( − 9 , − 8 , − 7) (C) ∅ (D) ( − z − 9 , − 6 z − 8 , z ) 23
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44. Given a Set of Vertex Points from a feasible region, Identify the vertex which will Maximize the given Optimizing Function. Vertex Points: (1 , 4) (2 , 2) (4 , 8) (5 , 6) z = − x − 2 y (A) (5 , 6) (B) (4 , 8) (C) (1 , 4) (D) (2 , 2) 24
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45. Given a system of linear inequalities, Use the graphing features of your calculator to determine which graph shows the feasible region and the correct vertex points. 7 x − 4 y ≥ 3 3 x − 7 y ≤ − 4 4 x + 3 y ≤ 44 (A) 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 (B) 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 (C) 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 (D) 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 25
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46. A restaurant sells two kinds of sandwiches, Veggie and Meaty. A Meaty sandwich use 2 slices of bread and 5 slices of cheese. A Veggie sandwich uses 4 slices of bread and 3 slices of cheese. Write the system of inequalities using v for Veggie and m for Meaty if there are 118 slices of bread available and 111 slices of cheese available. (A) m ≥ 0 v ≥ 0 2 m + 4 v ≤ 118 2 m + 4 v ≥ 111 (B) m ≥ 0 v ≥ 0 2 m + 4 v ≤ 118 5 m + 3 v ≤ 111 (C) m ≥ 0 v ≥ 0 2 m + 4 v ≥ 118 5 m + 3 v ≤ 111 (D) m ≥ 0 v ≥ 0 2 m + 4 v ≥ 118 2 m + 4 v ≥ 111 47. Simplify the Matrix Expression: 2 − 3 1 − 1 − 5 2 − 1 − 4 3 + 3 2 4 − 3 1 1 − 2 5 − 5 (A) ∅ (B) − 7 7 7 (C) − 5 11 − 9 − 13 8 − 7 − 2 − 1 (D) 0 14 − 11 − 7 7 − 8 7 − 9 26
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48. Simplify the Matrix Expression: 3 2 − 5 − 2 5 − 3 3 − 5 5 − 5 − 1 (A) ∅ (B) 0 − 4 − 8 − 6 − 47 − 15 (C) 50 16 26 12 (D) 25 − 10 − 25 10 − 5 2 49. Compute the inverse of the matrix: − 5 1 2 − 1 2 − 3 2 − 3 4 (A) − 1 − 10 − 7 − 2 − 24 − 17 − 1 − 13 − 8 (B) − 1 − 10 − 7 − 2 − 24 − 17 − 1 − 13 − 9 (C) − 1 − 10 − 7 − 2 − 23 − 17 − 1 − 13 − 9 (D) − 1 − 11 − 7 − 2 − 24 − 17 − 1 − 13 − 8 27
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50. Select the system of equations which corresponds to the given Matrix Equation: 1 2 3 6 3 4 2 − 1 0 x y z = − 9 − 7 3 (A) x + 2 y + 3 z = − 9 6 x + 3 y + 4 z = − 7 2 x − y = 3 (B) x + 2 y + 3 z = − 9 6 x + 3 y + 4 z = − 7 2 x + z = 3 (C) x + 2 y + 3 z = − 9 6 x + 3 y + 4 z = − 7 2 x + y = 3 (D) x + 2 y + 3 z = − 9 6 x + 3 y + 4 z = − 7 2 x − z = 3 28
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