MATH12550 Exam1 Review Questions Answers PDF

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MATH 12550.00 - Exam 1 Review - 1.1-1.7, 2.1-2.3, 3.1-3.6

FUNCTIONS 1) Evaluate the function f (x) = −2x2 + 3x at the indicated values: a) f (−3); b) f (2); c) f (−a); d) −f (a); e) f (a + h) . Answer: a) −27; b) −2; c) −2a2 − 3a; d) 2a2 − 3a; e) −2a2 − 4ah − 2h2 + 3a + 3h 2) Evaluate the function f (x) = 2|3x − 1| at the indicated values: a) f (−3); b) f (2); c) f (−a); d) −f (a); e) f (a + h). Answer: a) 20; b) 10; c) 2| − 3a − 1|; d) −2|3a − 1|; e) 2|3a + 3h − 1| 3) Is the following the graph of a function y = f (x) ? 7 y 6 5 4 3 2 1 −7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7

x 1 2 3 4 5 6 7

Answer: Yes 4) Is the following the graph of a function y = f (x) ? 7 y 6 5 4 3 2 1 −7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7

x 1 2 3 4 5 6 7

Answer: No

5) Is the following the graph of a function y = f (x) ?

1

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5 y 4 3 2 1 −5 −4 −3 −2 −−1 1 −2 −3 −4 −5

x 1

2

3

4

5

Answer: Yes

6) Is the following the graph of a function y = f (x) ? 5 y 4 3 2 1 −5 −4 −3 −2 −−1 1 −2 −3 −4 −5

x 1

2

3

4

5

Answer: No 7) Find the average rate of change of the function h(t) = −16t2 + 80t on the interval [1, 2]. Answer: 32 8) Find the average rate of change of the function f (x) = 2x2 − 9 on the interval [4, b]. Answer: 2(b + 4) 9) Find the average rate of change of the function f (x) = Answer:

−1 4(4 + h)

1 on the interval [1, 1 + h]. x+3

10) Find the difference quotient of the function f (x) = 3x − 5 and simplify. f (x + h) − f (x) Answer: =3 h 11) Find the difference quotient of the function f (x) = 3x2 − 5x + 4 and simplify. f (x + h) − f (x) Answer: = 6x + 3h − 5 h x−3 12) Find the domain of f (x) = 2 . x − 4x − 12 Answer: (−∞, −2) ∪ (−2, 6) ∪ (6, ∞) √ x−6 . 13) Find the domain of g(x) = √ x−4 Answer: [6, ∞) 2

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14) Graph the following piecewise function and find f (−3):  x+1 if x < −2 f (x) = −2x − 3 if x ≥ −2 5 y 4 3 2 1

Answer:

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

15) Graph the   x+1 −2x − 3 f (x) =  4

Answer:

x 1

2

3

4

5

; f (−3) = −2

following piecewise function and find f (1): if x < −2 if − 2 ≤ x ≤ 1 if x > 1 5 y 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1

2

3

4

5

; f (1) = −5

16) Identify the intervals on which the function is increasing and decreasing and has local extrema. 7 y 6 5 4 3 2 1

−7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7

x 1 2 3 4 5 6 7

Answer: Increasing on (2, ∞) and decreasing on (−∞, 2), local minimum at (2, −2) 17) Identify the intervals on which the function is increasing and decreasing and has local extrema. 3

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7 y 6 5 4 3 2 1 −7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7

x 1 2 3 4 5 6 7

Answer: Decreasing on (−2, 1) and increasing on (−∞, −2) ∪ (1, ∞), local maximum at (−2, 6) and local minimum at (1, −6) 18) Let f (x) = x2 + 2x and g(x) = 5x + 1, find f (g (x)) and g(f (x)). Answer: f (g(x)) = 25x2 + 20x + 3; g(f (x)) = 5x2 + 10x + 1 √ x+3 and g(x) = x − 1, find f (g(x)) and g(f (x)). r √2 x+3 1−x+3 ; g(f (x)) = 1 − Answer: f (g(x)) = 2 2 1 1 and g(x) = 20) Let f (x) = , find f (g (x)) and the domain of f (g(x)). x+3 x− 9    1 26 26 Answer: f (g(x)) = 1 ; domain: −∞, ∪ , 9 ∪ (9, ∞) +3 3 3 x−9 19) Let f (x) =

21) Use f (x) = x3 to sketch a graph of g(x) = (x + 4)3 . 7 y 6 5 4 3 2 1

Answer:

−7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7

22) Use f (x) =

√

x to sketch a graph of g(x) = 8 y 7 6 5 4 3 2 1

Answer:

x 1 2 3 4 5 6 7

−7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7

√

x + 5.

x 1 2 3 4 5 6 7

4

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23) Use f (x) =

√ 3

x to sketch a graph of g(x) = 7 y 6 5 4 3 2 1

Answer:

−7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7

√ 3

−x.

x 1 2 3 4 5 6 7

24) Use f (x) = |x| to sketch a graph of g(x) = 2|x − 3| + 1. 7 y 6 5 4 3 2 1

Answer:

−7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7

x 1 2 3 4 5 6 7

25) Is the function f (x) = 3x4 even, odd, or neither? Answer: Even 1 26) Is the function f (x) = + 3x even, odd, or neither? x Answer: Odd 27) Solve |x + 4| = 18. Answer: {−22, 14}       3 1 28) Solve  x + 5 =  x − 2. 4   3 36 84 Answer: − , 13 5    1  29) Solve  x − 2 ≤ 7. 3 Answer: [−15, 27] 30) Find the inverse f −1 (x) of f (x) = Answer:

2x 1−x

x x+2

31) Find the inverse f −1 (x) of f (x) = x3 − 5 5

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Answer:

√3

x+5

LINEAR FUNCTIONS 32) Write down the equation of a line, in slope-intercept form, that goes through the points (7, 5) and (3, 17). Answer: y = −3x + 26 33) Write down the equation of a line, in slope-intercept form, which has x-intercept (6, 0) and y-intercept (0, 10). 5 Answer: y = − x + 10 3 34) Write down the equation of the line below. 7 y 6 5 4 3 2 1 x −7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7

1 2 3 4 5 6 7

Answer: y = 3x − 3 35) Write down the equation of the line below. 7 y 6 5 4 3 2 1 x −7−6−5−4−3−2−1 −1 1 2 3 4 5 6 7 −2 −3 −4 −5 −6 −7 x Answer: y = − + 4 2 36) Graph the function f (x) = 2x − 5

6

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7 y 6 5 4 3 2 1 −7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7 Answer:

x 1 2 3 4 5 6 7

37) Are the following two lines given by equations below, parallel, perpendicular or neither? 

2x − 6y = 12 −x + 3y = 1

Answer: Parallel 38) Line 1 passes through (5, 11) and (10, 1). Line 2 passes through (−1, 3) and (−5, 11). Are the two lines parallel, perpendicular or neither? Answer: Parallel 39) Line 1 passes through (8, −10) and (0, −26). Line 2 passes through (2, 5) and (4, 4). Are the two lines parallel, perpendicular or neither? Answer: Perpendicular 40) Write an equation for a line perpendicular to y = 5x − 1 and passing through the point (5, 20). Answer: y = −51x + 21 41) Find a line parallel to y = 4, through the point (7, −2). Answer: y = −2 42) Does the following table represent a linear function? If so, find a linear equation that models the data. x -4 0 2 10 g(x) 18 -2 -12 -52

Answer: Yes. g(x) = −5x − 2 43) Does the following table represent a linear function? If so, find a linear equation that models the data.

7

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x 6 8 12 26 g(x) -8 -12 -18 -46

Answer: No. 44) On June 1st, a company has $4,000,000 profit. The company loses $150,000 per day, every day in June. Write down an expression P (n) which models the company’s profit on day n of June. Answer: P (n) = 4, 000, 000 − 150, 000n 45) The number of people getting a cold each year has dropped steadily by 50 since the year 2004 until 2010. In 2004, 875 people had colds. Find the linear function which models the number of people with colds as function of t time (in years). If this trend continues, when will there be no more people getting colds? Answer: P (t) = −50t + 875, middle of 2021. 46) In 2004, the school population was 1,700. In 2012, the population grew to 2,500. What was the average population growth per year? Find an equation P (t) for the school population t years after 2004. Answer: 100 students per year, P (t) = 100t + 1700 QUADRATIC FUNCTIONS 47) Subtract: (6 − 5i) − (10 + 3i) Answer: −4 − 8i 48) Multiply: (2 − 3i)(3 + 6i) Answer: 24 + 3i 2−i 49) Divide: 2+i 3 4 Answer: − i 5 5 50) Find all solutions of x2 − 4x + 5. Answer: x = 2 ± i 51) Find all solutions of x2 + 2x + 10. Answer: x = −1 ± 3i 52) Find the vertex, intercepts, and graph the function f (x) = x2 − 4x − 5 Answer: Vertex: (2, −9); y-intercept: (0, −5); x-intercepts: (−1, 0) and (5, 0);

8

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10 y 9 8 7 6 5 4 3 2 x 1 −10 −9−8−7−6−5−4−3−−1 2−1 1 2 3 4 5 6 7 8 910 −2 −3 −4 −5 −6 −7 −8 −9 −10 53) Find the vertex, intercepts, and graph the function f (x) = −2x2 − 4x Answer: Vertex: (−1, 2); y-intercept: (0, 0); x-intercepts: (0, 0) and (−2, 0); 7 y 6 5 4 3 2 1 x −7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7

1 2 3 4 5 6 7

√ √ 54) Find an equation of a quadratic function with x-intercepts (− 5, 0) and ( 5, 0) and y intercept (0, −10). Answer: f (x) = 2x2 − 10 55) Find an equation of a quadratic function with vertex (−2, 3) and a point on the graph is (3, 6). 3 Answer: f (x) = (x + 2)2 + 3 25 56) A rectangular plot of land is to be enclosed by a fence. One side is along a river, and does not need to be enclosed. If the total available fencing is 600 meters, find the dimensions of the plot to have the maximum area. Answer: Length: 300 meters, Width: 150 meters. 57) Sketch a graph of a quadratic function, whose graph opens downwards, and has one real root.

9

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y 5

x −6

−4

−2

2

4

6

−5

Answer:

58) Sketch a graph of a quadratic function, whose graph opens upwards, and has two imaginary roots. y 5

x −6

Answer:

−4

−2

2

4

6

−5

POLYNOMIAL FUNCTIONS 59) Determine if the function f (x) = 4x5 − 3x3 + 2x − 1 is a polynomial. If so, give the degree and leading coefficient. Answer: Yes; degree: 5; leading coefficient: 4 60) Determine if the function f (x) = 5x+1 − x2 is a polynomial. If so, give the degree and leading coefficient. Answer: No 61) Determine if the function f (x) = x2 (3 − 6x + x2 ) is a polynomial. If so, give the degree and leading coefficient. Answer: Yes; degree: 4; leading coefficient: 1 62) Determine the end behavior of the polynomial f (x) = 2x4 + 3x3 − 5x2 + 7 Answer: as x → −∞, f (x) → ∞; as x → ∞, f (x) → ∞ 63) Determine the end behavior of the polynomial f (x) = 2x2 (1 + 3x − x3 ) Answer: as x → −∞, f (x) → −∞; as x → ∞, f (x) → ∞ 64) Find all the zeros of the polynomial, noting multiplicities f (x) = (x + 3)2 (2x − 1)(x + 1)3 10

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1 Answer: x = −3, multiplicity 2; x = , multiplicity 1; x = −1, multiplicity 3 2 65) Find all the zeros of the polynomial, noting multiplicities f (x) = x5 + 4x4 + 4x3 Answer: x = 0, multiplicity 3; x = −2, multiplicity 2 66) Determine the zeros of the polynomial function below, noting the multiplicities, and write down possible algebraic expression: 7 y 6 5 4 3 2 1 x −7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7

1 2 3 4 5 6 7

Answer: x = −2, multiplicity 2; x = 2, multiplicity 1, f (x) = 21(x − 2)(x + 2)2 67) Determine the zeros of the polynomial function below, noting the multiplicities, and write down possible algebraic expression: 7 y 6 5 4 3 2 1 x −7−6−5−4−3−2−1 −1 1 2 3 4 5 6 7 −2 −3 −4 −5 −6 −7 1 Answer: x = , multiplicity 1; x = 3, multiplicity 3, f (x) = a(x − 21)(x − 3)3 2 68) Sketch a graph of a degree 3 polynomial with 1 real root.

11

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7 y 6 5 4 3 2 1 −7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7 Answer:

x 1 2 3 4 5 6 7

69) Sketch a graph of a degree 6 polynomial with 3 real roots, one of which has multiplicity 2 and one has multiplicity 3. 7 y 6 5 4 3 2 1 x −7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7 Answer:

1 2 3 4 5 6 7

70) Use the Intermediate Value Theorem to show f (x) = x3 − 5x + 1 has a zero between x = 2 and x = 3. Answer: f is a polynomial and since f (2) is negative and f (3) is positive, there is at least one real zero between x = 2 and x = 3. x3 − 2x2 + 4x + 4 x−2 Answer: Quotient is x2 + 4, remainder is 12 71) Divide:

3x4 − 4x2 + 4x + 8 x+1 Answer: Quotient is 3x3 − 3x2 − x + 5, remainder is 3 72) Divide:

73) Solve 2x3 − 3x2 − 18x − 8 = 0 1 Answer: −2, 4, − 2 74) Solve 3x3 + 11x2 + 8x − 4 = 0 1 Answer: −2, 3 12

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75) Solve 2x4 − 17x3 + 46x2 − 43x + 12 = 0 1 Answer: 1, 3, 4, 2 76) Construct a polynomial with the following √ information: roots √ x = −2 and x = 2 of multiplicity 1; root x = 0 of multiplicity 3; roots x = 3 and x = − 3) of multiplicity 2; root i of multiplicity 1. Answer: f (x) = x3 (x + 2)(x − 2)(x2 − 3)2 (x2 + 1) 77) Construct a polynomial with the following information: roots x = 3 and x = 5 of multiplic√ √ ity 1; root x = −1 of multiplicity 2; roots 2 and − 2) of multiplicity 1; root −2i of multiplicity 3. Answer: f (x) = (x − 3)(x − 5)(x + 1)2 (x2 − 2)(x2 + 4)3 78) Find the polynomial function which has a double zero at x = 3, zeros at x = 1 and x = −2, and a y-intercept at (0, 12). 2 Answer: f (x) = − (x − 3)2 (x − 1)(x + 2) 3 1 79) Find the polynomial function which has a zero of multiplicity 3 at x = , a zero at x = −3, 2 and contains the point (1, 8). Answer: f (x) = 2(2x − 1)3 (x + 3) 80) Sketch the polynomial f (x) = −

1 (x − 1)2 (x + 2)2 (x − 5)3 125

10 y 9 8 7 6 5 4 3 2 x 1 −1 1 2 3 4 5 6 7 −7−6−5−4−3−2−1 −2 −3 −4 −5 −6 −7 −8 −9 −10 Answer:

13

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