Pre-Calculus Semester 1 Practice Final

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1 Class: Date: Pre-Calculus Semester Practice Final Multiple Choice Identify the choice that best completes the statement or answers the question.. Evaluate the function at the specified value of the independent variable and simplify. q ( x) = q y ˆ a e 7 y 6 y 4 x x + b x + 6 x 4 c y + 6 y 4 d. Find all real values of x such that f (x) = 0. f ( x) = 6x 9. Find the domain of the function. q ( s) = 64 s a s 8 b s 8ors 8 c s 0 d all real numbers e 8 s 8 6. Let the function f be defined by the equation y = f (x), where x and f (x) are real numbers. Find the domain and range of the function f (x) = 6x. a domain: (, 4 ] [ 4 }, + ) b domain: (, 6 ) ( 6, + ) a 4 b ± 4 c ± d e c domain: (, 4 ] [ 4, + ). Find the value(s) of x for which f (x) = g (x). f ( x) = x + 7x + g ( x) = 6x 9 d domain: (, 6 ] [ 6, + ) e domain: (, 4 ) ( 4 }, + ) a 6, 7 b, 7, c 40, d 6, 7 7. Use the functions given by f ( x) = x 8 + and e, 6, 4. Find the domain of the function. q ( s) = 8s s 6 a s = 6, s = 0 b all real numbers s 6 c all real numbers s 6, s 0 d all real numbers e s = 6 g( x) = x to find the indicated value. f û gˆ ( ) a undefined b 6 67 e c d

2 8. A rectangle is bounded by the x-axis and the semicircle y= 4 x (see figure). Write the area A of the rectangle as a function of x and determine the domain of the function. y= 4 x a A( x) = x 4 x, x 0 b A( x) = x 4 x, x c A( x) = x 4 x, all real numbers d A( x) = x 4 x, x 0 e A( x) = x 4 x, x 9. Evaluate the difference quotient for the function. f (x) = x - 9 a b 9 c x + 9 d 9 e 9 0. Find the difference quotient and simplify your answer. f yˆ = 4y + 6y, f ( + h) f ( ), h 0 h a 4h b 6 4h c 6 4y + y e 4y + y. Use the graph of the function to find the domain and range of f. d 8 + h a domain: allrealnumbers range: (, )ß(, ) b domain: allrealnumbers range: allrealnumbers c domain: (, )ß(, ) range: (, )ß(, ) d domain: allrealnumbers range:, ] È, ) e domain: (, )ß(, ) range: (, )ß(, ). Determine whether the function is even, odd, or neither. f ( x) = 7x 4 a odd b even c neither. Describe the sequence of transformations from the related common function f ( x) = x to g. g ( x) = x + 6 a reflection in the y-axis; then horizontal shift 6 units right b reflection in the y-axis; then vertical shift 6 units up c reflection in the y-axis; then horizontal shift 6 units left d reflection in the x-axis; then vertical shift 6 units up e reflection in the x-axis; then vertical shift 6 units down

3 4. Find all vertical and horizontal asymptotes of the function. 7. Use the graphs of f and g to evaluate the function. f ( x ) = x a Horizontal : y =, Vertical : x = b Horizontal : y = 0, Vertical : x = c Horizontal : y =, Vertical : x = d Horizontal : y = 0, Vertical : x = e Horizontal : y =, Vertical : x =. Find g û f. f ( x) = x g ( x) = x a b c d e g û f g û f g û f g û f g û f ˆ ( x) = x 6x + 9 ˆ ( x) = x + 9 ˆ ( x) = x ˆ ( x) = x x + 9 ˆ ( x) = x 9 6. Let f ( x ) =, g (x) = x +. Find the composite x function which expresses the given correspondence correctly. x + a ( g û g ) ( x ) b ( f û g ) ( x ) c ( f û f ) ( x ) d ( g û f ) ( x ) e none of the above fû gˆ ( ) a b 9 c 0 d 4 e 8. Find the inverse function of f. f ( x) = x + a f c f e f ( x) = x + b f ( x) = x ( x) = x + ( x) = x + d f ( x) = x 9. The function f ( x ) = x is one-to-one on the domain ( x 0). Find f ( x ). a f ( x ) = x + b f ( x ) = x + c f ( x ) = x d f ( x ) = x e f ( x ) = x +

4 0. Find all real zeros of the polynomial f ( x) = x + x 6x 80 and determine the multiplicity of each. a x =, multiplicity; x = 4, multiplicity b x =, multiplicity c x = 4, multiplicity ; x =, multiplicity; x =, multiplicity d x = 4, multiplicity; x =, multiplicity e x = 4, multiplicity; x = 4, multiplicity; x =, multiplicity. Find all real zeros of the polynomial f ( x) = x 4 + 8x + 7x and determine the multiplicity of each. a x = 0,multiplicity;x = 7,multiplicity ;x = 7,multiplicity ; x =,multiplicity b x = 0, multiplicity; x = 7, multiplicity; x =, multiplicity c x = 7, multiplicity; x =, multiplicity d x = 0, multiplicity; x = 7, multiplicity ; x =, multiplicity e x = 7, multiplicity ; x =, multiplicity. An open box is to be made from a square piece of cardboard, inches on a side, by cutting equal squares with sides of length x from the corners and turning up the sides (see figure below). Determine the function, V, in terms of x, that represents the volume of the box. a V ( x) = 4x + 4x x b V ( x) = x + x c V ( x) = 4x 84x + 44x d V ( x) = 4x 4x + x e V ( x) = 4x + 4x. Do the operation and express the answer in a + bi form. ( 64) + (6 + 6) a 8 + i b 8 - i c i d i e i 4. Determine the domain of the function x f ( x) = x + 4x. a Domain: all real numbers b Domain: all real numbers except x = and c Domain: all real numbers except x = d Domain: all real numbers except x = and e Domain: all real numbers except x = and. Determine the zeros (if any) of the rational function f ( x) = x 49 x 4. a x = 7, x = 7 b x = 49, x = 49 c x = 7 4, x = 7 4 d no zeros e x = 4 4

5 6. Let Q represent a mass of radioactive radium ( 6 Ra) (in grams), whose half-life is 99 years. The quantity of radium present after t years is Q = ˆ t / 99 Determine the quantity present after 00 years. Round to the nearest hundredth of a gram. a 0.88 g b 0.47 g c.00 g d 0.00 g e 4.9 g 7. Rewrite the logarithmic equation log 4 6 = in exponential form. a 4 6 = b 4 = c 4 = 6 6 d 4 / 6 = e 6 ˆ = 4 8. Write the logarithmic equation ln = in exponential form. a.0e = b e = c.0 0 = d 0 = e e = 9. Rewrite the exponential equation = in logarithmic form. a log = b log = c log = d log = e log = 0. Write the exponential equation e / = in logarithmic form a ln( ) = b log 0 ( ) = c ln( ) = d ln e.0log ˆ = ˆ = Solve the equation log( x) = log( 0) for x using the One-to-One Property. a b 0 c d The equation has no solution. e 9. Identify the x-intercept of the function f(x) = ln( x-). a x = b x = 0 c x = d x = 4 e The function has no x-intercept.. A population growing at an annual rate r will triple in a time t given by the formula t = ln. If the r growth rate remains constant and equals 9% per year, how long will it take the population of the town to triple? a 6.6 years b years c. years d. years e. years 4. Simplify the expression log 0. a The expression cannot be simplified. b 6 c log 6 d + log 6 e 0 log 4 ˆ. Simplify the expression log 7. a 08 b 4 c d e The expression cannot be simplified. 6. Assume that x, y, z and b are positive numbers. Use the properties of logarithms to write the expression x 7 y log b 4 z 4 in terms of the logarithms of x, y, and z. a 8log b x + 8log b y 6log b z 7 b 4 log b x + log b y log b z 7 c 4 log b (x +y z) d log b x + log b y log b z e 7 4 log b x + log b y log b z

6 7. Condense the expression È log x + log 7 È 4 4 log 4 y to the logarithm of a single term. 7x ( 7x) a log 4 b log y 4 y d log 4 7x y e log 4 7x log 4 y 8. Find the exact value of log 8 64 a calculator. 64 a b c d 64 c log 4 e 6 7x y without using 9. Find the exact value of log 4 6 log 4 9 without using a calculator. a 9 b c 4 d 9 e 40. Find the exact value of lne.0 ln e without using a calculator. a b. c. d e 4. Solve the equation. x = a x = 0.4 b x =.868 c x = d x =.407 e x =.66 x ˆ 4. Solve = for x. a b - c - d - e no solution 4. Solve for x: 9 0 x ˆ =. Round to decimal places. a.407 b.6 c.6 d e no solution 44. Simplify the expression. log 6 a 6 b 6 c d 8 e none of these 4. Simplify the expression. 9 log 9 a b 8 c 4 d 9 e none of these 46. Use the One-to-One Property to solve the following equation for x. x = 8 a 7 b 7 c 64 d e Solve for x: 4 x / = Round to decimal places. a 0.8 b.794 c.9 d.9 e An initial investment of $9000 grows at an annual interest rate of % compounded continuously. How long will it take to double the investment? a.86 years b 4.40 years c.40 years d year e 4.86 years 49. Carbon dating presumes that, as long as a plant or animal is alive, the proportion of its carbon that is 4 C is constant. The amount of 4 C in an object made from harvested plants, like paper, will decline exponentially according to the equation A = A o e 0.000t, where A represents the amount of 4 C in the object, A o represents the amount of 4 C in living organisms, and t is the time in years since the plant was harvested. If an archeological artifact has 40% as much 4 C as a living organism, how old would you predict it to be? Round to the nearest year. a 7 years b 87 years c 74 years d 6,006 years e 0,4 years 0. Tritium, a radioactive isotope of hydrogen, has a half-life of.4 years. Of an initial sample of 69 grams, how much will remain after 7 years? a.848 grams b 6.89 grams c grams d.046 grams e 7.00 grams 6

7 . The chemical acidity of a solution is measured È in units of ph: ph = log H + È, where H+ is the hydrogen ion concentration in the solution. È What is H + if the ph = 8.8? a b c d e The chemical acidity of a solution is measured È in units of ph: ph = log H + È, where H+ is the hydrogen ion concentration in the solution. If a sample of rain has a ph of., how many È times higher is its H + than pure water's, which has a ph of 7? a 7 b c.6 0 d 6. 0 e Determine two coterminal angles (one positive and one negative) for θ = π. a 0π, 8π d π, 7π b 4π, 4π 9 e 7π, π c 8π, 4π 4. Find (if possible) the supplement of π. π a π e b π 6 c not possible d π 7. Find the angle, in radians, in the figure below if S = and r = 8. a π b c π d e π 8. Find the length of the arc, S, on a circle of radius 7 feet intercepted by a central angle of 0. Round to two decimal places. a S =.76 feet b S = 80.6 feet c S = 40. feet d S = 6.88 feet e S =. feet 9. The displacement from equilibrium of an oscillating weight suspended by a spring is given by y(t) = cos6t, where y is the displacement in centimeters and t is the time in seconds. Find the displacement when t =.4, rounding answer to four decimal places. a.784 cm b.497 cm c.87 cm d.60 cm e.4460 cm 60. Find the exact value of cscθ, using the triangle shown in the figure below, if a = 4 and b = 7.. Rewrite 4π 9 in degree measure. a b 0 c 60 d 40 e Rewrite 47.0 in radian measure. Round to three decimal places. a 0.64 b c 0.8 d 0. e 0.48 a 4 b 4 7 c 4 d 7 e 7 7

8 6. Given sin0 = and cos0 =, determine the following: csc 0 a undefined b csc 0 = c csc 0 = d csc 0 = e csc 0 = 6. Given the figure below, determine the value of sinθ. 6. Given sec θ = 0 and tanθ =, determine the following. csc( 90 θ) a csc( 90 θ) = 0 b undefined c csc( 90 θ) = e csc( 90 θ) = 0 0 d csc( 90 θ) = 0 6. If sinθ =, find the value of θ in degrees ( 0 < θ < 90 ) without the aid of a calculator. a θ = b θ = 90 c θ = 60 d θ = 7 e θ = Using the figure below, if θ = 6 and y = 8, determine the exact value of x. a sinθ = 4 d sinθ = 4 b sinθ = e sinθ = 4 c sinθ = The point (, ) is on the terminal side of an angle in standard position. Determine the exact value of tanθ. a tanθ = d tanθ = b tanθ = 7 e tanθ = c tanθ = 67. Given the equation below, determine two solutions such that 0 θ < 60. csc θ = a θ =, b θ = 0, 0 c θ = 0, 0 d θ = 60, 40 e θ = 4, 8 a x = tan6 8 d x = cot 6 4 b x = sin e x = tan4 c x = 6 csc 8 8

9 68. Given the equation below, determine two solutions such that 0 θ < π. sec θ = a θ = π 6, 7π 6 b θ = π 6, π 6 7. Use an inverse function to write θ as a function of x. c θ = 4π, π d θ = π, 4π e θ = π 4, π If a = and c = 0, determine the value of A. Round to two decimal places. a θ = arccos c θ = arccos x + 9 x + 9 b θ = arccos x d θ = arccos π x e θ = arccos x 7. Use an inverse function to write θ as a function of x. a 7.96 b 80. c 4.04 d 4.48 e 7. ˆ 70. Evaluate cos calculator. a π 6 b π 7. Evaluate arctan a π 4 b π c π 6 c π 6 without using a d π e π 4 without using a calculator. d π 6 e π 4 x + ˆ a θ = tan 4 c θ = tan x + ˆ e θ = sin ( x + ) b θ = tan 4 ˆ x + d θ = tan ˆ x Use the properties of inverse trigonometric È functions to evaluate sin arcsin( 0.6). a 0.6 b 0.40 c 0.9 d.07 e 0.9 9

10 7. Use the properties of inverse trigonometric È functions to evaluate arctan tan π ˆ If B = 9 and a = 9, determine the value of c. Round to two decimal places. a π 7 b π 9 c 7π 9 d π 9 e 9π 76. Find the exact value of sin arctan ˆ a b c d e A granular substance such as sand naturally settles into a cone-shaped pile when poured from a small aperture. Its height depends on the humidity and adhesion between granules. The angle of elevation of a pile, θ, is called the angle of repose. If the height of a pile of sand is feet and its diameter is approximately feet, determine the angle of repose. Round answer to nearest degree. a 0.9 b 4.99 c 8.6 d 4.6 e Find the altitude of the isosceles triangle shown below if θ = 7 and b = 7 meters. Round answer to two decimal places. a 7 b 8 c 6 d 0 e 9 a.84 meters b 6.4 meters c.8 meters d.8 meters e. meters 80. After leaving the runway, a plane's angle of ascent is and its speed is 76 feet per second. How many minutes will it take for the airplane to climb to a height of,000 feet? Round answer to two decimal places. a.9 minutes b.9 minutes c.0 minutes d.7 minutes e 0.9 minutes 0

11 Pre-Calculus Semester Practice Final Answer Section MULTIPLE CHOICE. C. D. D 4. B. B 6. A 7. C 8. B 9. A 0. A. C. C. D 4. D. A 6. B 7. E 8. D 9. E 0. E. B. C. B 4. E. A 6. E 7. C 8. E 9. E 0. C. E. D. E 4. D. C 6. B 7. C 8. B 9. E

12 40. E 4. C 4. C 4. A 44. A 4. A 46. B 47. E 48. A 49. C 0. D. B. D. C 4. D. E 6. C 7. B 8. C 9. B 60. D 6. B 6. A 6. C 64. A 6. D 66. A 67. C 68. B 69. D 70. C 7. D 7. B 7. A 74. A 7. B 76. D 77. D 78. A 79. B 80. C