MATH 165 Course Review with Answers

MATH 165 COURSE REVIEW MATH 165 Precalculus General Directions: When asked for EXACT SOLUTIONS, leave answers in fractional or radical form - not decimal form. That is, leave numbers like 2 , 3 , , and e as part of 3 your answer. 1. State the domain of each of the following functions. (a) f ( x)  7x  3 x 1  3x  2  (d) k ( x)  ln    x4  (b) g ( x)  2 (c) h( x)  4  x2 x2  4 x  5 2 x 2  13x  7 (f) m( x)  (e) sin x ex Assume x-scale and y-scale equal 1 2. The table on the right, generated on a graphing calculator, shows some points on the graphs of two functions. Use the table to answer these questions: (a) Where is y1’s x-intercept? (b) Where is y2’s y-intercept? (c) Give the coordinates of each point where y1 and y2 intersect. x y1 y2 2 0 6 1 1 1 0 2 1 1 3 3 2 4 2 3 5 1 4 6 6 3. Use the graph of function f at the right to answer the following: (a) What are the coordinates of the points P and Q? (b) Evaluate f(1). P (c) Evaluate f(0). (d) Solve f ( x)  6 for x. (e) Estimate the value x when f(x) = 0. Q Page 1 of 14 4. If f ( x)  3x 2  5 x  8 , evaluate and simplify f ( x  h)  f ( x ) , h  0. h 5. Given f ( x )  35 x 4  25 x 2 , use your graphing calculator to approximate the following. Round your answers to three decimal places. (a) Find any local minima/maxima. (b) Find intervals where f is increasing and/or decreasing. Use interval notation. 6. A media company is going to install cable from a house to their connection box B. The house is located at one end of a driveway 7 miles back from a road (see diagram). The nearest connection box is 25 miles down the road. The cost of installing the cable is $656 per mile off the road and $375 per mile along the road. cable Let x be the distance from where the driveway meets the road to where the cable comes to the road. 7 mi (a) Develop a function C(x) that expresses the total x installation cost as a function of x. (b) Now use your calculator to graph C. (c) Use the graph to determine the value of x that will produce the minimum cost. Round to the nearest three decimal places. (Tip: use a window [0, 20, 1] x [ 12,000 , 20,000 , 1000 ].) (d) State the minimum cost for that installation, rounded to the nearest cent. cable B 25 mi (b) A partial graph of of g is shown. Complete the graph if g ( x)   g ( x) . 7. (a) A partial graph of of f is shown. Complete the graph if f ( x)  f ( x) 8. On the right is a table of some values of a function f. Answer these questions about f: (a) Tell whether the function is even, odd, or neither. Explain how you know this. (b) List each x-intercept. (c) List each y-intercept. Page 2 of 14 x y 3 7 2 0 0 3 1 5 3 9 5 0 9. The complete graph of f(x) is shown at the left. On the graphs, xscl=1. 1 (a) Sketch the graph of  f  x  3 . (b) In terms of f(x), what is the formula for 2 the graph shown below? A D B E C 10. Use the functions f and g to evaluate the following. (a) (f + g ) (3) (d) (f g) (4) (b) (f  g ) (1) f (e)  g 2  11. If f  x   x 2  1 , and g  x   (a) f  g  f 3 y=f(x) y 4 (c)  f  (2)  g 3 2 (f)  g  f  2  1 x 0 0 1 2 3 4 x  5 , find and simplify (b) f  g x  and state the domain  (d) g 1  x  and state its domain  x 0 1 2 3 4 y  g ( x) 1 -2 0 -4 -3 (c)  g  f   x  and state the domain (e) f  g 1 2  12. Describe the behavior of the power function f  x   ax 88 , for a  0 as regards each of (a)-(d) below If there is insufficient information to decide, then answer “cannot be determined” (a) symmetries (b) domain (d) end behaviors: (c) range As x   , f  x   As x   , f  x   13. For each of these functions: I. Which end is this? left/right Which end is this? left/right f ( x)  5 x3  17 x 2  4 x  14 II. g ( x)  3x 4  20 x 3  51x 2  86 x  40 . (a) Find the zeros and identify them as rational, irrational, or non-real complex. (b) Factor the polynomial completely over the complex numbers. Page 3 of 14 14. The data below represents the number of students enrolled in grades 9 through 12 in private institutions. Year Enrollment in 1000’s 1300 1975 1980 1339 1981 1400 (a) Using a graphing utility, draw a scatter diagram of the data. Determine if the data appears to be linear, quadratic, cubic, exponential, logarithmic or sinusoidal. (Let 1975 correspond to t  0 ) 1982 1400 (b) Using a graphing utility, find the cubic function of best fit and graph. 1983 1400 1984 1400 1985 1362 1986 1336 1987 1247 1988 1206 1989 1193 1990 1137 1991 1125 1992 1163 1993 1191 1994 1236 1995 1260 1996 1297 (c) Use the function found in part (b) to predict the number of students enrolled in private institutions for grades 9 to 12 in the year 2000. Round your answer to the nearest thousand students. (d) Use the function found in part (b) to predict the year in which enrollment in private institutions for grades 9 through 12 will reach 2 ½ million students. (e) The function found in part (b) does approximate the data quite nicely for the years that we have information. Explain why, for this situation, a cubic function might not be a good model over the long term. Include the phrase "end behavior" in your discussion. 15. Following are some tables for the same rational function f. TABLE A x TABLE B x f(x) TABLE C x f(x) f(x) TABLE D x f(x) -4 8.25 -2 2.25 40 4.2505031 -30 4.251524158 -3.1 3104.25 -2.9 -2895.75 400 4.250006111 -300 4.250011451 -3.01 3010004.25 -2.99 -2989995.75 4000 4.250000062 -3000 4.250000111 -3.001 3001000004 -2.999 -2998999996 40000 4.250000001 -30000 4.250000001 (a) Name each table that suggests the existence of a horizontal asymptote (end behavior) . (b) Name each table that suggests the existence of a vertical asymptote (local behavior). Page 4 of 14 (c) Use the tables to complete each of the following statements: i. As x   , f  x   ii. As x   , f  x   iii. Write the equation of any horizontal asymptotes. (Write “none” for your answer if there isn’t one.) iv. Write the equation of any vertical asymptotes. (Write “none” for your answer if there isn’t one.) 16. Consider the rational function h( x)  (3x  2)( x  2) : x( x  2) (a) State the domain of the function. x   , h( x)  ______ . (b) Complete this statement: As (c) Write the equation of any vertical asymptotes. (d) Write the equation of any horizontal asymptotes. (e) Write the coordinates of any holes. (f) List any x-intercepts. (g) Graph and label any asymptotes, intercepts, and holes. (h) State the range of this function. 17. The Parks and Wildlife commission introduces 80,000 fish into a large man-made lake. The population of the fish, in thousands, is given by N (t )  204  3t  , 0  t , where t is the time in years. 1  0.05t (a) Approximate the population when t is 5, 10, and 25. (b) What is the limiting number of fish in the lake as time increase? 18. Find the exact solution of f ( x)  x(3 x  5)  0 . Give the exact values for answer(s) using interval (7 x  3)( x  4) notation. 19. (a) Solve for x: 1  214 x 32 (b) Simplify: Page 5 of 14 e 2ln( x )  ln 5 20. The value of an automobile depreciates. It is originally worth $40,000, but then it loses one-tenth of its value every year. (a) How much is it worth at the end of the first year? (b) How much is it worth after 3 years? (c) How much is it worth after t years? (d) How many years would pass before it was worth only half of its original value? (This would be its “half-life”.) Round this value to the nearest tenth of a year. 21. A function f is defined by the table below: Complete a table for f  1 x 2 1 0 1 2 3 x f(x) 3 1 0 6 5 7 f 1 22. Graph g x   1  5e x 3 . (a) Find all intercepts and asymptotes. Round to two decimal places. (b) State the domain and range of the function. 23. If y  f  x   13 log5 x  , find the inverse function f 1 x  . 24. The unlabeled sketch at right shows the graph of y  log 3 ( x  3) . The sketch correctly shows the shape of the graph, but it is not to scale. (a) (b) (c) Find the equation of any asymptotes and sketch. Find the x-intercept and label. Find the y-intercept and label. 25. Solve for x: log 5(8  18 x)  2 log5 x  1 26. Approximate the value of y to three decimal places: y = log5 40 27. Exponential Growth Applications: Assume the function Pt   Ce kt describes the population P of a certain country where the time is measured in t years. (a) What is the growth rate constant (k) if the population has tripled in 23 years? Approximate k to a tenth of a percent. (b) Use the exact value of k to find the population in 50 years if the initial population was 10,000, 28. Given sin   4 (a) tan  7 and      , find the EXACT values of the following: 2   (b) sec  (c) cos     (d) csc  2  2  Page 6 of 14 29. Graph two full periods of each of the following functions. F  x   3 cos4 x   5 G t   13 sin 2t    For each function determine the following: (a) period (b) range (c) amplitude (d) phase shift 30. Find a function of the form f ( x)  A cos( Bx)  D for the graph below. 31. Household electrical power in the US is provided in the form of alternating current. Typically voltage fluctuates smoothly between +155.6 volts and –155.6 volts. The voltage cycle 60 times per second. Use a cosine function to model the alternating voltage. 32. Establish the identity by verifying that csc csc   2 sec 2  1  csc 1  csc 33. Use trigonometric identities to find the exact value of sin     and cos     , given: tan   5 3 ,    12 2 1 3 sin    ,     2 2 34. Use trigonometric identities to find the exact value of tan(2), where (3, 4) is a point on the terminal side of . 35. Find, to the nearest degree, all angles  , 0    360 , such that sin    0.81 36. Find an algebraic expression in x (not involving trig functions) that is equivalent to for x >0. csc tan 1 3 x    37. Given cos 2 x   sin x : (a) Solve the equation algebraically for the exact value of the solution(s) on the interval [0, 2). (b) Verify the answer(s) in part (a) using the ZERO or INTERSECT features of your graphing calculator. Round answers to the nearest tenth of a radian. Page 7 of 14 38. Write TRUE or FALSE in the blank before each statement. [Recall that if a statement in mathematics is not always true, then it is considered false.] If it is false, write a corrected true statement in the blank. If it is already true, write “Identity” in the blank. (a) sin (A + B) = sin(A) + sin (B) (b) sin 2  A  sin A (c) (sin A + cos A) 2 = sin 2 A + cos 2 A (d) cos (A  B) = cos A  cos B (e) sin (2A) = 2 sin A (f) tan 1  A  2 1 tan( A) y 39. The graph at right shows two cycles of the graph of y  2 cos 2 t  cos t  1 . Algebraically find all zeros to the equation over the domain of all reals. Express your answer in terms of . -2  - 2 40. In traveling across flat land, you notice a mountain directly in front of you. The angle of elevation [to the peak] is 3.5. After you drive 13 miles closer to the mountain, the angle of elevation is 9. Approximate the height of the mountain to the nearest foot. (5280 feet = 1 mile) 41. Sketch the curve described by the parametric equations by hand, showing its orientation. Then verify using a graphing calculator: x  3 sin(t ) , y  5 cos(t ) ; 0  t  2 h Does this curve represent y as a function of x? 42. This question concerns the three graphs. (a) Find possible formulas for the three functions. (i) f (ii) 3 1 40 35 (4, 0) 2 (iii) g (2.5, 3) (-1.5, 0) 4 -5 -4 -3 -2 -1 1 -1 30 25 -2 20 15 -3 2 3 4 5 10 5 -4 -2 0 -5 0 2 4 (b) Estimate the average rate of change for function h between x = 2 and x = 0 ? (c) Where is function h increasing faster, between x = 4 and x = 2, or between x = 2 and x = 0 ? Page 8 of 14 43. For each function, select its most specific function type from the name bank below. Name Bank = {Exponential, linear, logarithmic, polynomial, quadratic, rational, sinusoidal} a. f ( x)  5 x3 b. d. f ( x)  x x2 e. c. f ( x)  5 x3 f. 44. The figure below shows the complete graphs of the functions f and g. Based on the graphs, the equation f ( x)  g ( x)  0 has how many solutions? 45. Use the graphs of the functions h and k below to answer the following: a. b. For what value(s) of x is h( x)  k ( x) ? Solve the inequality h( x)  k ( x) k(x) Page 9 of 14 h(x) MATH 165 Course Review Answer Key 1. a) b)  x x   and x  1  x  2  x  2 3 7  c)  x x  7 or x   b) at y  1 2. a) at x  2 3. a) b) c) d) e) 1  2 d)  ,  4    23 ,   e)  x x  2 or f)  ,   c)  2  x  5 (-1,1) and (4,6) P(2,6) Q(-1,-3) -1 -2 2 x  1.25 4. 6 x  3h  5 5. a) local minima: (-0.598, -4.464) and (0.598, -4.464) local maximum: (0, 0) b) Decreasing: Increasing: (-  , -0.598)  (0, 0.598) ( -0.598, 0)  (0.598,  ) 6. C  x   656 x 2  49  375(25  x) $13,142.74 7. a) b) 8. a) f is not odd because f   x    f  x  for all x f is not even because f   x   f  x  for all x b) (-2,0) (5,0) c) (0,3) Page 10 of 14 9. a) Translated points B’ E’ C’ A’ A B C D E D’ (0,-1) (1,0) (4,1) (5,0) (6,1) b)  f  x   2 10. a) -2 b) 4 c) undefined d) -12 d) g 1  x   x 2  5 domain:  x x  0 11. a) 6 b) x  4 , x  5 c) e) 4 f) 1 e) 82 x 2  4 , x   2 or x  2 12. a) symmetric with respect to the y-axis b) all reals c)  y y  0 d) as x   , f  x    right end as x   , f  x    left end 7 ; irrational: x  1  3 5 7  b) f  x   5  x   x  1  3 x  1  3 5  2 II. a) rational: x  , 4 ; nonreal complex: x  1  2i 3 2  b) g ( x)  3  x   ( x  4)( x  1  2i )( x  1  2i ) 3  13. I. a) rational: x    a) The data appears to fit a cubic or a sinusoidal model. b) If you elect to use year 1975 as x  0 , and rounding coefficients to 6 decimal places, the regression function is: f ( x)  0.307723x 3  9.790235 x 2  71.198866 x  1277.177287 . c) 1,746,000 students d) The closest year is 2004. e) Because the end behavior of this function is increasing as x, the enrollment at these institutions would increase without bounds as time goes by. Realistically, this is unlikely to happen. Page 11 of 14 # Students Enrolled in Grades 9-12 in Private Institutions Enrollment in 1000s 14.  1500 1400 1300 1200 1100 1000 1975 1985 Year 1995 2005 15. a) C, D b) A, B c) (i) as x   , f  x   4.25 (ii) as x   , f  x   4.25 (iii) y  4.25 (iv) x  3  x x  0 and x  2 16. a) b) as x   , h  x   3  2, 4  y 3 c) x  0 d) y  3 e)  2, 4   2  f)   , 0   3  g) See graph at right h)  y | y  3 and y  4   2 3 , 0 x0 17. a) t  5 t  10 t  25 304,000 fish 453,333 fish 702,222 fish  3 18.  , 4    ,  7 19. a) x  20. b) 1,200,000 fish  5  0   ,    3  3 2 b) 5 x 2 c) 40, 000  0.9  dollars d) 6.6 years t a) $36,000 b) $29,160 21. x f 1  x -3 -2 22. a) (1.39,0), (0,0.75) -1 -1 y 1 0 0 6 1 5 2 b) domain: range: 23. f 1 103 x  x  5 or f 1  x   0.2 10  7 3 all reals  y y  1 3x Page 12 of 14 x  3 24. a) No horizontal asymptote vertical asymptote at x  3 b) x-intercept at –2 c) y-intercept at 1  0,1  2, 0  25. x  0.4  2 5 26. 2.292 27. a) 4.8% 28. a) 29. F b) 108,948 4 33 a) b) 7 33 c)  4 7 d)  49 49 33  264 8 33  2 b)  2,8 c) 3 d) none  G a) 1  1 1 b)  ,   3 3 1 c) 3 1 d) 2   30. f  x   250 cos  x   250  10   2 1 1 1 1 3 31. f  t   155.6 cos 120 t  32. Solutions vary Page 13 of 14 3 1 2 2 2 1 3 2 5 3  12 , 26 33. a) 34. 12 3  5 26 24 7 35. 234o, 306o  37. a) 2 , 7 11 , 6 6 38. a) False b) True c) False d) False e) False f) False 39.  3  2n , x2  9 3 36. b) 1.6, 3.7, 5.8 sin  A  B   sin A cos B  cos A sin B  sin A  cos A  sin 2 A  2sin A cos A  cos 2 A  1  2sin A cos A cos  A  B   cos A cos B  sin A sin B sin  2 A   2sin A cos A 2 1  cot A tan A 5  2n ,   2n 3 40. 6839 feet 6 41. y is not a function of x. 42. a) i) 4 1 3 f  x     x   ( x  4) 2 2 2 -6 -4 -2 2 4 6 a) ii) g  x  -2 a) iii) h  x   -4 Start here, t=0 42. b) The rate of change is 43. a) rational 1 2 b) polynomial  , 11  3,   x6 x2 OR h  x   -6 c) Function h is increasing faster between x = 2 and x =0 . c) polynomial d) none e) exponential 44. two 45. a) 35 2 2  x  3  x  2  36 b.  11,3 or 11  x  3 Page 14 of 14 f) sinusoidal 4 1 x2