Math 107 Study Guide for Chapters 1 and 2

Transcription

1 Math 107 Study Guide for Chapters 1 and PRACTICE EXERCISES 1. Solve the following equations. (a) Solve for P: S = B + 1 PS (b) Solve for PV 1 1 PV T : = T T 1 (c) Solve for Q: L = Q (d) Solve for D: y = T + Da t. Find the equation of the line containing (-,3) and perpendicular to the line passing through (1, 1) and (3,1). Is the line y = + 5 parallel to the line you found? Eplain. 3. A driver going down a straight highway is traveling at 60 ft/sec (about 41 mph) on cruise control, and begins to accelerate at a rate of 5. ft/sec. The final velocity of the car is given by V = 6 t + 60 where V is the 5 velocity at time t. (a) Interpret the meaning of the slope and the y-intercept in this contet. (b) Determine the velocity of the car after 9.4 seconds. (c) If the car is traveling at 100 ft/sec, for how long did it accelerate? 4. For which of the following can y be epressed as a function of. Eplain. (a) 1 = 4 + y + 3y (b) (c) (d) y y < 1 5. If g() = 3, what is the value of g( 3)? 1

2 6. Which of the following graphs represent y as a function of? 7. Determine the domain of each function. State your answer in interval and set notation. (a) p() = 3 (b) F() = (c) f( ) = (d) h ( ) = If f() = 3, find f(b ) (a) 3b 1b + 1 (b) 3b 8 (c) 3b 4b + (d) 3b 4 (e) None of these 9. Find the average rate of change g(a+h) g(a) h = a + h. of the function g() = 3 between the points = a and 10. Evaluate and simplify the difference quotient F(+h) F() h 11. Determine whether each function is even, odd, or neither. (a) g ( ) = (b) f( t) = t 46 for the function F( ) = (b) F( ) = 08 (d) q ( ) = 1 +

3 1. Determine the intervals over which the function graphed below is increasing and decreasing. 13. State the domain and range for the f() graphed below. Then state the intervals where f is increasing or decreasing and intervals where f is positive or negative. Assume all endpoints have integer values. 14. Sketch a graph of the following piecewise-defined function. State its domain and range. 5 < h() = 1 1 < > From the tet, p. 199, #73 (Reading a graph and operations with functions).

4 Use the graphs below to answer the net two questions. 16. Using the graph of f and g, find (f g)(3). 17. Using the graph of f and g, find (f g)(1). 18. Suppose that a function f() has a domain of [ 10, 10] and a range of [ 4, 6]. What would be the domain and range of g() = f( 1) 3? 19. Describe, in order, the transformations used to go from y = f() to y = 1 f( 73) + π Let f( ) = and g() = + 1. Find: (a) ( f g)(1) (b) ( f g)() (c) ( f + g)( ) (d) Domain of (f + g)() in both interval and set notation (e) ( f g)( ) (f) Domain of ( f g)( ) in both interval and set notation f (g) ( ) g g (i) ( ) f (k) (f g)() f (h) Domain of ( ) in both interval and set notation g g (j) Domain of ( ) in both interval and set notation f (l) Domain of (f g)() in both interval and set notation 1. Given 4 f( ) = and g ( ) =, determine the following. Simplify your answers. (a) (g f)() (c) (f g)() (b) Domain of (g f)() in both interval and set notation (d) Domain of (f g)() in both interval and set notation

5 . Two functions g and f are defined in the figure below. Find the domain and range of the composition f g. 3. Given h() = 1 4. Simplify +4i 1 3i and g() = determine (g h)() and state its domain in interval notation. 5+1 into the form a + bi. 5. Solve 35 7d State your answer in interval notation. 6. Solve = Solve + 7 = Solve 36 = You should be able to check each of the following equations for symmetry with respect to both aes and the origin. (Hint: Use odd and even definitions.) (a) f() = (b) f() = (c) f() = 3 4 (d) f() = Find the equation of the line passing through the two points: (10,-3), (-8,6). Give answer in the following forms. Point-Slope form: Slope-Intercept form: Standard form: 31. Are the given lines parallel, perpendicular, or neither? L 1 : (0, 1), ( 10,5) L : 3 + 5y = Find the - and y-intercepts of the line 5y = 15.

6 33. Determine the slope and equation of each given line. a. b. 34. Find the slope and the - and y- intercepts of the line 3 6y 18 = 0. Graph the line. 35. A sales person receives a monthly salary of $3000 plus a commission of 5% of sales. Write a linear equation for the salesperson s monthly wage W as a function of the monthly sales S. Will a sales person make no money if they do not make any sales? Eplain. How much does a person need to sell in order to make $5000 in a month? 36. Suppose that the value of a new car has a linear depreciation in value. If the value of a new car is $18,000 and its worth is only $10,000 after 4 years, then how much is the car worth after 8 years? When will the car have no value at all? Write a linear equation for the value of the car as a function of its age in years, and use it to answer the questions. 37. Determine whether each equation represents y as a function of. (a) + y 7 = 4 (b) y = 4 (c) + y = 9 (d) y + y = 9 (e) y = Determine the domain and range of the following graphs. Use interval notation. Determine if the equation represents a function. Eplain. (6, 4)

7 39. Evaluate each of the following for f() = (a) f(5) (b) f(4) (c) f(8) (d) f( + 4) (e) f( + h) (f) f() + h (g) f() +f(h) (h) 5f() 40. Determine the domain of each of the following functions. Give answers in interval notation. 5 3 (a) f() = (b) g() = Find the average rate of change of the function f() = between the indicated points. (a) Between = and = 1 (b) Between = a and = a + h 4. Find the following difference quotients. Simplify completely. f (3 + h) f (3) (a) Let f() = Find the difference quotient, h 0. h g( a + h) g( a) (b) Let g() = 5. Find the difference quotient, h 0. h 43. Use the graphs of f and g to evaluate the following:. y = f() y = g() f( 6) = f( 3) = f(0) = g(8) = f(6) = f(3) = g(0) = g(1) = 44. Given that the point ( 1, ) is on the graph of the continuous function y = g(), what point MUST lie on the graph of the y = g( 1) +. The point on the graph of y = g( 1) + is (, ).

8 45. Given the graph of f() on the aes below, find the following. Use Interval notation. (a) Find the domain of f(): (b) Find the range of f(): (c) Over which interval(s) is the function increasing? (d) Over which interval(s) is the function decreasing? (e) Over which interval(s) is the function constant? (f) f(4) = f(0) = f(7) = f(1) = f( ) = (g) Graph the function y = f( + 6) + 3 on the same aes; label the points of the transformed graph. Describe the transformation verbally

9 46. Identify and graph the toolbo (parent) function for f()= Describe the transformation that occurs in the function, and then graph the function. Identify any intercepts Identify and graph the toolbo (parent) function for f()= Describe the transformation that occurs in the function, and then graph the function. Identify any intercepts. 48. Given the graph of h(), graph the following functions. Label the points that you graph clearly. (a) y = h() b) y = h( ) h() h() (3, 1) (3, 1) (, ) (, )

10 49. Graph the following piecewise function. Label the endpoints and the - and y-intercepts (if any) on the graph. + 1, 4 h() = 3, 4 < < , > 3 < Graph the following piecewise function: h ( ) = < Label the endpoints of each segment and any - and y-intercepts on the graph. 51. Given f ( ) = + 3 and g ( ) = 5, determine the domain of ( g f )( ) interval notation.. State your answer in 5. Use the given functions to find the following information. If it does not eist, write DNE (does not eist). Give the domain in interval notation. Simplify your answers completely. y Graph of f() y ( f )( 0) g = ( fg )(3) = g f (4) = ( g)( 8) f = Graph of g() ( f )( 4) f = Domain of ( g)( ) f + :