Download MATH 121 FINAL Exam Review

MATH 121 FINAL Exam Review
1. (Sections: 1.5, 2.1, 2.4). Let g(x) = 5(x + 23)2 − 20
(a) Give the coordinates of the vertex of g(x).
(b) Explain how g(x) is transformed from the graph of y = x2 .
(c) What are the zeros of g(x)?
2. (Section 3.2). Solve using the algebraic methods of your choice. Give
exact solutions (real or complex) in simplified form. Show work.
(a) 4x2 + 2x + 1 = 0.
(b) 3x2 + 6x + 5 = 0.
(c) 7x2 = −28
(d) −25 = x2 − 8x
3. (Sections: 4.2, 4.3). The cost function for a product is given by
C(x) = x2 + 12x + 2100, where x is the number of units produced and
sold and C(x) is the cost in dollars.
(a) What is the domain of this function in the context of the application?
(b) Is the function one-to-one for the domain in part (a)? Explain
how you know.
(c) Find the inverse of this function for the domain in part (a). Show
work. (Hint: Use the completing the square.)
(d) Use the inverse function in part (c) to find how many units are
produced and sold if the cost is $ 3,220. Show work.
4. (Section: 3.3). Graph by hand:
 2

 x −1
, x < −2
+ 2 , −2 ≤ x < 2
f (x) = 3x

 √
x+2 , x≥2
5. (Sections: 3.3, 6.1, also see page 434.). Sketch by hand the graph
of the function f (x) that satisfies the following conditions: f (0) = −3;
f (x) is concave up when x < 0; f (x) is concave down when x > 0.
6. (Sections: 3.3, 4.1, 6.1, also see page 434.). Sketch by hand the
graph of the function g(x) that satisfies the following conditions: g(x)
is an even function; range of g(x) is (−∞, 2].
7. (Sections: 4.1, 6.5). Sketch by hand the graph of the function h(x)
1
that satisfies the following conditions: h(x) is the graph of y = 2
x
shifted down 6 units and to the left 2 units (include and label any
asymptotes in your sketch).
8. (Section: 4.2). Let f (x) = 3x − 4 and g(x) =
x+4
3
(a) Find (f ◦ g)(x). (f [g(x)]).
(b) Find (g ◦ f )(x). (g[f (x)]).
(c) Are f (x) and g(x) inverse functions? Explain why or why not.
9. (Section: 4.3). Write the inverse of w(x) = 2x3 − 5.
10. Solve each of the following equations algebraically and check graphically:
(a) (Section: 4.4)
√
7x − 28 =
√
x2 − 4x.
(b) (Sections: 3.3, 4.4) | x − 18 |= x2 − 18x
(
)
(c) (Section: 5.3) 300 = 1200 2−0.1x .
11. (Section: 4.2). The profit for a product can be described by the
function P (x) = −0.4x2 + 280x − 24000 dollars, where x is the number
of units produced and sold.
(a) To maximize profit, how many units must be produced and sold?
(b) What is the maximum possible profit?
(c) Producing and selling how many units will result in a profit of at
least $ 9,000? Use graphical methods to solve.
12. (Sections: 3.1, 3.3, 5.1, 5.2). For each function listed below, first
identify the type of function and then give the domain and range using
interval notation.
Function
Type
Domain
Range
g(x) = 3x2 − 20
h(x) = 3 log7 x
k(x) =| 2x + 3 | −8
j(x) = 4 (6x )
13. (Sections: Chapter 5 Toolbox, 5.1). Use properties of exponents
1 x
to show that y = 3x − 4 is equivalent to y =
(3 ).
81
14. (Sections: Chapter 5 Toolbox, 5.1). Use
properties of exponents
( )x
1
to show that y = 6−x is equivalent to y =
.
6
15. (Section: 5.1). At the end of an advertising campaign, the weekly
sales (in dollars)
declined,
with weekly sales given by the equation
(
)
−0.08x
y = 12, 000 2
, where x is the number of weeks after the end
of the campaign.
(a) Determine the sales at the end of the campaign.
(b) Determine the sales 6 weeks after the end of the campaign.
(c) Does the model indicate that sales eventually reach $ 0? Explain.
16. (Section 5.2). Between the years 1976 and 1998, the percent of moms
who returned to the workforce within one year after they had a child is
given by the equation, w(x) = 1.11 + 16.94 ln x, where x is the number
of years past 1970.
(a) Find w−1 (x).
(b) Use w−1 (x) to estimate the year in which 50 percent of moms
returned to the workforce within one year.
17. (Section 5.3). Solve each equation algebraically. When necessary,
round answers to four decimal places.
(a) log4 x = −2.
(b) 4 + log x = 10.
(c) e(−2x+3) = 2
(d) ln (−2x + 3) = 10
(e) 2 ln x + 7 = ln (4x) + 10
18. (Sections: 5.2, 5.3). Rewrite 2 log x + 5 log y − 8 log z as a single
logarithm.
(
)
y 2 e3x
19. (Sections: 5.2, 5.3). Rewrite ln
as the sum, difference or
z3
product of logarithms and simplify if possible.
20. (Section: 5.5). Suppose $ 9,000 is invested for t years at 5.5% interest
compounded monthly.
(a) Write an equation that gives the future value, S, of the investment
after t years.
(b) Find the future value of the investment in 4 years.
(c) Find the number of years it will take the investment to double.
21. (Sections: 6.1, 6.3). Let f (x) = 3x3 + 18x2 − 12x − 72. Use this
function to answer each question.
(a) State the degree and leading coefficient of f (x).
(b) Describe the end behavior of the graph of f (x).
(c) Find all x such that f (x) = 0. Solve algebraically. Show work.
(d) Use your calculator to create a complete graph of f (x). Sketch
the graph and list your window settings.
(e) How many turning points does the graph of f (x) have?
(f) How many inflection points does the graph of f (x) have?
(g) At what point does a local maximum occur? round the coordinates to one decimal place.
22. (Section: 6.3). Solve algebraically.
(a) 0.2x3 − 20x = 0.
(b) x3 − 15x2 + 56x = 0.
(c) 2x4 − 3x3 − 20x2 = 0
23. (Sections: 7.1, 7.2). Solve the following system of equations algebraically.


 2x − 3y + z = 2
3x + 2y − z = 6


x − 4y + 2z = 2
24. (Section: 7.2). A car rental agency rent compact, midsize, and luxury
cars. Its goal is to purchase 80 cars for a total of $1,822,000 and to
earn a daily rental of $ 2,424 from all the cars. The compact cars cost
$16,000 each earn $ 20 per day in rental, the midsize cars cost $ 22,000
each earn $ 30 per day, and the luxury cars cost $ 38,000 each and earn
$ 52 per day. Your task will be to find the number of each type of car
the agency should purchase to meet its goal.
(a) Let x represent the number of compact cars purchased, y represent
the number of midsize cars purchased and z represent the number
of luxury cars purchased. Write a system of equations to model
this situation.
(b) Write the augmented matrix for your system of equations.
(c) Use your calculator to find the reduced-row echelon form of the
augmented matrix.
(d) Find the number of each type of car the agency should purchase
to meet its goal.
MATH 121 Final Exam Review Solutions
1. (a) (−23, −20)
(b) Shift down 20 units, left 23 units and stretch by a factor of 5.
(c) x = −25, x = −21
√
−1 ± i 3
2. (a) x =
4
√
−3 ± i 6
(b) x =
3
(c) x = ±2i
(d) x = 4 ± 3i
3. (a) x ≥ 0
(b) yes; give explanation
√
(c) C −1 (x) = −6 + x − 2064
(d) 28 units
4. For x < −2: decreasing, concave up with an open circle at (−2, 3);
For −2 ≤ x < 2: line segment with closed circle at (−2, −4) and an
open circle at (2, 8);
For x ≥ 2: increasing, concave down with a closed circle at (2, 2).
5. One possibility is the graph of f (x) = −x3 − 3.
6. One possibility is the graph of g(x) = −x2 + 2.
1
− 6; vertical asymptote at x = −2,
(x + 2)2
horizontal asymptote at y = −6.
7. Sketch the graph of h(x) =
8. (a) f [g(x)] = x
(b) g[f (x)] = x
(c) yes; give explanation
√
−1
9. w (x) =
3
x+5
2
10. (a) x = 4 and x = 7
(b) x = −1 and x = 18
(c) x = 20
11. (a) 350 units
(b) $ 25,000
(c) 150 ≤ x ≤ 550
12. (a) g(x): quadratic, Domain: (−∞, ∞), Range: [−20, ∞)
(b) h(x): logarithmic, Domain: (0, ∞), Range: (−∞, ∞)
(c) k(x): absolute value, Domain: (−∞, ∞), Range: (−8, ∞)
(d) j(x): exponential, Domain: (−∞, ∞), Range: (0, ∞)
13. y = 3x−4 = 3x · 3−4 =
( )x
14. y =
1
6
=
3x
3x
1 x
=
=
(3 )
4
81
81
3
1x
1
= x = 6−x
x
6
6
15. (a) $ 12,000
(b) ≈ $8604
(c) no; give explanation
16. (a) w−1 (x) = e(x − 1.11)/16.94
(b) 1988
17. (a) x =
1
16
(b) x = 1, 000, 000
(c) x ≈ 1.1534
(d) x ≈ −11011.7329
(e) x ≈ 80.3421
(
x2 y 5
18. log
z8
)
19. 2 ln y + 3x − 3 ln z
(
20. (a) S = 9000 1 +
0.055
12
(b) $ 11,209.06
(c) ≈ 12.6 or 13 years
)12t
21. (a) n = 3; a = 3
(b) concave down on left; concave up on right
(c) x = 2, x = −2, and x = −6
(d) possible window settings: [−10, 6, 2]x by [−100, 100, 25]y
(e) 2 turning points
(f) 1 inflection point
(g) −4.3, 73.9)
22. (a) x = 0 and x = 10 and x = −10
(b) x = 0 and x = 7 and x = 8
−5
(c) x = 0 and x =
and x = 4
2
23. x = 2, y = 2, z = 4


 x+y+z
= 80
24. (a)  16x + 22y + 38z = 1822

20x + 30y + 52z = 2424


1 1 1
80


(b)  16 22 38 1822 
20 30 52 2424


1 0 0 35


(c)  0 1 0 28 
0 0 1 17
(d) 35 compact, 28 midsize, and 17 luxury cars