MATH 91
Final Study Package
Name___________________________________
Solve the problem.
1) The width of a rectangle is 6 kilometers less than twice its length. If its area is 216 square
kilometers, find the dimensions of the rectangle.
1)
2) If the sides of a square are increased by 2 meters, the area becomes 25 square meters. Find
the length of a side of the original square.
2)
3) A triangular piece of glass is being cut so that the height of the triangle is 4 inches shorter
than twice the base. If the area of the triangle is 120 square inches, how long is the height of
the triangle?
3)
Find the domain and range.
4) {(9,3), (9,-7), (5,8), (-10,7), (-6,-6)}
4)
Decide whether the relation is a function.
5) {(-4, 4), (-1, 2), (4, 9), (4, 7)}
5)
Find the indicated function value.
x3 + 4
6) Find f(-4) when f(x) =
.
x2 + 2
6)
7)
7)
x
-4
-3
0
3
4
f(x)
-1
3
15
27
31
Find f(3)
Find the domain of the function.
1
8) f(x) =
x+3
8)
Find the indicated function value.
9) f(x) = 4x - 4, g(x) = -3x - 4
Find (f + g)(-5).
9)
Find the requested value.
10) f(x) = -3x2 - 7, g(x) = x + 4
10)
Find (f - g)(-3).
Find the composition.
11) If f(x) = 7x2 + 5x and g(x) = 3x, find (f ∘ g)(x).
11)
12) If f(x) = 6x + 5 and g(x) = x + 4, find (g ∘ f)(x).
12)
1
Find the inverse of the one-to-one function.
13) f(x) = 8x - 7
13)
Solve the inequality. Other than ∅, graph the solution set on a number line.
14) -4(5x - 12) < -24x + 24
14)
-9
15)
-8
-7
-6
-5
-4
-3
x
x
≥5+
3
18
15)
-20 -16 -12 -8
-4
0
4
8
12
16
20
Solve the compound inequality and graph the solution set on a number line. Except for the empty set, express the
solution set in interval notation.
16) -8x > -40 and x + 8 > 6
16)
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
17) -3x ≤ -9 or 3x > 9x - 6
-1
0
17)
1
2
3
4
5
Find the solution set for the equation.
4x - 3
18)
=4
8
18)
19) |4x + 7| + 2 = 10
19)
20) |2x - 4| = |x + 3|
20)
Solve and graph the solution set on a number line.
21) |4x - 3| - 4 < 0
0
1
2
3
4
5
6
7
8
9
21)
10
11
12
22) |x - 1| ≤ 0
-10
-8
22)
-6
-4
-2
0
2
4
6
8
10
23) 3(x + 1) + 6 ≥ 18
23)
-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
2
24) 3 + 1 -
x
≥5
2
24)
-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Solve and graph the solution set on a number line. Express the solution set in both set-builder and interval notations.
25) |2x + 1| - 4 < -8
25)
-2
-1
0
1
2
3
4
5
6
7
26) |7x - 4| - 1 > -6
0
2
26)
4
6
8
10
12
14
Graph the inequality.
27) y ≥ - 4x
27)
y
10
-10
10
x
-10
28) y ≤ -2x - 4
28)
y
10
5
-10
-5
5
10
x
-5
-10
Find the function value indicated for the function. If necessary, round to two decimal places. If the function value is not a
real number and does not exist, state so.
29) Evaluate p(x) = - x + 11 for p(-2)
29)
3
Find the domain of the square root function. Graph of the function.
30) f(x) = x - 2
30)
Use radical notation to rewrite the expression. Simplify, if possible.
31) (-8)1/3
31)
32) (-64)2/3
32)
33) 321/5 + 161/4
33)
Rewrite the expression with a positive rational exponent. Simplify, if possible.
34) 100 -3/2
34)
Simplify by factoring.
35) 600x
35)
36)
3
-27a 11b10
36)
Multiply and simplify. Assume that all variables in a radicand represent positive real numbers.
37) 24xy · 4xy2
37)
Add or subtract as indicated. You will need to simplify terms to identify like radicals.
38) -5 18 - 2 8 - 3 200
38)
39) 2
3
a+
3
64a
39)
Multiply and simplify. Assume that all variables represent positive real numbers.
40) (5 - x)(6 - x)
41) (9 7 - 2)2
40)
41)
Rationalize the denominator and simplify.
10x
42)
3
2x2
42)
Rationalize the denominator.
7+ 2
43)
7- 2
43)
Solve the equation.
44) x2 - 2x + 64 = x + 6
44)
4
45)
2x + 3 -
x+1=1
45)
Find each product. Write the result in the form a + bi.
46) (9 + 4i)(7 - 5i)
46)
Divide and simplify to the form a + bi.
6 + 4i
47)
8 - 3i
47)
Solve the equation by the square root property. If possible, simplify radicals or rationalize denominators. Express
imaginary solutions in the form a + bi.
48) 7x2 - 5 = 0
48)
49) (x - 5)2 = -81
49)
Complete the square for the binomial. Then factor the resulting perfect square trinomial.
50) x2 + 3x
51) x2 +
4
x
9
50)
51)
Solve the quadratic equation by completing the square.
52) x2 + 12x + 17 = 0
52)
Solve the formula for the specified variable. Assume all variables represent nonnegative numbers. If possible, simplify
radicals and rationalize denominators.
s2 h
53) V =
for s
53)
3
Find the distance between the pair of points. Give an exact answer.
54) (5, 3) and (-2, -6)
54)
Use the quadratic formula to solve the equation.
55) x2 + 7x + 7 = 0
55)
56) 4x2 = -8x - 1
56)
Solve the problem.
57) The length of a rectangular storage room is 10 feet longer than its width. If the area of the
room is 75 square feet, find its dimensions.
58) The hypotenuse of an isosceles right triangle is 4 feet longer than either of its legs. Find the
exact length of each side.
5
57)
58)
59) The area of a rectangular wall in a classroom is 341 square feet. Its length is 2 feet shorter
than three times its width. Find the length and width of the wall of the classroom.
59)
60) The area of a rectangular wall in a classroom is 198 square feet. Its length is 5 feet shorter
than three times its width. Find the length and width of the wall of the classroom.
60)
61) A ball is thrown downward with an initial velocity of 21 meters per second from a cliff that
is 90 meters high. The height of the ball is given by the quadratic equation
h = -4.9t2 - 21t + 150 where h is in meters and t is the time in seconds since the ball was
61)
thrown. Find the time that the ball will be 60 meters from the ground. Round your answer
to the nearest tenth of a second.
Sketch the graph of the quadratic function. Give the vertex and axis of symmetry.
62) f(x) = -(x + 1)2 + 4
62)
y
10
5
-10
-5
5
10
x
-5
-10
Sketch the graph of the quadratic function. Identify the vertex, intercepts, and the equation for the axis of symmetry.
63) f(x) = 4x2 + 24x + 41
63)
y
10
5
-10
-5
5
10
x
-5
-10
Determine whether the given quadratic function has a minimum value or maximum value. Then find the minimum or
maximum value and determine where it occurs.
64) f(x) = -x2 + 2x - 8
64)
Solve the problem.
65) You have 156 feet of fencing to enclose a rectangular region. Find the dimensions of the
rectangle that maximize the enclosed area.
6
65)
66) A developer wants to enclose a rectangular grassy lot that borders a city street for parking.
If the developer has 360 feet of fencing and does not fence the side along the street, what is
the largest area that can be enclosed?
Solve the equation by making an appropriate substitution.
67) x4 - 13x2 + 36 = 0
66)
67)
68) (x - 1)2 - 7(x - 1) + 10 = 0
68)
69) x-2 - 4x-1 + 3 = 0
69)
Graph the function by making a table of coordinates.
70) f(x) = 5 x
70)
y
6
4
2
-6
-4
-2
2
4
6 x
-2
-4
-6
Write the equation in its equivalent exponential form.
71) log2 32 = x
71)
Write the equation in its equivalent logarithmic form.
72) 132 = y
72)
Evaluate the expression without using a calculator.
73) log7 7
73)
74) log125 5
74)
Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
75) logb(yz 8 )
75)
76) logb
xy5
z6
76)
7
Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose
coefficient is 1. Where possible, evaluate logarithmic expressions.
1
77) 8 ln x - ln y
77)
4
78) 2 ln a - 9 ln b
78)
Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places
79) log6 3
79)
Solve the equation by expressing each side as a power of the same base and then equating exponents.
1
80) 3 x =
80)
81
81) 2 -x = 8
81)
82) 3 (1 + 2x) = 27
82)
Solve the logarithmic equation. Give an exact answer.
83) log 10 + log x = 1
3
3
83)
84) log (x + 2) - log x = 2
3
3
84)
85) ln 5 + ln(x - 1) = 0
85)
86) log(5 + x) - log(x - 4) = log 2
86)
Solve the problem.
87) The formula A = 121e0.031t models the population of a particular city, in thousands, t
years after 2011. When will the population of the city reach 150 thousand?
87)
88) The value of a particular investment follows a pattern of exponential growth. You invested
money in a money market account. The value of your investment t years after your initial
investment is given by the exponential growth model A = 4300e0.049t. When will the
88)
Solve.
account be worth $6683?
Write the standard form of the equation of the circle with the given center and radius.
89) Center (-6, -2), r = 1
8
89)
Give the center and radius of the circle described by the equation and graph the equation.
90) (x - 3)2 + (y - 1)2 = 25
90)
y
10
5
-10
-5
5
10
x
-5
-10
Complete the square and write the equation in standard form. Then give the center and radius of the circle and graph the
equation.
91) x2 + y2 - 6x - 8y - 11 = 0
91)
y
10
5
-10
-5
5
10
x
-5
-10
Find the standard form of the equation of the ellipse.
92)
92)
y
10
8
6
4
2
-10 -8 -6 -4 -2
-2
2
4
6
8
10
x
-4
-6
-8
-10
9
Sketch the ellipse for the equation.
(x - 1)2 (y + 1)2
93)
+
=1
4
9
93)
y
10
5
-10
-5
5
10
x
-5
-10
Find the vertices of the hyperbola with the given equation.
x2 y2
94)
=1
1
81
94)
Use vertices and asymptotes to graph the hyperbola.
95) 9y2 - 4x2 = 36
95)
y
10
5
-10
-5
5
10
x
-5
-10
Solve the system by the substitution method.
y=x+ 2
96)
y2 = 8x
96)
Solve the system by the addition method.
2
2
97) 3x + y = 9
3x2 - y2 = 9
97)
Write the first four terms of the sequence whose general term is given.
98) a n = 4n - 1
99) a n = (-4)n
98)
99)
10
100) a n =
n4
(n - 1)!
100)
Find the indicated sum.
5
101) ∑ (4i - 4)
i=2
4
102)
∑
101)
2i
102)
i=1
Find the common difference for the arithmetic sequence.
103) 8, 11, 14, 17, . . .
103)
Write the first five terms of the arithmetic sequence with the given first term, a 1 , and common difference, d.
104) a 1 = 4; d = 4
104)
Find the common ratio for the geometric sequence.
105) 1, -3, 9, -27, 81, . . .
105)
Write the first four terms of the geometric sequence with the given first term, a 1 , and common ratio, r.
106) a 1 = 5; r = 2
106)
11
Answer Key
Testname: MATH 91 FINAL STUDY GUIDE FALL 16
1)
2)
3)
4)
5)
length = 12 km, width = 18 km
3m
20 in.
domain = {-6, -10, 9, 5}; range = {-6, 7, -7, 8, 3}
not a function
10
6) 3
7) 27
8) (-∞, -3) or (-3, ∞)
9) -13
10) -35
11) 63x2 + 15x
12) 6x + 9
x+7
13) f-1 (x) =
8
14) (-∞, -6)
-9
-8
-7
-6
-5
-4
-3
15) [18, ∞)
-20 -16 -12 -8
-4
0
4
8
12
16
20
2
3
4
16) (-2, 5)
-5
-4
-3
-2
-1
0
1
5
6
7
17) (-∞, 1) ∪ [3, ∞)
-1
18)
35
29
,4
4
19)
1
15
,4
4
20) 7,
21) -
0
1
2
3
4
5
1
3
1 7
,
4 4
0
1
2
3
4
5
6
7
8
9
10
11
12
22) {1}
-10
-8
-6
-4
-2
0
2
4
6
8
10
23) (-∞, -9] ∪ [3, ∞)
-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
12
Answer Key
Testname: MATH 91 FINAL STUDY GUIDE FALL 16
24) (-∞, -2] ∪ [6, ∞)
-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
25) ∅
-2
-1
0
1
2
3
4
5
6
7
26) (-∞, ∞)
0
2
4
6
8
10
12
14
27)
y
10
-10
10
x
10
x
-10
28)
y
10
5
-10
-5
5
-5
-10
29) -3
13
Answer Key
Testname: MATH 91 FINAL STUDY GUIDE FALL 16
30) domain of f: [2, ∞)
y
10
5
-10
-5
5
10
x
-5
-10
31) -2
32) 16
33) 4
1
34)
1000
35) 10 6x
3
36) -3a 3 b3 a 2 b
37) 4xy 6y
38) -49 2
3
39) 6 a
40) 30 - 11 x + x
41) 571 - 36 7
3
42) 5 4x
9 + 2 14
43)
5
44) {2}
45) {3, -1}
46) 83 - 17i
36 50
47)
i
+
73 73
48) ±
35
7
49) {5 ± 9i}
9
9
3 2
50) ; x2 + 3x + = x +
4
4
2
51)
4 2 4
4
2 2
;x + x+
= x+
81
9
81
9
52) {-6 ±
53) s =
54)
19}
3Vh
h
130 units
14
Answer Key
Testname: MATH 91 FINAL STUDY GUIDE FALL 16
55)
-7 ± 21
2
56)
-2 ± 3
2
57) 5 ft by 15 ft
58) (4 + 4 2) ft, (4 + 4 2) ft, (8 + 4 2) ft
59) width = 11 ft; length = 31 ft
60) width = 9 ft; length = 22 ft
61) 2.6 sec
62) vertex: (- 1, 4)
axis of symmetry: x = - 1
y
10
5
-10
-5
5
10
x
10
x
-5
-10
63) vertex: (-3, 5)
x-intercepts: none
y-intercept: (0, 41)
axis of symmetry: x = -3
y
10
5
-10
-5
5
-5
-10
64) Maximum is - 7 at x = 1.
65) 39 ft by 39 ft
66) 16,200 sq ft
67) {-2, 2, -3, 3}
68) {3, 6}
1
69)
,1
3
15
Answer Key
Testname: MATH 91 FINAL STUDY GUIDE FALL 16
70)
y
6
4
2
-6
-4
-2
2
4
6 x
-2
-4
-6
71) 2 x = 32
72) log13y = 2
73)
1
2
74)
1
3
75) logb y + 8 logb z
76) logb x + 5 logb y - 6 logb z
77) ln
x8
4
y
78) ln
a2
b9
79) 0.6131
80) {-4}
81) {-3}
82) {1}
3
83)
10
84)
1
4
85)
6
5
86) {13}
87) 2018
88) 9 years after the initial investment
89) (x + 6)2 + (y + 2)2 = 1
16
Answer Key
Testname: MATH 91 FINAL STUDY GUIDE FALL 16
90) center (3, 1), r = 5
y
10
5
-10
-5
5
10
x
5
10
x
5
10
x
-5
-10
91) (x - 3)2 + (y - 4)2 = 36
center (3, 4), r = 6
y
10
5
-10
-5
-5
-10
92)
x2 y2
+
=1
49 25
93)
y
10
5
-10
-5
-5
-10
94) (-1, 0), (1, 0)
17
Answer Key
Testname: MATH 91 FINAL STUDY GUIDE FALL 16
95)
y
10
5
-10
-5
5
10
x
-5
-10
96) {(2, 4)}
97) {( 3, 0), (- 3, 0)}
98) 3, 7, 11, 15
99) -4, 16, -64, 256
81 128
100) 1, 16,
,
2
3
101)
102)
103)
104)
105)
106)
40
30
3
4, 8, 12, 16, 20
-3
5, 10, 20, 40, . . .
18