Recall that
N
=
{
1
,
2
,
3
,
…
}
{\displaystyle \mathbb {N} =\{1,2,3,\ldots \}}
1. List 4 smallest elements of the set
{
y
∣
y
=
2
x
and
x
∈
N
}
{\displaystyle \{y\mid y=2x{\mbox{ and }}x\in \mathbb {N} \}}
2. List 4 smallest elements of the set
{
z
∣
z
=
|
x
+
2
|
and
x
is a negative integer
}
{\displaystyle \{z\mid z=|x+2|{\mbox{ and }}x{\mbox{ is a negative integer}}\}}
3. If
A
=
{
0
,
1
,
2
,
3
,
4
,
5
,
6
}
{\displaystyle A=\{0,1,2,3,4,5,6\}}
,
B
=
{
1
,
3
,
5
,
7
,
9
,
11
}
{\displaystyle B=\{1,3,5,7,9,11\}}
, and
C
=
{
4
,
5
,
6
,
7
,
8
}
{\displaystyle C=\{4,5,6,7,8\}}
, find
A
∪
B
{\displaystyle A\cup B}
A
∩
B
{\displaystyle A\cap B}
(
A
∪
B
)
∩
C
{\displaystyle (A\cup B)\cap C}
(
A
∩
B
)
∪
(
B
∩
C
)
{\displaystyle (A\cap B)\cup (B\cap C)}
A
∪
B
=
{
0
,
1
,
2
,
3
,
4
,
5
,
6
,
7
,
9
,
11
}
{\displaystyle A\cup B=\{0,1,2,3,4,5,6,7,9,11\}}
A
∩
B
=
{
1
,
3
,
5
}
{\displaystyle A\cap B=\{1,3,5\}}
(
A
∪
B
)
∩
C
=
{
4
,
5
,
6
,
7
}
{\displaystyle (A\cup B)\cap C=\{4,5,6,7\}}
(
A
∩
B
)
∪
(
B
∩
C
)
=
{
1
,
3
,
5
,
7
}
{\displaystyle (A\cap B)\cup (B\cap C)=\{1,3,5,7\}}
A
∪
B
=
{
0
,
1
,
2
,
3
,
4
,
5
,
6
,
7
,
9
,
11
}
{\displaystyle A\cup B=\{0,1,2,3,4,5,6,7,9,11\}}
A
∩
B
=
{
1
,
3
,
5
}
{\displaystyle A\cap B=\{1,3,5\}}
(
A
∪
B
)
∩
C
=
{
4
,
5
,
6
,
7
}
{\displaystyle (A\cup B)\cap C=\{4,5,6,7\}}
(
A
∩
B
)
∪
(
B
∩
C
)
=
{
1
,
3
,
5
,
7
}
{\displaystyle (A\cap B)\cup (B\cap C)=\{1,3,5,7\}}
10
{\displaystyle 10}
−
4
{\displaystyle -4}
8
{\displaystyle 8}
1
{\displaystyle 1}
10
{\displaystyle 10}
−
4
{\displaystyle -4}
8
{\displaystyle 8}
1
{\displaystyle 1}
30. Rewrite the expression
(
x
2
−
1
)
(
2
x
+
3
)
{\displaystyle (x^{2}-1)(2x+3)}
as a polynomial in the standard form and state the degree of the polynomial.
2
x
3
+
3
x
2
−
2
x
−
3
{\displaystyle 2x^{3}+3x^{2}-2x-3}
2
x
3
+
3
x
2
−
2
x
−
3
{\displaystyle 2x^{3}+3x^{2}-2x-3}
31. Rewrite the expression
5
x
17
(
4
x
−
3
−
2
x
−
2
)
{\displaystyle 5x^{17}(4x^{-3}-2x^{-2})}
as a polynomial in the standard form and state the degree of the polynomial.
−
10
x
15
+
20
x
14
{\displaystyle -10x^{15}+20x^{14}}
−
10
x
15
+
20
x
14
{\displaystyle -10x^{15}+20x^{14}}
32. Rewrite the expression
6
(
y
3
−
y
)
2
−
2
y
4
(
3
y
2
−
6
)
{\displaystyle 6(y^{3}-y)^{2}-2y^{4}(3y^{2}-6)}
as a polynomial in the standard form and state the degree of the polynomial.
6
y
2
{\displaystyle 6y^{2}}
6
y
2
{\displaystyle 6y^{2}}
We are only interested in factors that have integer coefficients.
42. Show that the polynomial
3
x
2
+
5
x
+
3
{\displaystyle 3x^{2}+5x+3}
is irreducible (that is, cannot be factored using real coefficients)
b
2
−
4
a
c
=
5
2
−
4
⋅
3
⋅
3
=
−
11
<
0
{\displaystyle b^{2}-4ac=5^{2}-4\cdot 3\cdot 3=-11<0}
b
2
−
4
a
c
=
5
2
−
4
⋅
3
⋅
3
=
−
11
<
0
{\displaystyle b^{2}-4ac=5^{2}-4\cdot 3\cdot 3=-11<0}
50. You want to install hardwood floor tile. The delivery fee is $80, and the installation costs $6 per square foot (parts and labor). How many square feet can you tile on a $1100 budget?
51. Your distant relative's will stipulates that her money is to be divided between you, your sibling, and your child in such a way that your sibling gets twice as much as you do, and your child gets half as much as you do. How much money will you get if the total inheritance is seven hundred thousand dollars?
52. Solve the equation
3
−
2
x
=
x
+
9
{\displaystyle 3-2x=x+9}
.
x
=
−
2
{\displaystyle x=-2}
x
=
−
2
{\displaystyle x=-2}
53. Solve the equation
2
(
x
+
7
)
=
−
3
(
x
+
1
−
2
(
x
−
1
)
)
{\displaystyle 2(x+7)=-3(x+1-2(x-1))}
.
x
=
23
{\displaystyle x=23}
x
=
23
{\displaystyle x=23}
54. Determine whether the equation
2
(
x
−
1
)
=
−
2
(
1
−
x
)
{\displaystyle 2(x-1)=-2(1-x)}
is conditional, a contradiction, or an identity.
55. Solve the equation
|
4
−
x
2
|
=
3
{\displaystyle \displaystyle \left|{\frac {4-x}{2}}\right|=3}
.
{
−
2
,
10
}
{\displaystyle \{-2,10\}}
{
−
2
,
10
}
{\displaystyle \{-2,10\}}
56. Solve the equation
|
1
−
2
(
3
−
x
)
|
=
15
{\displaystyle |1-2(3-x)|=15}
.
{
−
5
,
10
}
{\displaystyle \{-5,10\}}
{
−
5
,
10
}
{\displaystyle \{-5,10\}}
m
=
E
c
−
2
{\displaystyle m=Ec^{-2}}
m
=
E
c
−
2
{\displaystyle m=Ec^{-2}}
F
=
9
5
C
+
32
{\displaystyle F={\frac {9}{5}}C+32}
F
=
9
5
C
+
32
{\displaystyle F={\frac {9}{5}}C+32}
70. Solve the equation
x
2
−
6
x
+
5
=
0
{\displaystyle x^{2}-6x+5=0}
by factoring.
{
1
,
5
}
{\displaystyle \{1,5\}}
{
1
,
5
}
{\displaystyle \{1,5\}}
71. Solve the equation
3
(
x
+
1
)
2
−
48
=
0
{\displaystyle 3(x+1)^{2}-48=0}
using the square root procedure.
{
−
5
,
3
}
{\displaystyle \{-5,3\}}
{
−
5
,
3
}
{\displaystyle \{-5,3\}}
72. Solve the equation
x
2
−
6
x
=
16
{\displaystyle x^{2}-6x=16}
by completing the square.
{
−
2
,
8
}
{\displaystyle \{-2,8\}}
{
−
2
,
8
}
{\displaystyle \{-2,8\}}
73. Solve the equation
3
x
2
+
2
x
=
1
{\displaystyle 3x^{2}+2x=1}
by using the quadratic formula.
{
−
1
,
1
3
}
{\displaystyle \left\{-1,{\frac {1}{3}}\right\}}
{
−
1
,
1
3
}
{\displaystyle \left\{-1,{\frac {1}{3}}\right\}}
80. Solve the equation
x
3
+
3
x
2
−
4
x
−
12
=
0
{\displaystyle x^{3}+3x^{2}-4x-12=0}
.
{
−
3
,
−
2
,
2
}
{\displaystyle \{-3,-2,2\}}
{
−
3
,
−
2
,
2
}
{\displaystyle \{-3,-2,2\}}
81. Solve the equation
2
x
+
1
x
+
4
+
3
=
−
2
x
+
4
{\displaystyle \displaystyle {\frac {2x+1}{x+4}}+3={\frac {-2}{x+4}}}
.
x
=
−
3
{\displaystyle x=-3}
x
=
−
3
{\displaystyle x=-3}
82. Solve the equation
x
=
2
x
+
3
{\displaystyle x={\sqrt {2x+3}}}
.
x
=
3
{\displaystyle x=3}
x
=
3
{\displaystyle x=3}
83. Solve the equation
4
x
4
/
5
−
54
=
270
{\displaystyle 4x^{4/5}-54=270}
.
x
=
±
243
{\displaystyle x=\pm 243}
x
=
±
243
{\displaystyle x=\pm 243}
84. Solve the equation
6
x
6
+
x
3
−
15
=
0
{\displaystyle 6x^{6}+x^{3}-15=0}
.
{
3
2
3
,
−
5
3
3
}
{\displaystyle \left\{{\sqrt[{3}]{\frac {3}{2}}},{\sqrt[{3}]{-{\frac {5}{3}}}}\right\}}
{
3
2
3
,
−
5
3
3
}
{\displaystyle \left\{{\sqrt[{3}]{\frac {3}{2}}},{\sqrt[{3}]{-{\frac {5}{3}}}}\right\}}
90. Find the distance between points
(
3
,
5
)
{\displaystyle (3,5)}
and
(
−
2
,
17
)
{\displaystyle (-2,17)}
.
91. Find the midpoint of the line segment connecting points
(
−
4
,
7
)
{\displaystyle (-4,7)}
and
(
2
,
−
1
)
{\displaystyle (2,-1)}
.
(
−
1
,
3
)
{\displaystyle (-1,3)}
(
−
1
,
3
)
{\displaystyle (-1,3)}
92. Find the intercepts of the graph of the equation
3
y
=
12
−
4
x
{\displaystyle 3y=12-4x}
.
x -intercept at
(
3
,
0
)
{\displaystyle (3,0)}
y -intercept at
(
0
,
4
)
{\displaystyle (0,4)}
x -intercept at
(
3
,
0
)
{\displaystyle (3,0)}
y -intercept at
(
0
,
4
)
{\displaystyle (0,4)}
93. Sketch the graph of the function
y
=
x
2
−
x
−
2
{\displaystyle y=x^{2}-x-2}
and find its intercepts.
101. Find the domain of
f
(
x
)
=
x
+
x
−
4
{\displaystyle f(x)=x+{\sqrt {x-4}}}
.
x
≥
4
{\displaystyle x\geq 4}
x
≥
4
{\displaystyle x\geq 4}
103. Find the domain of
A
(
x
)
=
x
x
−
1
{\displaystyle \displaystyle A(x)={\frac {\sqrt {x}}{x-1}}}
.
[
0
,
1
)
∪
(
1
,
∞
)
{\displaystyle [0,1)\cup (1,\infty )}
[
0
,
1
)
∪
(
1
,
∞
)
{\displaystyle [0,1)\cup (1,\infty )}
104. Find the zeroes of
f
(
x
)
=
(
x
−
1
)
2
−
4
{\displaystyle f(x)=(x-1)^{2}-4}
.
{
−
1
,
3
}
{\displaystyle \{-1,3\}}
{
−
1
,
3
}
{\displaystyle \{-1,3\}}
105. Find the zeroes of
g
(
x
)
=
x
2
+
1
{\displaystyle g(x)=x^{2}+1}
.
∅
{\displaystyle \emptyset }
∅
{\displaystyle \emptyset }
106. Find the zeroes of
h
(
r
)
=
x
−
1
/
x
{\displaystyle h(r)=x-1/x}
.
{
−
1
,
1
}
{\displaystyle \{-1,1\}}
{
−
1
,
1
}
{\displaystyle \{-1,1\}}
110. Find the slope of the line that passes through the points
(
−
1
,
3
)
{\displaystyle (-1,3)}
and
(
2
,
−
4
)
{\displaystyle (2,-4)}
.
−
7
/
3
{\displaystyle -7/3}
−
7
/
3
{\displaystyle -7/3}
111. Graph the function
f
(
x
)
=
−
3
x
−
2
{\displaystyle f(x)=-3x-2}
by finding the slope and the
y
{\displaystyle y}
-intercept.
112. Graph the function given by
3
y
+
2
x
=
2
{\displaystyle 3y+2x=2}
by finding the slope and the
y
{\displaystyle y}
-intercept.
In the next four exercises, state the answer in the form
y
=
m
x
+
b
{\displaystyle y=mx+b}
113. Find an equation for the line with the slope
m
=
2
{\displaystyle m=2}
and containing the point
(
2
,
1
)
{\displaystyle (2,1)}
.
y
=
2
x
−
3
{\displaystyle y=2x-3}
y
=
2
x
−
3
{\displaystyle y=2x-3}
114. Find an equation for the line containing the points
(
−
1
,
4
)
{\displaystyle (-1,4)}
and
(
2
,
1
)
{\displaystyle (2,1)}
.
y
=
−
x
+
3
{\displaystyle y=-x+3}
y
=
−
x
+
3
{\displaystyle y=-x+3}
115. Find an equation of the line parallel to the line
y
=
−
2
x
−
1
{\displaystyle y=-2x-1}
and containing the point
(
2
,
2
)
{\displaystyle (2,2)}
.
y
=
−
2
x
+
6
{\displaystyle y=-2x+6}
y
=
−
2
x
+
6
{\displaystyle y=-2x+6}
116. Find an equation of the line perpendicular to the line
y
=
−
2
x
−
1
{\displaystyle y=-2x-1}
and containing the point
(
−
1
,
−
1
)
{\displaystyle (-1,-1)}
.
y
=
x
/
2
−
1
/
2
{\displaystyle y=x/2-1/2}
y
=
x
/
2
−
1
/
2
{\displaystyle y=x/2-1/2}
120. Complete the square to find the standard form of the quadratic function
f
(
x
)
=
x
2
+
6
x
−
1
{\displaystyle f(x)=x^{2}+6x-1}
and use it to sketch its graph.
121. Complete the square to find the standard form of the quadratic function
f
(
x
)
=
−
2
x
2
−
4
x
+
5
{\displaystyle f(x)=-2x^{2}-4x+5}
and use it to sketch its graph.
122. Use the vertex formula to find the vertex of the graph of the quadratic function
f
(
x
)
=
x
2
−
6
x
{\displaystyle f(x)=x^{2}-6x}
, and write the function in the standard form.
123. Use the vertex formula to find the vertex of the graph of the quadratic function
f
(
x
)
=
−
5
x
2
−
6
x
+
3
{\displaystyle f(x)=-5x^{2}-6x+3}
, and write the function in the standard form.
124. Find the maximum or minimum value of the function
f
(
x
)
=
2
x
2
+
6
x
−
5
{\displaystyle f(x)=2x^{2}+6x-5}
and state the range of the function.
125. Find the maximum or minimum value of the function
f
(
x
)
=
−
x
2
−
6
x
−
2
{\displaystyle f(x)=-x^{2}-6x-2}
and state the range of the function.
130. Use polynomial long division to divide
P
(
x
)
=
−
x
4
+
2
x
3
−
3
x
2
−
1
{\displaystyle P(x)=-x^{4}+2x^{3}-3x^{2}-1}
by
D
(
x
)
=
x
+
1
{\displaystyle D(x)=x+1}
.
−
x
3
+
3
x
2
−
6
x
+
6
+
−
7
x
+
1
{\displaystyle -x^{3}+3x^{2}-6x+6+{\frac {-7}{x+1}}}
−
x
3
+
3
x
2
−
6
x
+
6
+
−
7
x
+
1
{\displaystyle -x^{3}+3x^{2}-6x+6+{\frac {-7}{x+1}}}
131. Use polynomial long division to divide
P
(
x
)
=
2
x
5
+
x
4
+
x
−
3
{\displaystyle P(x)=2x^{5}+x^{4}+x-3}
by
D
(
x
)
=
x
2
−
x
+
2
{\displaystyle D(x)=x^{2}-x+2}
.
2
x
3
+
3
x
2
−
x
−
7
+
−
4
x
+
11
x
2
−
x
+
2
{\displaystyle 2x^{3}+3x^{2}-x-7+{\frac {-4x+11}{x^{2}-x+2}}}
2
x
3
+
3
x
2
−
x
−
7
+
−
4
x
+
11
x
2
−
x
+
2
{\displaystyle 2x^{3}+3x^{2}-x-7+{\frac {-4x+11}{x^{2}-x+2}}}
Polynomial Functions of Higher Degrees [ edit | edit source ] 140. Use the
Intermediate Value Theorem to show that the polynomial function
Failed to parse (syntax error): {\displaystyle p(x) = x^6 − 3x^5 + 6x^2 − 1}
has a zero in the interval
[
−
1
,
1
]
{\displaystyle [-1,1]}
.
p
(
0
)
<
0
<
p
(
1
)
{\displaystyle p(0)<0<p(1)}
p
(
0
)
<
0
<
p
(
1
)
{\displaystyle p(0)<0<p(1)}
141. Given that the function
Failed to parse (syntax error): {\displaystyle \displaystyle f(x) = \frac{x^6 − 10}{x^5+10x^2−x}}
is continuous on the interval
[
1
,
3
]
{\displaystyle [1,3]}
, prove that it has a zero in that interval.
f
(
1
)
<
0
<
f
(
3
)
{\displaystyle f(1)<0<f(3)}
f
(
1
)
<
0
<
f
(
3
)
{\displaystyle f(1)<0<f(3)}
Zeros of Polynomial Functions [ edit | edit source ] 150. Find all zeroes of a polynomial function
p
(
x
)
=
(
x
+
4
)
3
(
x
2
−
9
)
2
{\displaystyle p(x)=(x+4)^{3}(x^{2}-9)^{2}}
and state the multiplicity of each zero.
151. Use the Descarte's Rule of Signs to state the possible numbers of positive and negative zeroes of the polynomial function
p
(
x
)
=
x
4
−
3
x
3
+
x
2
−
1
{\displaystyle p(x)=x^{4}-3x^{3}+x^{2}-1}
.
3 or 1 positive zeroes, 1 negative zero.
3 or 1 positive zeroes, 1 negative zero.
Fundamental Theorem of Algebra [ edit | edit source ] 160. Find a polynomial function of the lowest degree with roots
1
,
−
1
,
2
+
3
i
,
2
−
3
i
{\displaystyle 1,\ -1,\ 2+3i,\ 2-3i}
.
p
(
x
)
=
x
4
−
4
x
3
+
12
x
2
+
4
x
−
13
{\displaystyle p(x)=x^{4}-4x^{3}+12x^{2}+4x-13}
p
(
x
)
=
x
4
−
4
x
3
+
12
x
2
+
4
x
−
13
{\displaystyle p(x)=x^{4}-4x^{3}+12x^{2}+4x-13}
161. Find all zeroes of a polynomial function
p
(
x
)
=
4
x
4
−
4
x
3
+
13
x
2
−
12
x
+
3
{\displaystyle p(x)=4x^{4}-4x^{3}+13x^{2}-12x+3}
, given that it has a root
1
/
2
{\displaystyle 1/2}
of multiplicity
2
{\displaystyle 2}
. State the answer by rewriting the polynomial as a product of linear factors.
p
(
x
)
=
(
x
−
1
/
2
)
2
(
x
+
i
3
)
(
x
−
i
3
)
{\displaystyle p(x)=(x-1/2)^{2}(x+i{\sqrt {3}})(x-i{\sqrt {3}})}
p
(
x
)
=
(
x
−
1
/
2
)
2
(
x
+
i
3
)
(
x
−
i
3
)
{\displaystyle p(x)=(x-1/2)^{2}(x+i{\sqrt {3}})(x-i{\sqrt {3}})}
170. Use composition of functions to determine whether
f
(
x
)
=
x
2
−
3
2
{\displaystyle \displaystyle f(x)={\frac {x}{2}}-{\frac {3}{2}}}
and
g
(
x
)
=
2
x
+
3
{\displaystyle g(x)=2x+3}
are inverses of each other.
171. Find the inverse of the function
f
=
{
(
−
2
,
4
)
,
(
−
1
,
1
)
,
(
0
,
0
)
,
(
1
,
1
)
,
(
2
,
4
)
}
{\displaystyle f=\{(-2,4),(-1,1),(0,0),(1,1),(2,4)\}}
or prove that it does not exist.
This function is not injective, since
f
(
−
1
)
=
f
(
1
)
=
1
{\displaystyle f(-1)=f(1)=1}
This function is not injective, since
f
(
−
1
)
=
f
(
1
)
=
1
{\displaystyle f(-1)=f(1)=1}
173. Given
f
(
x
)
=
4
−
x
{\displaystyle \displaystyle f(x)={\sqrt {4-x}}}
find the inverse of
f
{\displaystyle f}
, and state the domains and the ranges for both
f
{\displaystyle f}
and
f
−
1
{\displaystyle f^{-1}}
.
f
−
1
(
x
)
=
4
−
x
2
{\displaystyle \displaystyle f^{-1}(x)=4-x^{2}}
d
o
m
(
f
)
=
r
a
n
(
f
−
1
)
=
(
−
∞
,
4
]
{\displaystyle \mathop {\mathrm {dom} } (f)=\mathop {\mathrm {ran} } (f^{-1})=(-\infty ,4]}
r
a
n
(
f
)
=
d
o
m
(
f
−
1
)
=
[
0
,
∞
)
{\displaystyle \mathop {\mathrm {ran} } (f)=\mathop {\mathrm {dom} } (f^{-1})=[0,\infty )}
f
−
1
(
x
)
=
4
−
x
2
{\displaystyle \displaystyle f^{-1}(x)=4-x^{2}}
d
o
m
(
f
)
=
r
a
n
(
f
−
1
)
=
(
−
∞
,
4
]
{\displaystyle \mathop {\mathrm {dom} } (f)=\mathop {\mathrm {ran} } (f^{-1})=(-\infty ,4]}
r
a
n
(
f
)
=
d
o
m
(
f
−
1
)
=
[
0
,
∞
)
{\displaystyle \mathop {\mathrm {ran} } (f)=\mathop {\mathrm {dom} } (f^{-1})=[0,\infty )}
182. Sketch the graph of the function
f
(
x
)
=
3
x
{\displaystyle f(x)=3^{x}}
.
183. Sketch the graph of the function
f
(
x
)
=
2
−
x
+
1
{\displaystyle f(x)=2^{-x}+1}
.
190. Rewrite the equation
log
4
(
x
+
1
)
=
3
y
{\displaystyle \log _{4}(x+1)=3y}
in exponential form.
4
3
y
=
x
+
1
{\displaystyle 4^{3y}=x+1}
4
3
y
=
x
+
1
{\displaystyle 4^{3y}=x+1}
192. Evaluate
log
0.3
(
100
/
9
)
{\displaystyle \log _{0.3}(100/9)}
without using a computer.
−
2
{\displaystyle -2}
−
2
{\displaystyle -2}
193. Find the domain of the function
f
(
x
)
=
ln
(
x
+
2
)
{\displaystyle f(x)=\ln(x+2)}
.
(
−
2
,
∞
)
{\displaystyle (-2,\infty )}
(
−
2
,
∞
)
{\displaystyle (-2,\infty )}
194. Find the domain of the function
f
(
x
)
=
ln
(
x
2
−
9
)
{\displaystyle f(x)=\ln(x^{2}-9)}
.
|
x
|
>
3
{\displaystyle |x|>3}
|
x
|
>
3
{\displaystyle |x|>3}
200. Use a computer to estimate
log
π
(
1.618
)
{\displaystyle \log _{\pi }(1.618)}
.
0.4203532
{\displaystyle 0.4203532}
0.4203532
{\displaystyle 0.4203532}
201. Rewrite the expression
log
5
(
z
4
x
125
y
)
{\displaystyle \displaystyle \log _{5}\left({\frac {z^{4}{\sqrt {x}}}{125y}}\right)}
in a way that leaves the arguments of
log
{\displaystyle \log }
function as simple as possible.
4
log
5
(
z
)
+
1
2
log
5
(
x
)
−
3
−
log
5
(
y
)
{\displaystyle \displaystyle 4\log _{5}(z)+{\frac {1}{2}}\log _{5}(x)-3-\log _{5}(y)}
4
log
5
(
z
)
+
1
2
log
5
(
x
)
−
3
−
log
5
(
y
)
{\displaystyle \displaystyle 4\log _{5}(z)+{\frac {1}{2}}\log _{5}(x)-3-\log _{5}(y)}
202. Rewrite the expression
1
2
log
3
(
x
)
−
log
3
(
y
2
)
+
2
log
3
(
x
+
2
)
{\displaystyle \displaystyle {\frac {1}{2}}\log _{3}(x)-\log _{3}(y^{2})+2\log _{3}(x+2)}
as a single logarithm with coefficient
1
{\displaystyle 1}
.
log
3
(
(
x
+
2
)
2
x
y
)
{\displaystyle \displaystyle \log _{3}\left({\frac {(x+2)^{2}{\sqrt {x}}}{y}}\right)}
log
3
(
(
x
+
2
)
2
x
y
)
{\displaystyle \displaystyle \log _{3}\left({\frac {(x+2)^{2}{\sqrt {x}}}{y}}\right)}
Exponetial and Logarithmic Equations [ edit | edit source ] 210. Solve the equation
10
7
−
x
=
1000
{\displaystyle 10^{7-x}=1000}
.
x
=
4
{\displaystyle x=4}
x
=
4
{\displaystyle x=4}
211. Solve the equation
log
10
(
x
2
+
19
)
=
2
{\displaystyle \log _{10}(x^{2}+19)=2}
.
x
=
±
9
{\displaystyle x=\pm 9}
x
=
±
9
{\displaystyle x=\pm 9}
212. Solve the equation
log
3
(
x
)
+
log
3
(
x
+
6
)
=
3
{\displaystyle \log _{3}(x)+\log _{3}(x+6)=3}
.
{
3
,
−
9
}
{\displaystyle \{3,\ -9\}}
{
3
,
−
9
}
{\displaystyle \{3,\ -9\}}
213. Solve the equation
5
3
x
=
3
x
+
4
{\displaystyle 5^{3x}=3^{x+4}}
.
x
=
4
log
5
(
3
)
3
−
log
5
(
3
)
{\displaystyle \displaystyle x={\frac {4\log _{5}(3)}{3-\log _{5}(3)}}}
x
=
4
log
5
(
3
)
3
−
log
5
(
3
)
{\displaystyle \displaystyle x={\frac {4\log _{5}(3)}{3-\log _{5}(3)}}}
222. Convert the degree measure
18
∘
{\displaystyle 18^{\circ }}
into the exact radian measure.
π
/
10
{\displaystyle \pi /10}
π
/
10
{\displaystyle \pi /10}
223. Convert the radian measure
5
π
6
{\displaystyle \displaystyle {\frac {5\pi }{6}}}
into the exact degree measure.
150
∘
{\displaystyle 150^{\circ }}
150
∘
{\displaystyle 150^{\circ }}
Right Triangle Trigonometry [ edit | edit source ] 230. Find the values of
sin
(
θ
)
{\displaystyle \sin(\theta )}
,
cos
(
θ
)
{\displaystyle \cos(\theta )}
,
tan
(
θ
)
{\displaystyle \tan(\theta )}
,
cot
(
θ
)
{\displaystyle \cot(\theta )}
,
sec
(
θ
)
{\displaystyle \sec(\theta )}
, and
csc
(
θ
)
{\displaystyle \csc(\theta )}
if
θ
=
∠
A
{\displaystyle \theta =\angle A}
in the right triangle
A
B
C
{\displaystyle ABC}
with
∠
B
=
π
/
2
{\displaystyle \angle B=\pi /2}
and side lengths
A
B
=
5
{\displaystyle AB=5}
and
A
C
=
13
{\displaystyle AC=13}
.
sin
(
θ
)
=
12
/
13
{\displaystyle \sin(\theta )=12/13}
cos
(
θ
)
=
5
/
13
{\displaystyle \cos(\theta )=5/13}
tan
(
θ
)
12
/
5
{\displaystyle \tan(\theta )12/5}
cot
(
θ
)
=
5
/
12
{\displaystyle \cot(\theta )=5/12}
sec
(
θ
)
=
13
/
12
{\displaystyle \sec(\theta )=13/12}
csc
(
θ
)
=
13
/
5
{\displaystyle \csc(\theta )=13/5}
sin
(
θ
)
=
12
/
13
{\displaystyle \sin(\theta )=12/13}
cos
(
θ
)
=
5
/
13
{\displaystyle \cos(\theta )=5/13}
tan
(
θ
)
12
/
5
{\displaystyle \tan(\theta )12/5}
cot
(
θ
)
=
5
/
12
{\displaystyle \cot(\theta )=5/12}
sec
(
θ
)
=
13
/
12
{\displaystyle \sec(\theta )=13/12}
csc
(
θ
)
=
13
/
5
{\displaystyle \csc(\theta )=13/5}
Trigonometric Functions of any Angle [ edit | edit source ] 240. Find the value of
sin
(
x
)
{\displaystyle \sin(x)}
,
cos
(
x
)
{\displaystyle \cos(x)}
,
tan
(
x
)
{\displaystyle \tan(x)}
,
csc
(
x
)
{\displaystyle \csc(x)}
,
sec
(
x
)
{\displaystyle \sec(x)}
, and
cot
(
x
)
{\displaystyle \cot(x)}
for the angle, in standard position,
whose terminal side passes through the point
(
−
3
,
−
10
)
{\displaystyle (-3,-10)}
.
sin
(
x
)
=
−
10
/
109
{\displaystyle \sin(x)=-10/{\sqrt {109}}}
cos
(
x
)
=
−
3
/
109
{\displaystyle \cos(x)=-3/{\sqrt {109}}}
tan
(
x
)
=
10
/
3
{\displaystyle \tan(x)=10/3}
csc
(
x
)
=
−
109
/
10
{\displaystyle \csc(x)=-{\sqrt {109}}/10}
sec
(
x
)
=
−
109
/
3
{\displaystyle \sec(x)=-{\sqrt {109}}/3}
cot
(
x
)
=
0.3
{\displaystyle \cot(x)=0.3}
sin
(
x
)
=
−
10
/
109
{\displaystyle \sin(x)=-10/{\sqrt {109}}}
cos
(
x
)
=
−
3
/
109
{\displaystyle \cos(x)=-3/{\sqrt {109}}}
tan
(
x
)
=
10
/
3
{\displaystyle \tan(x)=10/3}
csc
(
x
)
=
−
109
/
10
{\displaystyle \csc(x)=-{\sqrt {109}}/10}
sec
(
x
)
=
−
109
/
3
{\displaystyle \sec(x)=-{\sqrt {109}}/3}
cot
(
x
)
=
0.3
{\displaystyle \cot(x)=0.3}
241. Find
cos
(
θ
)
{\displaystyle \cos(\theta )}
if
cot
(
θ
)
=
−
1
{\displaystyle \cot(\theta )=-1}
and
π
/
2
<
θ
<
π
{\displaystyle \pi /2<\theta <\pi }
.
−
1
/
2
{\displaystyle -1/{\sqrt {2}}}
−
1
/
2
{\displaystyle -1/{\sqrt {2}}}
Trigonometric Functions of Real Numbers [ edit | edit source ] 250. Find the exact value of
cot
(
2
π
/
3
)
{\displaystyle \cot(2\pi /3)}
.
−
3
{\displaystyle -{\sqrt {3}}}
−
3
{\displaystyle -{\sqrt {3}}}
251. Find the exact value of
sec
(
−
5
π
/
6
)
{\displaystyle \sec(-5\pi /6)}
.
−
0.5
{\displaystyle -0.5}
−
0.5
{\displaystyle -0.5}
252. Find the exact value of
tan
(
12
π
)
{\displaystyle \tan(12\pi )}
.
0
{\displaystyle 0}
0
{\displaystyle 0}
253. Rewrite
cot
(
t
)
sin
(
t
)
{\displaystyle \cot(t)\sin(t)}
in terms of a single trigonometric function or a constant.
cos
(
t
)
{\displaystyle \cos(t)}
cos
(
t
)
{\displaystyle \cos(t)}
254. Rewrite
1
−
sin
2
(
t
)
cot
2
(
t
)
{\displaystyle \displaystyle {\frac {1-\sin ^{2}(t)}{\cot ^{2}(t)}}}
in terms of a single trigonometric function or a constant.
sin
2
(
t
)
{\displaystyle \sin ^{2}(t)}
sin
2
(
t
)
{\displaystyle \sin ^{2}(t)}
255. Rewrite
1
1
−
sin
(
t
)
+
1
1
+
sin
(
t
)
{\displaystyle \displaystyle {\frac {1}{1-\sin(t)}}+{\frac {1}{1+\sin(t)}}}
in terms of a single trigonometric function or a constant.
2
sec
2
(
t
)
{\displaystyle 2\sec ^{2}(t)}
2
sec
2
(
t
)
{\displaystyle 2\sec ^{2}(t)}
Verification of Trigonometric Identities [ edit | edit source ] 270. Verify the identity
tan
(
x
)
(
1
−
cot
(
x
)
)
=
tan
(
x
)
−
1
{\displaystyle \tan(x)(1-\cot(x))=\tan(x)-1}
.
{\displaystyle }
{\displaystyle }
271. Verify the identity
sin
4
(
x
)
−
cos
4
(
x
)
=
sin
2
(
x
)
−
cos
2
(
x
)
{\displaystyle \sin ^{4}(x)-\cos ^{4}(x)=\sin ^{2}(x)-\cos ^{2}(x)}
.
{\displaystyle }
{\displaystyle }
272. Verify the identity
(
tan
(
x
)
+
1
)
2
=
sec
2
(
x
)
+
2
tan
(
x
)
{\displaystyle (\tan(x)+1)^{2}=\sec ^{2}(x)+2\tan(x)}
.
{\displaystyle }
{\displaystyle }
273. Verify the identity
sec
(
x
)
csc
(
x
)
=
cot
(
x
)
+
tan
(
x
)
{\displaystyle \sec(x)\csc(x)=\cot(x)+\tan(x)}
.
{\displaystyle }
{\displaystyle }
![{\displaystyle [0,1]\cup (2,3)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/363693cc935878131d23e3d033c4e7f27f478a48)
![{\displaystyle (-\infty ,-1]\cup (1,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e3aaf180dd39c005bbcd901690ef2478fbaf05e)
![{\displaystyle (-\infty ,1]\cap (-1,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/daa73225b222a4d459f44ab0d64f0fd65d4767ee)
![{\displaystyle \left\{{\sqrt[{3}]{\frac {3}{2}}},{\sqrt[{3}]{-{\frac {5}{3}}}}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4aa074cb7d7e289978162b694b5dc82340245a5)
![{\displaystyle [-1,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51e3b7f14a6f70e614728c583409a0b9a8b9de01)
![{\displaystyle [1,3]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0fcfee68a647afbfbe0440e15fc9fd260abbdc7)
![{\displaystyle \mathop {\mathrm {dom} } (f)=\mathop {\mathrm {ran} } (f^{-1})=(-\infty ,4]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/808e6da8bf9150e1edbc6611319df6dc3933595e)
