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Math Problems
Algebra 1
Power rule with rational exponents
What is
(
f
∗
g
)
(
x
)
(f * g)(x)
(
f
∗
g
)
(
x
)
?
\newline
f
(
x
)
=
4
x
+
1
f(x) = 4x + 1
f
(
x
)
=
4
x
+
1
\newline
g
(
x
)
=
2
x
g(x) = 2x
g
(
x
)
=
2
x
\newline
Write your answer as a polynomial or a rational function in simplest form.
\newline
______
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What is
(
f
∗
g
)
(
x
)
(f * g)(x)
(
f
∗
g
)
(
x
)
?
\newline
f
(
x
)
=
3
x
f(x) = 3x
f
(
x
)
=
3
x
\newline
g
(
x
)
=
3
x
2
+
4
x
g(x) = 3x^2 + 4x
g
(
x
)
=
3
x
2
+
4
x
\newline
Write your answer as a polynomial or a rational function in simplest form.
\newline
______
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What is
(
f
∗
g
)
(
x
)
(f * g)(x)
(
f
∗
g
)
(
x
)
?
\newline
f
(
x
)
=
x
2
−
4
f(x) = x^2 - 4
f
(
x
)
=
x
2
−
4
\newline
g
(
x
)
=
−
2
x
g(x) = -2x
g
(
x
)
=
−
2
x
\newline
Write your answer as a polynomial or a rational function in simplest form.
\newline
______
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What is
(
f
∗
g
)
(
x
)
(f * g)(x)
(
f
∗
g
)
(
x
)
?
\newline
f
(
x
)
=
3
x
+
3
f(x) = 3x + 3
f
(
x
)
=
3
x
+
3
\newline
g
(
x
)
=
−
3
x
g(x) = -3x
g
(
x
)
=
−
3
x
\newline
Write your answer as a polynomial or a rational function in simplest form.
\newline
______
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Your answer is incorrect. The expression can be simplified further.
\newline
Write the following expression in simplified radical form.
\newline
80
s
10
t
16
4
\sqrt[4]{80 s^{10} t^{16}}
4
80
s
10
t
16
\newline
Assume that all of the variables in the expression represent positive real numbers.
\newline
2
s
2
r
3
5
s
2
t
4
2 s^{2} r^{3} \sqrt{5 s^{2} t^{4}}
2
s
2
r
3
5
s
2
t
4
\newline
□
□
□
□
\sqrt{\square} \quad \sqrt[\square]{\square} \quad \square
□
□
□
□
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The list below shows the temperature Bao recorded at
11
11
11
:
00
00
00
p.m. for four days last January.
\newline
Monday:
−
7
∘
F
-7^{\circ} \mathrm{F}
−
7
∘
F
\newline
Tuesday:
0
∘
F
0^{\circ} \mathrm{F}
0
∘
F
\newline
Wednesday:
7
∘
F
7^{\circ} \mathrm{F}
7
∘
F
\newline
Thursday:
−
2
5
∘
F
-25^{\circ} \mathrm{F}
−
2
5
∘
F
\newline
On which of the days did Bao record the lowest (coldest) temperature?
\newline
Monday
\newline
Tuesday
\newline
Wednesday
\newline
Thursday
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Is
π
\pi
π
a natural number?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
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Is
π
\pi
π
an integer?
\newline
Choices:
\newline
(A) yes
\newline
(B) no
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Let
f
(
x
)
=
1
x
−
2
f(x)=\frac{1}{x-2}
f
(
x
)
=
x
−
2
1
.
\newline
Can we use the mean value theorem to say the equation
f
′
(
x
)
=
1
2
f^{\prime}(x)=\frac{1}{2}
f
′
(
x
)
=
2
1
has a solution where
1
<
x
<
4
1<x<4
1
<
x
<
4
?
\newline
Choose
1
1
1
answer:
\newline
(A) No, since the function is not differentiable on that interval.
\newline
B No, since the average rate of change of
f
f
f
over the interval
1
≤
x
≤
4
1 \leq x \leq 4
1
≤
x
≤
4
isn't equal to
1
2
\frac{1}{2}
2
1
.
\newline
(C) Yes, both conditions for using the mean value theorem have been met.
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Write the expression in simplest form.
\newline
(
9
3
+
7
)
=
(\frac{9}{\sqrt{3}+\sqrt{7}})=
(
3
+
7
9
)
=
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exttt{ extdagger} exttt{
0
0
0
//
1
1
1
pt}
\newline
53
i
g
h
t
l
e
f
t
h
a
r
p
o
o
n
s
1
53 ightleftharpoons 1
53
i
g
h
tl
e
f
t
ha
r
p
oo
n
s
1
\newline
(i) Details
\newline
\vdots
\newline
Consider the discrete random variable
\newline
X
X
X
given in the table below. Calculate the mean, variance, and standard deviation of
\newline
X
X
X
. Round answers to two decimal places.
\newline
tured snippets
\newline
Feedback
\newline
;
\newline
ordered universe
\newline
e exttt{ces} of the universe ne every piece
\newline
exttt{k} of relationships and
\newline
\newline
\newline
X
X
X
\newline
1
1
1
\newline
11
11
11
\newline
15
15
15
\newline
17
17
17
\newline
\newline
P
(
X
)
P(X)
P
(
X
)
\newline
0.68
0.68
0.68
\newline
X
X
X
0
0
0
\newline
X
X
X
1
1
1
\newline
X
X
X
2
2
2
\newline
\newline
\newline
\{:[\mu=
39
39
39
.
74
74
74
],[\sigma^{
2
2
2
}=]:}
\newline
\sigma=$\(\newline\)What is the expected value of\(\newline\)\(X\)? \(X\)\(4\)\(\newline\)Submit Question
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Suppose that the functions
f
f
f
and
g
g
g
are defined as follows.
\newline
f
(
x
)
=
x
−
1
g
(
x
)
=
6
x
f(x)=x-1\quad g(x)=\frac{6}{x}
f
(
x
)
=
x
−
1
g
(
x
)
=
x
6
\newline
Find
g
f
\frac{g}{f}
f
g
. Then, give its domain using an interval or union of intervals.
\newline
Simplify your answers.
\newline
(
g
f
)
(
x
)
=
□
\left(\frac{g}{f}\right)(x)=\square
(
f
g
)
(
x
)
=
□
\newline
Domain of
g
f
\frac{g}{f}
f
g
:\square
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The system has no solution.
\newline
The system has a unique solution:
\newline
(
x
,
y
)
=
(
□
,
□
)
(x,y)=(\square,\square)
(
x
,
y
)
=
(
□
,
□
)
\newline
{
−
x
−
4
y
+
4
=
0
x
+
4
y
=
4
\begin{cases} -x-4y+4=0 \ x+4y=4 \end{cases}
{
−
x
−
4
y
+
4
=
0
x
+
4
y
=
4
\newline
The system has infinitely many solutions.
\newline
They must satisfy the following equation:
\newline
y
=
□
y=\square
y
=
□
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Given:
\newline
∠
A
B
D
≅
∠
C
D
B
\angle ABD \cong \angle CDB
∠
A
B
D
≅
∠
C
D
B
and
\newline
A
B
‾
≅
C
D
‾
\overline{AB} \cong \overline{CD}
A
B
≅
C
D
\newline
Prove:
\newline
∠
A
′
≅
∠
C
\angle A' \cong \angle C
∠
A
′
≅
∠
C
\newline
List the three congruencies from the given statement or the diagram:
\newline
List Remaining Congruent Parts:
\newline
because...
\newline
Therefore...
\newline
(What you wanted to prove)
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destion
1
1
1
of
9
9
9
, Step
2
2
2
of
2
2
2
\newline
Correct
\newline
Consider the following functions.
\newline
f
(
x
)
=
−
4
x
and
g
(
x
)
=
x
3
f(x)=-4 x \text { and } g(x)=\sqrt[3]{x}
f
(
x
)
=
−
4
x
and
g
(
x
)
=
3
x
\newline
Step
2
2
2
of
2
2
2
: Find the formula for
(
f
g
)
(
x
)
\left(\frac{f}{g}\right)(x)
(
g
f
)
(
x
)
and simplify your answer. Then find the domain for
(
f
g
)
(
x
)
\left(\frac{f}{g}\right)(x)
(
g
f
)
(
x
)
. Round your answer to two decimal places, if
n
n
n
\newline
Answer
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The formula for the density of a bacteria population is
\newline
N
A
\frac{N}{A}
A
N
where there's an average of
\newline
N
N
N
bacteria in an area size
\newline
A
A
A
.
\newline
Guang and Orion are doing an experiment in school for which they need to measure the density of a bacteria population after a certain solution has been introduced. Guang reports a density of
4500
4500
4500
bacteria per square centimeter and Orion reports a density of
50
50
50
bacteria per square millimeter.
\newline
Which one reports a higher density?
\newline
Choose
1
1
1
answer:
\newline
(A) Guang reports a higher density than Orion.
\newline
(B) Orion reports a higher density than Guang.
\newline
(C) Guang and Orion report equal densities.
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n
=
(
1.96
σ
E
)
2
n=\left(\frac{1.96 \sigma}{E}\right)^{2}
n
=
(
E
1.96
σ
)
2
\newline
Elon is planning a study of hours of internet use per month per household. He uses the given equation to determine the minimum number of randomly selected households,
n
n
n
, that he needs to include in his sample given a standard deviation of
σ
\sigma
σ
hours and a maximum allowable error of
E
E
E
hours. Which of the following equations correctly gives the standard deviation in terms of the maximum allowable error and the minimum number of randomly selected households?
\newline
Choose
1
1
1
answer:
\newline
(A)
σ
=
E
n
1.96
\sigma=\sqrt{\frac{E n}{1.96}}
σ
=
1.96
E
n
\newline
(B)
σ
=
E
n
1.96
\sigma=\frac{E \sqrt{n}}{1.96}
σ
=
1.96
E
n
\newline
(C)
σ
=
1.96
n
σ
\sigma=\sqrt{\frac{1.96 n}{\sigma}}
σ
=
σ
1.96
n
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Fully simplify using only positive exponents.
\newline
27
x
y
5
27
x
y
4
\frac{27 x y^{5}}{27 x y^{4}}
27
x
y
4
27
x
y
5
\newline
Answer:
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A
(
q
)
=
86
(
0.9
)
q
4
A(q) = 86(0.9)^{\frac{q}{4}}
A
(
q
)
=
86
(
0.9
)
4
q
\newline
The function models
A
A
A
, the number of active players, in thousands, of a mobile game
q
q
q
quarter years after
2018
2018
2018
. Based on the function, how many quarter years does it take for the number of active players to decreases by
10
%
10\%
10%
?
\newline
Choose
1
1
1
answer:
\newline
(A)
0.1
0.1
0.1
\newline
(B)
0.225
0.225
0.225
\newline
(C)
0.9
0.9
0.9
\newline
(D)
4
4
4
Get tutor help
Find the average value of the function
f
(
x
)
=
4
2
x
−
11
f(x)=\frac{4}{2 x-11}
f
(
x
)
=
2
x
−
11
4
from
x
=
6
x=6
x
=
6
to
x
=
10
x=10
x
=
10
. Write your answer as the logarithm of a single number in simplest form.
\newline
Answer:
ln
(
□
)
\ln (\square)
ln
(
□
)
Get tutor help
(
1
sec
x
−
tan
x
)
−
(
1
sec
x
+
tan
x
)
\left(\frac{1}{\sec x - \tan x}\right) - \left(\frac{1}{\sec x + \tan x}\right)
(
sec
x
−
tan
x
1
)
−
(
sec
x
+
tan
x
1
)
Get tutor help
Nayeli lights a
4
m
2
4m^{2}
4
m
2
space with
12
12
12
candles. Citlali lights a
10
m
2
10m^{2}
10
m
2
space with
30
30
30
of the same type of candles.
\newline
Whose space is lit brighter?
\newline
Choose
1
1
1
answer:
\newline
(A) Nayeli's area
\newline
(B) Citlali's area
\newline
(C) The spaces are equally bright.
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Which of the following is a correct interpretation of the expression
\newline
−
7
−
(
−
11
)
-7-(-11)
−
7
−
(
−
11
)
?
\newline
Choose
1
1
1
answer:
\newline
(A) Start at
−
7
-7
−
7
on the number line and move
11
11
11
to the left.
\newline
(B) Start at
−
7
-7
−
7
on the number line and move
11
11
11
to the right.
\newline
(C) Start at
−
11
-11
−
11
on the number line and move
7
7
7
to the left.
\newline
(D) Start at
−
11
-11
−
11
on the number line and move
7
7
7
to the right.
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Which of the following is a correct interpretation of the expression
−
7
−
(
−
11
)
-7-(-11)
−
7
−
(
−
11
)
?
\newline
Choose
1
1
1
answer:
\newline
(A) Start at
−
7
-7
−
7
on the number line and move
11
11
11
to the left.
\newline
(B) Start at
−
7
-7
−
7
on the number line and move
11
11
11
to the right.
\newline
(C) Start at
−
11
-11
−
11
on the number line and move
7
7
7
to the left.
\newline
(D) Start at
−
11
-11
−
11
on the number line and move
7
7
7
to the right.
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Write the expression
−
1
2
ln
4
+
ln
5
-\frac{1}{2} \ln 4+\ln 5
−
2
1
ln
4
+
ln
5
as a single logarithm in simplest form without any negative exponents.
\newline
Answer:
ln
(
□
)
\ln (\square)
ln
(
□
)
Get tutor help
Write the expression
−
1
4
ln
16
+
3
ln
5
-\frac{1}{4} \ln 16+3 \ln 5
−
4
1
ln
16
+
3
ln
5
as a single logarithm in simplest form without any negative exponents.
\newline
Answer:
ln
(
□
)
\ln (\square)
ln
(
□
)
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Write the expression
1
2
ln
9
+
ln
3
\frac{1}{2} \ln 9+\ln 3
2
1
ln
9
+
ln
3
as a single logarithm in simplest form without any negative exponents.
\newline
Answer:
ln
(
□
)
\ln (\square)
ln
(
□
)
Get tutor help
Write the expression
1
3
ln
27
−
3
ln
5
\frac{1}{3} \ln 27-3 \ln 5
3
1
ln
27
−
3
ln
5
as a single logarithm in simplest form without any negative exponents.
\newline
Answer:
ln
(
□
)
\ln (\square)
ln
(
□
)
Get tutor help
Write the expression
−
ln
2
+
ln
5
-\ln 2+\ln 5
−
ln
2
+
ln
5
as a single logarithm in simplest form without any negative exponents.
\newline
Answer:
ln
(
□
)
\ln (\square)
ln
(
□
)
Get tutor help
Write the expression
1
3
ln
8
+
ln
3
\frac{1}{3} \ln 8+\ln 3
3
1
ln
8
+
ln
3
as a single logarithm in simplest form without any negative exponents.
\newline
Answer:
ln
(
□
)
\ln (\square)
ln
(
□
)
Get tutor help
g
(
x
)
=
2
x
−
9
g
′
(
x
)
=
?
\begin{array}{l} g(x)=\sqrt{2 x-9} \\ g^{\prime}(x)=? \end{array}
g
(
x
)
=
2
x
−
9
g
′
(
x
)
=
?
\newline
Choose
1
1
1
answer:
\newline
(A)
2
x
−
9
2
\frac{\sqrt{2 x-9}}{2}
2
2
x
−
9
\newline
(B)
1
2
x
−
9
\frac{1}{\sqrt{2 x-9}}
2
x
−
9
1
\newline
(C)
1
2
2
x
−
9
\frac{1}{2 \sqrt{2 x-9}}
2
2
x
−
9
1
\newline
(D)
1
x
\frac{1}{\sqrt{x}}
x
1
Get tutor help
Let
f
(
x
)
=
1
∣
x
∣
f(x)=\frac{1}{|x|}
f
(
x
)
=
∣
x
∣
1
.
\newline
Can we use the mean value theorem to say the equation
f
′
(
x
)
=
−
1
4
f^{\prime}(x)=-\frac{1}{4}
f
′
(
x
)
=
−
4
1
has a solution where
2
<
x
<
4
2<x<4
2
<
x
<
4
?
\newline
Choose
1
1
1
answer:
\newline
(A) No, since the function is not differentiable on that interval.
\newline
(B) No, since the average rate of change of
f
f
f
over the interval
2
≤
x
≤
4
2 \leq x \leq 4
2
≤
x
≤
4
isn't equal to
−
1
4
-\frac{1}{4}
−
4
1
.
\newline
(C) Yes, both conditions for using the mean value theorem have been met.
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Simplify. Assume all variables are positive.
\newline
(
2
r
1
3
)
8
(2r^{\frac{1}{3}})^8
(
2
r
3
1
)
8
\newline
\newline
Write your answer in the form
A
A
A
or
A
B
\frac{A}{B}
B
A
, where
A
A
A
and
B
B
B
are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.
\newline
______
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