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Math Problems
Algebra 1
Domain and range of square root functions: equations
Is the function
h
(
x
)
=
4
5
+
8
9
x
h(x)=\frac{4}{5}+\frac{8}{9} \sqrt{x}
h
(
x
)
=
5
4
+
9
8
x
linear or nonlinear?
\newline
linear
\newline
nonlinear
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Select the expressions that are equivalent to
4
v
+
6
v
4v + 6v
4
v
+
6
v
.
\newline
Multi-select Choices:
\newline
(A)
10
v
10v
10
v
\newline
(B)
v
+
10
v + 10
v
+
10
\newline
(C)
7
v
+
3
v
7v + 3v
7
v
+
3
v
\newline
(D)
9
v
9v
9
v
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Suppose that the functions
f
f
f
and
g
g
g
are defined as follows,
{
f
(
x
)
=
x
+
2
g
(
x
)
=
3
x
+
4
\begin{cases} f(x) = x + 2 \ g(x) = \sqrt{3x + 4} \end{cases}
{
f
(
x
)
=
x
+
2
g
(
x
)
=
3
x
+
4
Find
f
g
\frac{f}{g}
g
f
and
f
+
g
f + g
f
+
g
. Then, give their domains using interval notation.
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Graphs and Functions composition of two functions Advanced For the real-valued functions
f
(
x
)
=
4
x
+
3
f(x)=4x+3
f
(
x
)
=
4
x
+
3
and
g
(
x
)
=
x
−
3
g(x)=\sqrt{x-3}
g
(
x
)
=
x
−
3
, find the composition
f
∘
g
f\circ g
f
∘
g
and specify its domain using interval notation.
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Suppose that the functions
f
f
f
and
g
g
g
are defined as follows.
\newline
f
(
x
)
=
2
x
g
(
x
)
=
x
−
3
f(x)=\frac{2}{x}\quad g(x)=x-3
f
(
x
)
=
x
2
g
(
x
)
=
x
−
3
\newline
Find
f
g
\frac{f}{g}
g
f
. Then, give its domain using an interval or union of intervals Simplify your answers.
\newline
(
f
g
)
(
x
)
=
0
\left(\frac{f}{g}\right)(x)=\boxed{\phantom{0}}
(
g
f
)
(
x
)
=
0
\newline
Domain of
f
g
\frac{f}{g}
g
f
:
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Select the expressions that are equivalent to
5
(
c
+
1
)
5(c + 1)
5
(
c
+
1
)
.
\newline
Multi-select Choices:
\newline
(A)
1
(
c
+
5
)
1(c + 5)
1
(
c
+
5
)
\newline
(B)
5
(
1
+
c
)
5(1 + c)
5
(
1
+
c
)
\newline
(C)
6
c
6c
6
c
\newline
(D)
5
c
5c
5
c
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Solve
h
(
x
)
=
e
1
2
h(x)=e^{\frac{1}{2}}
h
(
x
)
=
e
2
1
for values of
x
x
x
in the domain of
h
h
h
.
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Select the expressions that are equivalent to
4
(
5
d
)
4(5d)
4
(
5
d
)
.
\newline
Multi-select Choices:
\newline
(A)
4
(
d
+
4
d
)
4(d + 4d)
4
(
d
+
4
d
)
\newline
(B)
4
(
5
+
d
)
4(5 + d)
4
(
5
+
d
)
\newline
(C)
d
+
9
d + 9
d
+
9
\newline
(D)
5
d
+
4
5d + 4
5
d
+
4
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Select the expressions that are equivalent to
2
(
3
j
)
2(3j)
2
(
3
j
)
.
\newline
Multi-select Choices:
\newline
(A)
2
+
3
j
2 + 3j
2
+
3
j
\newline
(B)
(
2
j
)
3
(2j)3
(
2
j
)
3
\newline
(C)
2
(
2
j
+
j
)
2(2j + j)
2
(
2
j
+
j
)
\newline
(D)
6
j
6j
6
j
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What could be the value of
x
x
x
in the following equation? Select all that apply.
\newline
x
2
=
1
9
x^2 = \frac{1}{9}
x
2
=
9
1
\newline
Multi-select Choices:
\newline
(A)
−
1
9
-\sqrt{\frac{1}{9}}
−
9
1
\newline
(B)
−
1
3
-\frac{1}{3}
−
3
1
\newline
(C)
1
3
\frac{1}{3}
3
1
\newline
(D)
1
9
\sqrt{\frac{1}{9}}
9
1
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The function
f
f
f
is defined as
f
(
x
)
=
4
−
x
f(x)=\sqrt{4-x}
f
(
x
)
=
4
−
x
.
\newline
What is the
x
x
x
-coordinate of the point on the function's graph that is closest to the origin?
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Let
g
(
x
)
=
x
sin
(
x
)
g(x)=\sqrt{x} \sin (x)
g
(
x
)
=
x
sin
(
x
)
.
\newline
g
′
(
x
)
=
g^{\prime}(x)=
g
′
(
x
)
=
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Find the domain of
\newline
y
=
x
+
3
4
−
x
2
−
9
y=\frac{x+3}{4-\sqrt{x^{2}-9}}
y
=
4
−
x
2
−
9
x
+
3
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solve for
x
x
x
cos
x
+
sin
x
=
2
\cos x +\sin x = \sqrt{2}
cos
x
+
sin
x
=
2
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Find the Domain
\newline
g
(
x
)
=
−
1
23
x
2
+
7
g(x)=\frac{-1}{23} x^{2}+7
g
(
x
)
=
23
−
1
x
2
+
7
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Find the range of
{
tan
x
+
sin
x
}
\{\tan x + \sin x\}
{
tan
x
+
sin
x
}
where
{
.
}
\{.\}
{
.
}
represents the fractuonal part of function
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What is a rule for
h
′
(
x
)
h'(x)
h
′
(
x
)
if
h
(
x
)
=
tan
−
1
x
h(x)=\tan^{-1}\sqrt{x}
h
(
x
)
=
tan
−
1
x
?
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Make
y
y
y
the subject of the formula:
\newline
x
=
y
p
+
p
2
x=\sqrt{y p+p^{2}}
x
=
y
p
+
p
2
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Given that
l
=
2
r
+
1
2
π
r
l=2 r+\frac{1}{2} \pi r
l
=
2
r
+
2
1
π
r
, make
r
r
r
the subject of the relation.
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g
(
x
)
=
(
5
−
2
x
)
(
14
+
2
x
)
g(x)=(5-2 x)(14+2 x)
g
(
x
)
=
(
5
−
2
x
)
(
14
+
2
x
)
\newline
The function
g
g
g
is defined by the given equation. For what value of
x
x
x
does
g
(
x
)
g(x)
g
(
x
)
reach its maximum?
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Eduardo is selling candy bars for a school fundraiser. He is given a box containing
80
80
80
candy bars to sell. Each bar sells for
$
3
\$3
$3
. The function
F
(
b
)
F(b)
F
(
b
)
represents the amount of money Eduardo raises from selling
b
b
b
candy bars.
\newline
What is the domain of
F
(
b
)
F(b)
F
(
b
)
?
\newline
Choices:
\newline
(A) all whole numbers from
3
3
3
to
80
80
80
\newline
(B) all whole numbers from
0
0
0
to
80
80
80
\newline
(C) all whole numbers
\newline
(D) all whole numbers from
0
0
0
to
3
3
3
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Eduardo is selling candy bars for a school fundraiser. He is given a box containing
80
80
80
candy bars to sell. Each bar sells for
$
3
\$3
$3
. The function
F
(
b
)
F(b)
F
(
b
)
represents the amount of money Eduardo raises from selling
b
b
b
candy bars.
\newline
What is the domain of
F
(
b
)
F(b)
F
(
b
)
?
\newline
Choices:
\newline
(A)all whole numbers from
3
3
3
to
80
80
80
\newline
(B)all whole numbers from
0
0
0
to
80
80
80
\newline
(C)all whole numbers from
0
0
0
to
3
3
3
\newline
(D)all whole numbers
\newline
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y
=
6
x
−
12
cos
x
,
x
∈
[
−
π
2
;
π
]
y=6 x-12 \cos x, x \in\left[-\frac{\pi}{2} ; \pi\right]
y
=
6
x
−
12
cos
x
,
x
∈
[
−
2
π
;
π
]
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Use the following function rule to find
f
(
88
)
f(88)
f
(
88
)
.
\newline
f
(
x
)
=
10
56
+
x
f(x)=10 \sqrt{56+x}
f
(
x
)
=
10
56
+
x
\newline
f
(
88
)
=
□
f(88)=\square
f
(
88
)
=
□
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Which of the following is a rational number?
\newline
Choices:
\newline
(A)
0
0
0
\newline
(B)
π
\pi
π
\newline
(C)
5
\sqrt{5}
5
\newline
(D)
8
\sqrt{8}
8
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Which of the following is a rational number?
\newline
Choices:
\newline
(A)
10
\sqrt{10}
10
\newline
(B)
8
\sqrt{8}
8
\newline
(C)
0
0
0
\newline
(D)
π
\pi
π
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Which of the following are whole numbers?
\newline
Multi-select Choices:
\newline
(A)
0
0
0
\newline
(B)
3
3
3
\newline
(C)
7
\sqrt{7}
7
\newline
(D)
9
9
9
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The derivative of the function
f
f
f
is defined by
f
′
(
x
)
=
x
2
−
5
−
4
sin
(
2
x
)
f^{\prime}(x)=x^{2}-5-4 \sin (2 x)
f
′
(
x
)
=
x
2
−
5
−
4
sin
(
2
x
)
. What is the
x
x
x
-coordinate of the absolute maximum value of the function
f
f
f
on the closed interval
[
−
4
,
3
]
[-4,3]
[
−
4
,
3
]
? You may use a calculator and round your answer to the nearest thousandth.
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Evaluate
lim
x
→
1
+
(
x
−
1
x
2
−
1
)
\lim_{x \to 1^{+}}\left(\frac{\sqrt{x-1}}{x^{2}-1}\right)
x
→
1
+
lim
(
x
2
−
1
x
−
1
)
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Differentiate
f
(
x
)
=
1
−
cos
(
4
x
)
1
+
cos
(
4
x
)
f(x)=\sqrt{\frac{1-\cos(4x)}{1+\cos(4x)}}
f
(
x
)
=
1
+
c
o
s
(
4
x
)
1
−
c
o
s
(
4
x
)
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Determine the
x
x
x
-intercepts of the following equation.
\newline
(
−
x
+
1
)
(
x
+
3
)
=
y
(-x+1)(x+3)=y
(
−
x
+
1
)
(
x
+
3
)
=
y
\newline
(
0
,
1
)
(0,1)
(
0
,
1
)
and
(
0
,
−
3
)
(0,-3)
(
0
,
−
3
)
\newline
(
0
,
−
3
)
(0,-3)
(
0
,
−
3
)
\newline
(
0
,
3
)
(0,3)
(
0
,
3
)
\newline
(
3
,
0
)
(3,0)
(
3
,
0
)
\newline
(
1
,
0
)
(1,0)
(
1
,
0
)
and
(
3
,
0
)
(3,0)
(
3
,
0
)
\newline
(
1
,
0
)
(1,0)
(
1
,
0
)
and
(
−
3
,
0
)
(-3,0)
(
−
3
,
0
)
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2
2
2
)
8
log
5
a
+
2
log
5
b
8 \log _{5} a+2 \log _{5} b
8
lo
g
5
a
+
2
lo
g
5
b
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Find the
y
y
y
-coordinate of the
y
y
y
-intercept of the polynomial function defined below.
\newline
f
(
x
)
=
−
(
x
−
4
)
(
4
x
−
2
)
(
x
−
2
)
3
f(x)=-(x-4)(4 x-2)(x-2)^{3}
f
(
x
)
=
−
(
x
−
4
)
(
4
x
−
2
)
(
x
−
2
)
3
\newline
Answer:
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What is the midline equation of the function
g
(
x
)
=
3
sin
(
2
x
−
1
)
+
4
g(x)= 3\sin(2x-1)+4
g
(
x
)
=
3
sin
(
2
x
−
1
)
+
4
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Find the derivative of
\newline
y
=
1
+
x
y=\sqrt{1+\sqrt{x}}
y
=
1
+
x
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Given the function
\newline
y
=
sin
(
6
−
4
x
2
)
,
y=\sin(\sqrt{6-4x^{2}}),
y
=
sin
(
6
−
4
x
2
)
,
find
\newline
(
d
y
d
x
)
.
(\frac{dy}{dx}).
(
d
x
d
y
)
.
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Given the function
\newline
f
(
x
)
=
2
cos
(
4
−
4
x
)
f(x)=2\cos(4-4x)
f
(
x
)
=
2
cos
(
4
−
4
x
)
, find
\newline
f
′
(
x
)
f^{\prime}(x)
f
′
(
x
)
.
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Given the function
\newline
f
(
x
)
=
cos
(
4
x
)
,
f(x)=\cos(4\sqrt{x}),
f
(
x
)
=
cos
(
4
x
)
,
find
\newline
f
′
(
x
)
.
f^{\prime}(x).
f
′
(
x
)
.
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Given the function
y
=
27
x
3
sin
x
y=\sqrt[3]{27x}\sin x
y
=
3
27
x
sin
x
, find
d
y
d
x
\frac{dy}{dx}
d
x
d
y
in any form.
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y
=
(
sec
2
x
−
1
sec
2
x
+
1
)
,
y=\sqrt{\left(\frac{\sec 2x-1}{\sec 2x+1}\right)},
y
=
(
s
e
c
2
x
+
1
s
e
c
2
x
−
1
)
,
then find
d
y
d
x
\frac{dy}{dx}
d
x
d
y
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Assume that
x
x
x
and
y
y
y
are both differential
\newline
y
=
x
y=\sqrt{x}
y
=
x
\newline
(a) Find
d
y
d
t
\frac{dy}{dt}
d
t
d
y
, given
x
=
16
x=16
x
=
16
and
d
x
d
t
=
6
\frac{dx}{dt}=6
d
t
d
x
=
6
.
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y
=
4
x
−
3
x
y=\sqrt{\frac{4}{x}}-\sqrt{3x}
y
=
x
4
−
3
x
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The differentiable functions
x
x
x
and
y
y
y
are related by the following equation:
\newline
y
=
x
y=\sqrt{x}
y
=
x
\newline
Also,
d
x
d
t
=
12
\frac{d x}{d t}=12
d
t
d
x
=
12
.
\newline
Find
d
y
d
t
\frac{d y}{d t}
d
t
d
y
when
x
=
9
x=9
x
=
9
.
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Let
y
=
ln
(
x
)
y=\sqrt{\ln (x)}
y
=
ln
(
x
)
.
\newline
Find
d
y
d
x
\frac{d y}{d x}
d
x
d
y
.
\newline
Choose
1
1
1
answer:
\newline
(A)
1
2
x
ln
(
x
)
\frac{1}{2 x \sqrt{\ln (x)}}
2
x
l
n
(
x
)
1
\newline
(B)
1
x
\frac{1}{\sqrt{x}}
x
1
\newline
(C)
1
x
\sqrt{\frac{1}{x}}
x
1
\newline
(D)
(
1
2
x
)
x
\frac{\left(\frac{1}{2 \sqrt{x}}\right)}{\sqrt{x}}
x
(
2
x
1
)
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Let
y
=
ln
(
x
)
y=\sqrt{\ln (x)}
y
=
ln
(
x
)
.
\newline
Find
d
y
d
x
\frac{d y}{d x}
d
x
d
y
.
\newline
Choose
1
1
1
answer:
\newline
(A)
(
1
2
x
)
x
\frac{\left(\frac{1}{2 \sqrt{x}}\right)}{\sqrt{x}}
x
(
2
x
1
)
\newline
(B)
1
x
\frac{1}{\sqrt{x}}
x
1
\newline
(C)
1
2
x
ln
(
x
)
\frac{1}{2 x \sqrt{\ln (x)}}
2
x
l
n
(
x
)
1
\newline
(D)
1
x
\sqrt{\frac{1}{x}}
x
1
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Let
y
=
e
x
y=\sqrt{e^{x}}
y
=
e
x
.
\newline
Find
d
y
d
x
\frac{d y}{d x}
d
x
d
y
.
\newline
Choose
1
1
1
answer:
\newline
(A)
x
e
x
−
1
x \sqrt{e^{x-1}}
x
e
x
−
1
\newline
(B)
1
2
e
x
\frac{1}{2 \sqrt{e^{x}}}
2
e
x
1
\newline
(C)
e
x
\sqrt{e^{x}}
e
x
\newline
(D)
e
x
2
\frac{\sqrt{e^{x}}}{2}
2
e
x
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Let
g
(
x
)
=
5
x
−
1
g(x)=\sqrt{5 x-1}
g
(
x
)
=
5
x
−
1
and let
c
c
c
be the number that satisfies the Mean Value Theorem for
g
g
g
on the interval
[
1
,
10
]
[1,10]
[
1
,
10
]
.
\newline
What is
c
c
c
?
\newline
Choose
1
1
1
answer:
\newline
(A)
2
2
2
.
25
25
25
\newline
(B)
4
4
4
.
25
25
25
\newline
(C)
6
6
6
.
5
5
5
\newline
(D)
8
8
8
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Let
f
(
x
)
=
x
+
9
f(x)=\sqrt{x+9}
f
(
x
)
=
x
+
9
and let
c
c
c
be the number that satisfies the Mean Value Theorem for
f
f
f
on the interval
[
0
,
16
]
[0,16]
[
0
,
16
]
.
\newline
What is
c
c
c
?
\newline
Choose
1
1
1
answer:
\newline
(A)
16
16
16
\newline
(B)
43
43
43
\newline
(C)
55
55
55
\newline
(D)
7
7
7
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lim
x
→
−
1
x
+
1
x
+
5
−
2
=
\lim _{x \rightarrow-1} \frac{x+1}{\sqrt{x+5}-2}=
lim
x
→
−
1
x
+
5
−
2
x
+
1
=
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Find
lim
x
→
−
2
3
−
6
x
+
21
x
+
2
\lim _{x \rightarrow-2} \frac{3-\sqrt{6 x+21}}{x+2}
lim
x
→
−
2
x
+
2
3
−
6
x
+
21
.
\newline
Choose
1
1
1
answer:
\newline
(A)
−
1
-1
−
1
\newline
(B)
−
2
-2
−
2
\newline
(C)
−
3
-3
−
3
\newline
(D) The limit doesn't exist
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1
2
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