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Math Problems
Algebra 1
Domain and range of square root functions: equations
lim
x
→
3
2
x
−
5
−
1
x
−
3
=
\lim _{x \rightarrow 3} \frac{\sqrt{2 x-5}-1}{x-3}=
lim
x
→
3
x
−
3
2
x
−
5
−
1
=
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lim
x
→
−
4
x
+
4
3
x
+
13
−
1
=
\lim _{x \rightarrow-4} \frac{x+4}{\sqrt{3 x+13}-1}=
lim
x
→
−
4
3
x
+
13
−
1
x
+
4
=
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Which of the following functions are continuous for all real numbers?
\newline
f
(
x
)
=
e
x
f(x)=e^{x}
f
(
x
)
=
e
x
\newline
g
(
x
)
=
x
g(x)=\sqrt{x}
g
(
x
)
=
x
\newline
Choose
1
1
1
answer:
\newline
(A)
f
f
f
only
\newline
(B)
g
g
g
only
\newline
(C) Both
f
f
f
and
g
g
g
\newline
(D) Neither
f
f
f
nor
g
g
g
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Which of the following functions are continuous for all real numbers?
\newline
h
(
x
)
=
sin
(
x
)
h(x)=\sin (x)
h
(
x
)
=
sin
(
x
)
\newline
f
(
x
)
=
cos
(
x
)
f(x)=\cos (x)
f
(
x
)
=
cos
(
x
)
\newline
Choose
1
1
1
answer:
\newline
(A)
h
h
h
only
\newline
(B)
f
f
f
only
\newline
(C) Both
h
h
h
and
f
f
f
\newline
(D) Neither
h
h
h
nor
f
f
f
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Find
lim
x
→
6
h
(
x
)
\lim _{x \rightarrow 6} h(x)
lim
x
→
6
h
(
x
)
for
\newline
h
(
x
)
=
5
x
+
6
.
h(x)=\sqrt{5 x+6} \text {. }
h
(
x
)
=
5
x
+
6
.
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Find
lim
x
→
−
3
g
(
x
)
\lim _{x \rightarrow-3} g(x)
lim
x
→
−
3
g
(
x
)
for
\newline
g
(
x
)
=
7
x
+
22
.
g(x)=\sqrt{7 x+22} \text {. }
g
(
x
)
=
7
x
+
22
.
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lim
x
→
π
4
csc
(
x
)
=
?
\lim _{x \rightarrow \frac{\pi}{4}} \csc (x)=?
x
→
4
π
lim
csc
(
x
)
=
?
\newline
Choose
1
1
1
answer:
\newline
(A)
2
2
\frac{\sqrt{2}}{2}
2
2
\newline
(B)
3
2
\frac{\sqrt{3}}{2}
2
3
\newline
(C)
2
\sqrt{2}
2
\newline
(D) The limit doesn't exist.
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lim
x
→
π
6
sin
(
x
)
=
?
\lim _{x \rightarrow \frac{\pi}{6}} \sin (x)=?
x
→
6
π
lim
sin
(
x
)
=
?
\newline
Choose
1
1
1
answer:
\newline
(A)
1
2
\frac{1}{2}
2
1
\newline
(B)
2
2
\frac{\sqrt{2}}{2}
2
2
\newline
(C)
2
\sqrt{2}
2
\newline
(D) The limit doesn't exist.
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Let
h
(
x
)
=
x
−
2
x
+
7
−
3
h(x)=\frac{x-2}{\sqrt{x+7}-3}
h
(
x
)
=
x
+
7
−
3
x
−
2
when
x
≠
2
x \neq 2
x
=
2
.
\newline
h
h
h
is continuous for all
x
>
−
7
x>-7
x
>
−
7
.
\newline
Find
h
(
2
)
h(2)
h
(
2
)
.
\newline
Choose
1
1
1
answer:
\newline
(A)
6
6
6
\newline
(B)
2
2
2
\newline
(C)
4
4
4
\newline
(D)
−
3
-3
−
3
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Let
g
(
x
)
=
x
−
5
x
−
4
−
1
g(x)=\frac{x-5}{\sqrt{x-4}-1}
g
(
x
)
=
x
−
4
−
1
x
−
5
when
x
≠
5
x \neq 5
x
=
5
.
\newline
g
g
g
is continuous for all
x
>
4
x>4
x
>
4
.
\newline
Find
g
(
5
)
g(5)
g
(
5
)
.
\newline
Choose
1
1
1
answer:
\newline
(A)
2
2
2
\newline
(B)
8
8
8
\newline
(C)
10
10
10
\newline
(D)
5
5
5
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Let
f
(
x
)
=
x
−
1
−
2
x
−
5
f(x)=\frac{\sqrt{x-1}-2}{x-5}
f
(
x
)
=
x
−
5
x
−
1
−
2
when
x
≠
5
x \neq 5
x
=
5
.
\newline
f
f
f
is continuous for all
x
>
1
x>1
x
>
1
.
\newline
Find
f
(
5
)
f(5)
f
(
5
)
.
\newline
Choose
1
1
1
answer:
\newline
(A)
1
1
1
\newline
B)
1
2
\frac{1}{2}
2
1
\newline
(C)
1
10
\frac{1}{10}
10
1
\newline
(D)
1
4
\frac{1}{4}
4
1
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Find
lim
x
→
1
f
(
x
)
\lim _{x \rightarrow 1} f(x)
lim
x
→
1
f
(
x
)
for
\newline
f
(
x
)
=
51
−
2
x
.
f(x)=\sqrt{51-2 x} \text {. }
f
(
x
)
=
51
−
2
x
.
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f
(
x
)
=
{
7
+
x
for
−
7
≤
a
x
2
−
5
for
x
>
−
3
f(x)=\left\{\begin{array}{ll} \sqrt{7+x} & \text { for }-7 \leq a \\ x^{2}-5 & \text { for } x>-3 \end{array}\right.
f
(
x
)
=
{
7
+
x
x
2
−
5
for
−
7
≤
a
for
x
>
−
3
\newline
Find
lim
x
→
−
3
f
(
x
)
\lim _{x \rightarrow-3} f(x)
lim
x
→
−
3
f
(
x
)
.
\newline
Choose
1
1
1
answer:
\newline
(A)
−
3
-3
−
3
\newline
(B)
2
2
2
\newline
(C)
4
4
4
\newline
(D) The limit doesn't exist.
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The component form of vector
v
⃗
\vec{v}
v
is
v
⃗
=
(
4
,
3
)
\vec{v}=(4,3)
v
=
(
4
,
3
)
.
\newline
Find
4
v
⃗
4 \vec{v}
4
v
.
\newline
4
v
⃗
=
(
□
,
□
)
4 \vec{v}=(\square, \square)
4
v
=
(
□
,
□
)
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f
(
x
)
=
x
2
−
6
x
f(x)=x^{2}-6 x
f
(
x
)
=
x
2
−
6
x
\newline
g
(
x
)
=
x
g(x)=\sqrt{x}
g
(
x
)
=
x
\newline
Evaluate.
\newline
f
(
g
(
25
)
)
=
f(g(25))=
f
(
g
(
25
))
=
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f
(
n
)
=
10
n
f(n)=\sqrt{10 n}
f
(
n
)
=
10
n
\newline
g
(
n
)
=
n
2
−
n
g(n)=n^{2}-n
g
(
n
)
=
n
2
−
n
\newline
Evaluate.
\newline
(
f
∘
g
)
(
10
)
=
(f \circ g)(10)=
(
f
∘
g
)
(
10
)
=
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f
(
c
)
=
3
c
−
8
f(c)=3 c-8
f
(
c
)
=
3
c
−
8
\newline
g
(
c
)
=
14
−
c
g(c)=\sqrt{14-c}
g
(
c
)
=
14
−
c
\newline
Evaluate.
\newline
(
f
∘
g
)
(
−
11
)
=
(f \circ g)(-11)=
(
f
∘
g
)
(
−
11
)
=
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(
y
−
k
)
y
=
1
3
(y-k) y=\frac{1}{3}
(
y
−
k
)
y
=
3
1
\newline
In the given equation,
k
k
k
is a constant. One of the solutions to the equation is:
\newline
3
+
9
+
4
(
1
3
)
2
\frac{3+\sqrt{9+4\left(\frac{1}{3}\right)}}{2}
2
3
+
9
+
4
(
3
1
)
\newline
What is the value of
k
k
k
?
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Solve the following equation for
w
w
w
.
\newline
5
−
w
4
=
w
+
8
w
=
\begin{array}{l} \sqrt{5-\frac{w}{4}}=\sqrt{w+8} \\ w= \end{array}
5
−
4
w
=
w
+
8
w
=
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Solve the following equation for
y
y
y
.
\newline
2
y
+
5
=
15
−
2
y
y
=
\begin{array}{l} 2 y+5=\sqrt{15-2 y} \\ y= \\ \end{array}
2
y
+
5
=
15
−
2
y
y
=
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Solve the following equation for
z
z
z
.
\newline
4
z
+
9
=
z
+
1
z
=
□
\begin{array}{l} \quad \sqrt{4 z+9}=z+1 \\ z=\square \end{array}
4
z
+
9
=
z
+
1
z
=
□
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Solve the following equation for
y
y
y
.
\newline
2
y
−
3
=
3
y
2
−
10
x
y
=
□
\begin{array}{l} 2 y-3=\sqrt{3 y^{2}-10 x} \\ y=\square \end{array}
2
y
−
3
=
3
y
2
−
10
x
y
=
□
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Solve the following equation for
x
x
x
.
\newline
5
x
−
4
=
x
−
2
x
=
\begin{array}{l} \sqrt{5 x-4}=x-2 \\ x= \end{array}
5
x
−
4
=
x
−
2
x
=
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What is the midline equation of the function
\newline
h
(
x
)
=
−
4
sin
(
x
−
π
4
)
?
h(x)=-4 \sin \left(x-\frac{\pi}{4}\right) ?
h
(
x
)
=
−
4
sin
(
x
−
4
π
)
?
\newline
y
=
y=
y
=
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Solve the following equation for
y
y
y
.
\newline
y
2
+
4
=
4
y
y
=
\begin{array}{l} \sqrt{y^{2}+4}=\sqrt{4 y} \\ y= \end{array}
y
2
+
4
=
4
y
y
=
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If
5
a
=
5
5^{a}=\sqrt{5}
5
a
=
5
, what is the value of
a
a
a
?
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h
(
x
)
=
4
x
h(x)=\sqrt{4 x}
h
(
x
)
=
4
x
\newline
The function
h
h
h
is defined. What is the value of
h
(
9
)
h(9)
h
(
9
)
?
\newline
Choose
1
1
1
answer:
\newline
(A)
3
3
3
\newline
(B)
6
6
6
\newline
(C)
12
12
12
\newline
(D)
36
36
36
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f
(
x
)
=
x
−
1
f(x)=\sqrt{x-1}
f
(
x
)
=
x
−
1
\newline
The function
f
f
f
is defined. What is the value of
f
(
10
)
f(10)
f
(
10
)
?
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The altitude at which we boil an egg affects how long it takes for the egg to achieve perfect hardness.
\newline
It takes
198
198
198
seconds to boil a perfect egg at the lowest place possible, the edge of the Dead Sea, which has an altitude of
−
418
-418
−
418
meters.
\newline
The highest place possible is the summit of Mount Everest which has an altitude of
8848
8848
8848
meters. It takes
209
209
209
seconds to boil a perfect egg there.
\newline
T
(
a
)
T(a)
T
(
a
)
models the time (in seconds) it takes to boil a perfect egg at an altitude of
a
a
a
meters.
\newline
Which number type is more appropriate for the domain of
T
T
T
?
\newline
Choose
1
1
1
answer:
\newline
(A) Integers
\newline
(B) Real numbers
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Alice studies the relationship between climate and heart disease around the world.
\newline
H
(
t
)
H(t)
H
(
t
)
models the probability for the occurrence of heart disease (in percents relative to the global average) at an area where the temperature is
t
t
t
degrees Celsius.
\newline
According to Alice's model, when the temperature is
−
5
∘
C
-5^{\circ}C
−
5
∘
C
(which is the lowest temperature included in the model), the probability is
10
%
10\%
10%
above average. Then the probability decreases until the temperature reaches
3
0
∘
C
30^{\circ}C
3
0
∘
C
(which is the highest temperature included in the model), where the probability is
20
%
20\%
20%
below average.
\newline
Which number type is more appropriate for the domain of
H
H
H
?
\newline
Choose
1
1
1
answer:
\newline
(A) Integers
\newline
(B) Real numbers
Get tutor help
Find the derivative of
p
(
x
)
p(x)
p
(
x
)
.
\newline
p
(
x
)
=
ln
(
x
)
p(x) = \ln(\sqrt{x})
p
(
x
)
=
ln
(
x
)
\newline
p
′
(
x
)
=
p' (x) =
p
′
(
x
)
=
______
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What is the domain of this function?
\newline
y
=
x
y = \sqrt{x}
y
=
x
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What is the domain of this function?
\newline
y
=
x
y = \sqrt{x}
y
=
x
\newline
Choices:
\newline
{
x
∣
x
≥
0
}
\{x|x \geq 0\}
{
x
∣
x
≥
0
}
\newline
{
x
∣
x
≤
0
}
\{x|x \leq 0\}
{
x
∣
x
≤
0
}
\newline
{
x
∣
x
<
0
}
\{x | x < 0\}
{
x
∣
x
<
0
}
\newline
{
x
∣
x
>
0
}
\{x|x>0\}
{
x
∣
x
>
0
}
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