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Let’s check out your problem:
If
f
′
(
x
)
=
−
f
(
x
)
f^{\prime}(x)=-f(x)
f
′
(
x
)
=
−
f
(
x
)
and
f
(
1
)
=
2
f(1)=2
f
(
1
)
=
2
, then
f
(
3
)
=
m
e
n
f(3)=m e^{n}
f
(
3
)
=
m
e
n
for some integers
m
m
m
and
n
n
n
.
\newline
What are
m
m
m
and
n
n
n
?
\newline
m
=
□
n
=
□
\begin{array}{l} m= \square \\ n= \square \end{array}
m
=
□
n
=
□
View step-by-step help
Home
Math Problems
Algebra 2
Transformations of quadratic functions
Full solution
Q.
If
f
′
(
x
)
=
−
f
(
x
)
f^{\prime}(x)=-f(x)
f
′
(
x
)
=
−
f
(
x
)
and
f
(
1
)
=
2
f(1)=2
f
(
1
)
=
2
, then
f
(
3
)
=
m
e
n
f(3)=m e^{n}
f
(
3
)
=
m
e
n
for some integers
m
m
m
and
n
n
n
.
\newline
What are
m
m
m
and
n
n
n
?
\newline
m
=
□
n
=
□
\begin{array}{l} m= \square \\ n= \square \end{array}
m
=
□
n
=
□
Find
C
C
C
value:
Now, we use the initial condition
f
(
1
)
=
2
f(1) = 2
f
(
1
)
=
2
to find the value of
C
C
C
. Plugging
x
=
1
x = 1
x
=
1
into
f
(
x
)
=
C
e
−
x
f(x) = Ce^{-x}
f
(
x
)
=
C
e
−
x
, we get
f
(
1
)
=
C
e
−
1
=
2
f(1) = Ce^{-1} = 2
f
(
1
)
=
C
e
−
1
=
2
.
Solve for C:
Solve for C:
C
e
−
1
=
2
Ce^{-1} = 2
C
e
−
1
=
2
implies
C
=
2
e
C = 2e
C
=
2
e
.
Simplify function:
Now we have the function
f
(
x
)
=
2
e
⋅
e
−
x
f(x) = 2e \cdot e^{-x}
f
(
x
)
=
2
e
⋅
e
−
x
. Simplify this to get
f
(
x
)
=
2
e
1
−
x
f(x) = 2e^{1-x}
f
(
x
)
=
2
e
1
−
x
.
Substitute
x
=
3
x=3
x
=
3
:
To find
f
(
3
)
f(3)
f
(
3
)
, substitute
x
=
3
x = 3
x
=
3
into
f
(
x
)
=
2
e
1
−
x
f(x) = 2e^{1-x}
f
(
x
)
=
2
e
1
−
x
. So,
f
(
3
)
=
2
e
1
−
3
f(3) = 2e^{1-3}
f
(
3
)
=
2
e
1
−
3
.
Calculate
f
(
3
)
f(3)
f
(
3
)
:
Calculate
f
(
3
)
f(3)
f
(
3
)
:
f
(
3
)
=
2
e
−
2
f(3) = 2e^{-2}
f
(
3
)
=
2
e
−
2
. This is the same as
f
(
3
)
=
2
e
2
f(3) = \frac{2}{e^2}
f
(
3
)
=
e
2
2
.
Express in
m
e
n
me^{n}
m
e
n
:
Now, we need to express
2
e
2
\frac{2}{e^2}
e
2
2
in the form of
m
e
n
me^{n}
m
e
n
. Since
e
−
2
e^{-2}
e
−
2
is the same as
1
e
2
\frac{1}{e^2}
e
2
1
, we can write
f
(
3
)
=
2
⋅
e
−
2
f(3) = 2 \cdot e^{-2}
f
(
3
)
=
2
⋅
e
−
2
.
Final result:
Thus, we have
f
(
3
)
=
2
⋅
e
(
−
2
)
f(3) = 2 \cdot e^{(-2)}
f
(
3
)
=
2
⋅
e
(
−
2
)
, which means
m
=
2
m = 2
m
=
2
and
n
=
−
2
n = -2
n
=
−
2
.
More problems from Transformations of quadratic functions
Question
The function
h
h
h
is defined over the real numbers. This table gives a few values of
h
h
h
.
\newline
\begin{tabular}{lllll}
\newline
x
x
x
&
−
6
-6
−
6
.
1
1
1
&
−
6
-6
−
6
.
01
01
01
&
−
6
-6
−
6
.
001
001
001
&
−
5
-5
−
5
.
9
9
9
\\
\newline
\hline
h
(
x
)
h(x)
h
(
x
)
&
−
0
-0
−
0
.
25
25
25
&
−
0
-0
−
0
.
74
74
74
&
−
0
-0
−
0
.
98
98
98
&
−
1
-1
−
1
.
0
0
0
\newline
\end{tabular}
\newline
What is a reasonable estimate for
lim
x
→
−
6
h
(
x
)
\lim _{x \rightarrow-6} h(x)
lim
x
→
−
6
h
(
x
)
?
\newline
Choose
1
1
1
answer:
\newline
(A)
−
6
-6
−
6
\newline
(B)
−
2
-2
−
2
\newline
(C)
−
1
-1
−
1
\newline
(D) The limit doesn't exist
Get tutor help
Posted 4 months ago
Question
What is the amplitude of
\newline
g
(
x
)
=
−
2
sin
(
π
2
x
−
3
)
+
5
?
g(x)=-2 \sin \left(\frac{\pi}{2} x-3\right)+5 ?
g
(
x
)
=
−
2
sin
(
2
π
x
−
3
)
+
5
?
\newline
units
Get tutor help
Posted 3 months ago
Question
What is the amplitude of
y
=
5
sin
(
4
x
−
2
)
−
3
y=5 \sin (4 x-2)-3
y
=
5
sin
(
4
x
−
2
)
−
3
?
\newline
units
Get tutor help
Posted 3 months ago
Question
What is the amplitude of
y
=
−
3
cos
(
π
x
+
2
)
−
6
?
y=-3 \cos (\pi x+2)-6 ?
y
=
−
3
cos
(
π
x
+
2
)
−
6
?
\newline
units
Get tutor help
Posted 3 months ago
Question
What is the amplitude of
y
=
3
sin
(
2
x
−
1
)
+
4
y=3 \sin (2 x-1)+4
y
=
3
sin
(
2
x
−
1
)
+
4
?
\newline
units
Get tutor help
Posted 3 months ago
Question
What is the period of the function
h
(
x
)
=
−
3
cos
(
π
x
+
2
)
−
6
?
h(x)=-3 \cos (\pi x+2)-6 ?
h
(
x
)
=
−
3
cos
(
π
x
+
2
)
−
6
?
\newline
Give an exact value.
\newline
units
Get tutor help
Posted 3 months ago
Question
What is the period of the function
\newline
h
(
x
)
=
5
sin
(
4
x
−
2
)
−
3
?
h(x)=5 \sin (4 x-2)-3 \text { ? }
h
(
x
)
=
5
sin
(
4
x
−
2
)
−
3
?
\newline
Give an exact value.
\newline
units
Get tutor help
Posted 3 months ago
Question
What is the period of the function
g
(
x
)
=
2
cos
(
7
x
+
5
)
+
1
?
g(x)=2 \cos (7 x+5)+1 ?
g
(
x
)
=
2
cos
(
7
x
+
5
)
+
1
?
\newline
Give an exact value.
\newline
units
Get tutor help
Posted 3 months ago
Question
What is the period of the function
f
(
x
)
=
3
sin
(
2
x
−
1
)
+
4
f(x)=3 \sin (2 x-1)+4
f
(
x
)
=
3
sin
(
2
x
−
1
)
+
4
?
\newline
Give an exact value.
\newline
units
Get tutor help
Posted 3 months ago
Question
What is the period of the function
f
(
x
)
=
−
4
cos
(
5
x
−
9
)
−
7
f(x)=-4 \cos (5 x-9)-7
f
(
x
)
=
−
4
cos
(
5
x
−
9
)
−
7
?
\newline
Give an exact value.
\newline
units
Get tutor help
Posted 3 months ago