Which function is the inverse of if the domain of f(x) is x ≥ 0?

(a) The differential equation

[tex]y' + \dfrac14 y = 3 + 2 \cos(2x)[/tex]

is linear, so we can use the integrating factor method. We have I.F.

[tex]\mu = \displaystyle \exp\left(\int \frac{dx}4\right) = e^{x/4}[/tex]

so that multiplying both sides by [tex]\mu[/tex] gives

[tex]e^{x/4} y' + \dfrac14 e^{x/4} y = 3e^{x/4} + 2 e^{x/4} \cos(2x)[/tex]

[tex]\left(e^{x/4} y\right)' = 3e^{x/4} + 2 e^{x/4} \cos(2x)[/tex]

Integrate both sides. (Integrate by parts twice on the right side; I'll omit the details.)

[tex]e^{x/4} y = 12 e^{x/4} + \dfrac8{65} e^{x/4} (8\sin(2x) + \cos(2x)) + C[/tex]

Solve for [tex]y[/tex].

[tex]y = 12 + \dfrac8{65} (\sin(2x) + \cos(2x)) + Ce^{-x/4}[/tex]

Given that [tex]y(0)=0[/tex], we find

[tex]0 = 12 + \dfrac8{65} (\sin(0) + \cos(0)) + Ce^0 \implies C = -\dfrac{788}{65}[/tex]

and the particular solution to the initial value problem is

[tex]\boxed{y = 12 + \dfrac8{65} (\sin(2x) + \cos(2x)) - \dfrac{788}{65} e^{-x/4}}[/tex]

As [tex]x[/tex] gets large, the exponential term will converge to 0. We have

[tex]\sin(2x) + \cos(2x) = \sqrt2 \sin\left(2x + \dfrac\pi4\right)[/tex]

which means the trigonometric terms will oscillate between [tex]\pm\sqrt2[/tex]. So overall, the solution will oscillate between [tex]12\pm\sqrt2[/tex] for large [tex]x[/tex].

(b) We want the smallest [tex]x[/tex] such that [tex]y=12[/tex], i.e.

[tex]0 = \dfrac8{65} (\sin(2x) + \cos(2x)) - \dfrac{788}{65} e^{-x/4}[/tex]

[tex]\dfrac{788}{65} e^{-x/4} = \dfrac{8\sqrt2}{65} \sin\left(2x + \dfrac\pi4\right)[/tex]

[tex]\dfrac{197}{\sqrt2} e^{-x/4} = \sin\left(2x + \dfrac\pi4\right)[/tex]

Using a calculator, the smallest solution seems to be around [tex]\boxed{x\approx21.909}[/tex]

Answer:

(-2,-1,0,1,2)

(1,4,6,7)

Answer:

14-2a

Step-by-step explanation:

Since 15 and 21 are not included: we will have

=15-2a-1

=14-2a

20 gallons of gas was consumed by the car that has efficiency of 20 miles per gallon  and 30 gallons of gas was consumed by the car that has  efficiency of 30 miles per gallon .

The first car has a fuel efficiency of 20 miles per gallon of gas and the second has a fuel efficiency of 30 miles per gallon of gas.

Therefore,

let

x = number of gallons for car that consume 20 miles per gallon of gas

y = number of gallons for car that consume  30 miles per gallon of gas

Hence,

x + y = 50

20x + 30y = 1300

20x + 20y = 1000

20x + 30y = 1300

10y = 300

y = 300 / 10

y = 30

Therefore,

x = 50 - 30

x = 20

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The true statement about the graph of [tex]f(x) = \frac{x^2 + 2x + 1}{2x^2 - 10x+12}[/tex]  is (c) its vertical asymptote are x = 3 and x = 2

The function is given as:

[tex]f(x) = \frac{x^2 + 2x + 1}{2x^2 - 10x+12}[/tex]

Set the denominator to 0

2x^2 - 10x + 12 = 0

Divide through by 2

x^2 - 5x + 6 = 0

Expand

x^2 - 2x - 3x + 6 = 0

Factorize

(x - 2)(x - 3) = 0

Solve for x

x = 2 or x = 3

The above represents the vertical asymptote of the graph

Hence, the vertical asymptote are x = 3 and x = 2

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Answer:

4 slices for all the brothers

Step-by-step explanation:

[tex]\huge\text{Hey there!}[/tex]

[tex]\huge\textbf{Equation:}[/tex]

[tex]\mathbf{8-|2(-3.5)-5|}[/tex]

[tex]\huge\textbf{Solving:}[/tex]

[tex]\mathbf{8-|2(-3.5)-5|}[/tex]

[tex]\mathbf{= 8 - |-7 - 5|}[/tex]

[tex]\mathbf{= -8 - |-12|}[/tex]

[tex]\mathbf{= 8 - 12}[/tex]

[tex]\mathbf{= -4}[/tex]

[tex]\huge\textbf{Answer:}[/tex]

[tex]\huge\boxed{\mathsf{-4}}\huge\checkmark[/tex]

[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]

Answer:

-4

Step-by-step explanation:

resolve the absolute value

I 2(-3.5) -5 I = I -7 -5 I = I -12 I = 12 The absolute value of any number is always positive

Then:

8 - 12 = -4

Hope this helps

Answer:

just multiply 20.50 by 8 and how many days she work in a week

Answer:

9

Step-by-step explanation:

22.5÷2.5=9 this is how you solve it

The Pink Party Punch has a stronger lemon-lime flavor because it has a higher volume of 6 liters.

Volume can be defined as the total quantity of a substance that a container can hold at a given period of time. Also, the volume of a container is measured in:

Based on the information provided in the table above, we can infer and logically deduce that the Pink Party Punch has a stronger lemon-lime flavor because it has a higher volume of 6 liters.

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probability//

Step-by-step explanation:

Answer:

Step-by-step explanation:

Answer:

3800grams

Step-by-step explanation:

multiply by 1000

Answer:

Step-by-step explanation:

Domain is the set of all inputs for which the given function is defined.

graphically it represents the shadow of the graph taken on the x axis from above and below the x axis

Here is the shadow will be formed from -3 till +∞, hence we have our domain  as [-3 , ∞ ]

The relations from the following sets of ordered pairs which is a function is; {(8,1),(−4,1),(3,5),(0,4),(−1,2)}

It follows from the definition of a function that a function from an array of X-values to a set Y-values assigns only one Y-value to each X-value.

Hence, it follows from the definition above that the relation; {(8,1),(−4,1),(3,5),(0,4),(−1,2)} is that which is a function.

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Side a, angle B and angle C of triangle ABC are 26.6 units, 61.9 degrees and 8.1 degrees respectively.

Given that;

First we determine the length of side a

[tex]a = \sqrt{b^2-c^2-2bc*cos(A)}[/tex]

We substitute our values into the above equation[tex]a = \sqrt{25^2-4^2-(2*25*4*cos(110)}\\\\a = 26.6[/tex]

Side a has a length of 26.6 units.

[tex]B = cos^{-1}(\frac{a^2+c^2-b^2}{2ac}) \\\\B = cos^{-1}(\frac{26.63464^2+4^2-25^2}{2*26.63464*4}) \\\\B = 61.9[/tex]

Angle B is 61.9 degrees.

[tex]C = cos^{-1}(\frac{a^2+b^2-c^2}{2ab} )\\\\C = cos^{-1}(\frac{26.63464^2+25^2-4^2}{2*26.63464*25} )\\\\C =8.1[/tex]

Angle C is 8.1 degrees.

Therefore, side a, angle b and angle c of the triangle ABC are 26.6 units, 61.9 degrees and 8.1 degrees respectively.

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Answer: Choice A) It has a range of 17

Reason:

The smallest value is 8 and the largest is 14. The range is the distance between these min and max

range = max - min = 14 - 8 = 6

So the range is 6 instead of 17.

The range of a data set is the difference between maximum and minimum thus the range of the given data set 8, 10, 10, 12, 12, 12, 12, 14, 14,16 is 8 so option (A) does not describe the data set.

Mode is the highest frequency number while the median is the middle value of a data set after writing in either an increasing or decreasing manner.

For example;

Set { 1,2,2,3,4,5,6} here mode is 2 and median is 3.

As per the given data set.

8, 10, 10, 12, 12, 12, 12, 14, 14,16

The range = maximum - minimum

Range = 16 - 8 = 8

The median of the data set is 12.

The mode is the highest frequency thus 12 is the mode of the data set.

Hence "The range of a data set is the difference between maximum and minimum thus the range of the given data set 8, 10, 10, 12, 12, 12, 12, 14, 14,16 is 8".

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The answer depends on how many total teams there are. The team players always end up with multiple of 6: 6, 12, 18 24... etc.

Hope it helps!

Volume

Volume

Answer:

Step-by-step explanation:

As the length AB increases, the measure of C increases

As the length AB increases, the measure of C increases.

An angle is formed when two lines or rays meet at a single point. The common point is referred to as the vertex. The length of the angle formed when two rays or arms intersect at a common vertex is known as an angle measure in geometry.

Angles and sides in relation to one another:

The biggest side and angle of any triangle are in opposition to one another.

The smallest side and angle in every triangle are on opposing sides.

The middle side and middle angle are always in opposition to one another in triangles.

When we extend vertex A further to the left of vertex B,

we increase the side length of AB.

As the length AB increases, the measure of opposite angle increases.

Let C be the angle opposite to AB.

Therefore, as the length AB increases, the measure of C increases.

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Answer:

62 cm^2

Step-by-step explanation:

Surface area of a rectangular prism is the area of each of the 6 faces added up. And area is length x width. There are three unique sets of sides, since each has another identical area opposite from it.

2(5 x 2) + 2(2 x 3) + 2(5 x 3)

2(10) + 2(6) + 2(15)

20 + 12 + 30

62 cm^2

Answer:

Step-by-step explanation:

SA=2lw+2lh+2hw

2 * 2 * 3 + 2 * 2 * 5 + 2 * 3 * 5 =

62 cm²

The equation of parabola passing through coordinates A (-2,0), B (-1.5,-2.5), C (0,5), D (1, 0), E (1.5, 2.5), and F (2, 0) is y = -2.5(2x+4)², and the graph is attached.

Each element of X receives exactly one element of Y when a function from one set to the other is used. The sets X and Y are collectively referred to as the function's domain and co-domain, respectively.

Given:

The coordinates of the point of parabola A (-2,0), B (-1.5,-2.5), C (0,5), D (1, 0), E (1.5, 2.5), and F (2, 0).

Calculate the equation of parabola by the formula given below,

[tex]y=a(x-h)^2+k[/tex]

Here (h, k) are the coordinates,

Put A (-2,0)

y = a[x- (-2)] ² + 0

y = a(2x + 4)²

Put B (-1.5,-2.5)

y = a[x- (-1.5)]² + (-2.5)

y = a(x + 1.5)² - 2.5

y = 2ax + 3a -2.5

Solve the above equation

a = -2.5

Thus, the equation of the parabola is,

y = -2.5(2x+4)²

Therefore, the equation of parabola passing through coordinates A (-2,0), B (-1.5,-2.5), C (0,5), D (1, 0), E (1.5, 2.5), and F (2, 0) is y = -2.5(2x+4)², and the graph is attached.

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Step-by-step explanation:

a).

n! is

[tex]n(n - 1)(n - 2)(n - 3)(n - 4).........(1)[/tex]

(n-3)! is

[tex](n - 3)(n - 4)(n - 5)..........(1)[/tex]

If we divide the two, everything behind (n-3) will just cancel out so we would be left with only

[tex](n)(n - 1)(n - 2)[/tex]

So the answer to a is

[tex]n(n - 1)(n - 2)[/tex]

b

Permutations formula of P(6,4) is

6!/(6-4)!, or 6!/2!.

[tex] \frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{2 \times 1} = 360[/tex]

Then we divide by 4!

[tex] \frac{360}{4 \times 3 \times 2 \times 1} = 15[/tex]

The answer to b is 15

Answer:

Step-by-step explanation:

Let the sides be a, b and c.

The length of one side of a triangle is 2 feet less than three times the length of its second side

The length of the third side is 3/4 of the sum of the lengths of the first two sides:

The perimeter of the triangle is 17.5 feet:

Substitute a with b in the second equation:

Now substitute a and c with b in the third equation and solve for b:

Find the value of a:

Find the value of c:

The sides of the triangle are: