Consider a function f defined as follows: f(x)=(x-3)^2, x belongs to [1,2]. Is f invertible in the domain [1,2]? If yes, find the inverse of the function.

(Need a detailed explanation showing the function is invertible, then determining the inverse.)

a. The value of k from augmented matrix [tex]\left(\begin{array}{ccc}3&- 5&k\\- 9&15&3\end{array}\right)[/tex] is - 1.

b. The value of k from augmented matrix [tex]\left(\begin{array}{ccc}k&1&- 6\\5&- 1&6\end{array}\right)[/tex] is - 5.

The complete question is in the attachment. The augmented matrix is a consistent linear system if

a. The augmented matrix [tex]\left(\begin{array}{ccc}3&- 5&k\\- 9&15&3\end{array}\right)[/tex]

Making the second row become zero R₂ = R₂ + 3R₁

[tex]\left(\begin{array}{ccc}3&- 5&k\\- 9 \:+\: (3 \times 3)&15 \:+\: (3 \times - 5)&3 \:+\: (3 \times k)\end{array}\right)[/tex]

[tex]\left(\begin{array}{ccc}3&- 5&k\\- 9 \:+\: 9&15 \:+\: (- 15)&3 \:+\: 3k\end{array}\right)[/tex]

[tex]\left(\begin{array}{ccc}3&- 5&k\\0&0&3 \:+\: 3k\end{array}\right)[/tex]

From the second row

0x + 0y = 3 + 3k

For the system to be consistent, then

3 + 3k = 0

3k = - 3

k = - 3/3

k = - 1

b. The augmented matrix [tex]\left(\begin{array}{ccc}k&1&- 6\\5&- 1&6\end{array}\right)[/tex]

Making the second row become zero R₂ = R₂ + R₁

[tex]\left(\begin{array}{ccc}k&1&- 6\\5 \:+\: k&- 1 \:+\: 1&6 \:+\: (- 6)\end{array}\right)[/tex]

[tex]\left(\begin{array}{ccc}k&1&- 6\\k \:+\: 5&0&0\end{array}\right)[/tex]

From the second row

(k + 5)x + 0y = 0

For the system to be consistent, then

k + 5 = 0

k = - 5

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Answer: 1

Step-by-step explanation:

1 + 0 = 1

Answer:

10/10/10

Step-by-step explanation:

The dimensional analysis conversion of 100 inch to feet is 8.33 feet.

Dimensional analysis is the study of the relationships between different physical quantities in engineering and science by identifying their base quantities and units of measurement and tracking these dimensions as calculations or comparisons are performed. when there is insufficient information to set up precise equations, a method of analysis in which physical quantities are expressed in terms of their fundamental dimensions. Dimensional Analysis (also known as the Factor-Label Method or the Unit Factor Method) is a problem-solving strategy that takes advantage of the fact that any number or expression can be multiplied by one without changing its value.

Here,

100 inch to feet,

=100/12

=8.33 feet

The conversion of 100 inch to feet using dimensional analysis is 8.33 feet.

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Answer:

Step-by-step explanation:oc (6,5), (3,2), (2,4) (6, 3), (1, 0)

The migration matrix that has been found is

                [0.98033     0.00179

                 0.01967      0.99821 ]       by matrix method.

The 2012 populations of California and the US minus California are 38,041,430 and 275,872,610, respectively, according to the exercise statement. Let's represent these quantities by the variables c and u. What we want are two vectors m1 and m2 such that

cm 1+um 2 ​=     [ 2013 population of California

                       2013 population of US minus California ]

m1 = [ 0.98033   0.01967 ]      and    m2 = [ 0.00179   0.99821]

Now that we have the two vectors for equation ,we use the fact that a vector equation of the form

                             au + bv = w

is equivalent to a matrix equation of the form  

                              [ u  v ] [a b] = w

to get our migration matrix for this system.

[0.98033     0.00179

0.01967      0.99821 ]

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The complete question is :

In 2012 the population of California was 38,041,430, and the population living in the United States but outside California was 275,872,610. During the year, it is estimated that 748,252 persons moved from California to elsewhere in the United States, while 493,641 persons moved to California from elsewhere in the United States. Set up the migration matrix for this situation, using five decimal places for the migration rates into and out of California. Let your work show how you produced the migration matrix.

​

Answer:  Option (E)

4,  4√3

===========================================================

Reason:

The hypotenuse is 8. Half of that is 8/2 = 4, which is the value of "s".

This is the short leg of the right triangle. The short leg is opposite the 30 degree angle. Smallest side opposite the smallest angle.

Then use this 30-60-90 triangle template formula

[tex]\text{long leg} = (\text{short leg})*\sqrt{3}\\\\\text{long leg} = 4\sqrt{3}\\\\[/tex]

The line represents 1 foot, so 4 feet would require a line that is 4 times as long as the 1/4 inch line, which is 1 inch.

The formula for finding the length of a line to represent a certain number of feet is L = (1/4)*F, where L is the length of the line, and F is the number of feet. In this case, to find the length of a line to represent 4 feet, the calculation would be L = (1/4)*4 = 1. Therefore, the line would need to be 1 inch long to represent 4 feet. This is because 1/4 inch of the line represents 1 foot, so 4 feet would require a line that is 4 times as long as the 1/4 inch line, which is 1 inch.Therefore, to represent a length of 4 feet on the same drawing, a line 1 inch long needs to be drawn. The calculation is as follows: 1/4 inch line = 1 foot, therefore 4/4 inch line = 4 feet. Therefore, 1 inch line = 4 feet.

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Answer:  To find the mean test score, we need to calculate the sum of the product of each score and its corresponding frequency, and divide it by the total number of students.

To estimate the mean mathematics test score, we can use the midpoint of each data class as the value for each score and multiply it by the corresponding frequency.

For the first class, 550 to 599, the midpoint is (550+599)/2 = 574.5

For the second class, 600 to 649, the midpoint is (600+649)/2 = 624.5

For the third class, 650 to 699, the midpoint is (650+699)/2 = 674.5

For the fourth class, 700 to 749, the midpoint is (700+749)/2 = 724.5

For the fifth class, 750 to 799, the midpoint is (750+799)/2 = 774.5

The mean test score is:

(5574.5 + 11624.5 + 14674.5 + 12724.5 + 7*774.5) / 49 = 669.3

Therefore, the estimate of the mean mathematics test score for the students in the study is 669.3.

Step-by-step explanation:

The length in yards, of the carpet needed for this room is 128 yards.

length of the room = 32 ft

breadth of the room = 27 ft

so we find area now,

area = length X breadth

area = 32 X 27

area = 864ft^2

The width of the carpet in feet is = 2.25ft

so , to find the length of the carpet needed, you must apply the formula for calculate the area of a rectangle and solve for the length, as following,

area = length X width

864 = length X 2.25

length = 864/ 2.25

length = 384ft

length = 128yards

so the length, in yards, of the carpet needed for this room is 128 yards.

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Answer:

The volume of the prism is 6.186

Step-by-step explanation:

1. The volume of a cylinder can be calculated using the formula V = πr^2h where r is the radius of the cylinder base, h is the height and π is a mathematical constant.

2. The diameter of the cylinder base is given as 19.8 cm and the base area of the cylinder is given as 307.8 cm^2.

3. The volume of the cylinder is given as 9.234 cm^3.

4. If the cylinder and the prism share the same height, the volume of the prism can be calculated by using the base area of the prism.

5. The base area of the prism is given as 196 cm^2.

6. The volume of the prism can be calculated by multiplying the base area with the height of the prism.

7. The volume of the prism is 196cm^2 * h = 6.186 cm^3

8. So the volume of the prism is 6.186 cm^3

The radius of curvature is 1410 ft, the velocity is 5.48 ft/s, the normal acceleration is 0.0213 ft/s2, and the magnitude of the acceleration is 0.500 ft/s2 after the particle travels 30 ft along the path to (2.806, 142).

The radius of curvature, r, can be calculated using the formula r = v2/a, where v is the velocity and a is the normal acceleration. The velocity is calculated by integrating the acceleration with respect to time. The normal acceleration is calculated by taking the derivative of the velocity with respect to time. The magnitude of the acceleration is the same as the acceleration given in the problem.

1. Calculate the velocity, v, by integrating the acceleration (0.5 ft/s2) with respect to time:

v = 0.5 * t = 0.5 * 30 s = 15 ft/s

2. Calculate the normal acceleration, a, by taking the derivative of the velocity with respect to time:

a = dv/dt = 0.5 ft/s2

3. Calculate the radius of curvature, r, using the formula r = v2/a:

r = 15 ft/s2/0.5 ft/s2 = 30 ft = 1410 ft

4. The magnitude of the acceleration is the same as the acceleration given in the problem, 0.5 ft/s2.

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Answer: 38 toys

If each of the smaller boxes has 10 toys, and Martin's mom added 28 more toys into only one of the boxes, that box would have a total of 38 toys.

10 + 28 = 38 toys

Answer:

The equation Jared can use to find the rate at which he drove is:

rate (r) = distance (d) / time (t)

So in this case, Jared can plug in the values he knows into the equation:

r = 15 miles / 0.5 hours

Solving for r, he can find the rate at which he drove:

r = 30 miles per hour.

So Jared drove at a rate of 30 miles per hour.

Step-by-step explanation:

PLEASE GIVE BRAINLIEST

Answer: 7/12 can't be simplified anymore. Have a nice day ;)

Correct option is C, The information about the acids that are created from the sugar in sweets and bacteria from teeth comes from Jaws.

gum, lollipops, and hard sweets without additional sugar.

Gum with a high Xylitol content, like Icebreaker Ice Cubes, makes you salivate more, which helps kill dangerous bacteria. Hard sweets and sugar-free lollipops can also stimulate the salivary glands.

Black chocolate offers more antioxidants than milk chocolate and less sugar.

As long as it is eaten in moderation and leaves your teeth bushed, the occasional candy piece is acceptable.

The greatest strategy to stop sugar's harmful effects on teeth is to always pay close attention. We advise you to limit your sugar intake, regularly brush away the bacteria-filled plaque on your teeth, and consume nutritional meals to maintain strong teeth.

Combining routine dental exams with fluoride treatments from our dental team will give you and your family the best chance of preventing tooth decay.

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The equation of the graph that passes through the points (4, 24) and (6, 30) is y = 3x + 12

The slope-intercept form is expresses as;

y = mx + b

Where m is slope and b is y-intercept.

The point-slope form is expressed as;

y - y₁ = m( x - x₁ )

Given that the the graph passes through the points (4, 24) and (6, 30).

First, determine the slope.

Slope m = ( y₂ - y₁ )/( x₂ - x₁ )

Slope m = ( 30 - 24 )/( 6 - 4 )

Slope m = ( 6 )/( 2 )

Slope m = 3

Next, plug the slope m = 3 and point 1 ( 4,24) into the point-slope form and simplify.

y - y₁ = m( x - x₁ )

y - 24 = 3( x - 4 )

y - 24 = 3x - 12

y = 3x - 12 + 24

y = 3x + 12

Therefore, the equation of line is y = 3x + 12,

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The accurate of the function is calculated using the fomula O(n).

Function in math is known as an equation that has only one answer for y for every x and it assigns exactly one output to each input of a specified type.

Here we have given that maximum distance is less than 10%.

As we all know that the basic approach is to check only consecutive pairs of x and y.

Here we have given that for every element x or y, then we have to check the index of the previous occurrence of x or y and if the previous occurring element is not similar to current element update the minimum distance.

Here we have a question arises what if an x is preceded by another x and that is preceded by a y for that situation we have to know how to get the minimum distance between pairs.

Hereby analyzing closely it can be seen that every x followed by a y or vice versa can only be the closest pair so ignore all other pairs.

Complete question:

Write the function to calculate the accuracy? show that maximum distance is less than 10%.

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The set of all differentiable real-valued functions defined on the real line. Proved that V is a vector space with the operations of addition and scalar multiplication defined as:

(f + g)(s) = f(s) + g(s)

(c f)(s) = c[f(s)].

Let V denote the set of all differentiable real-valued functions defined on the real line. Prove that V is a vector space with the operations of addition and scalar multiplication defined as:

(f + g)(s) = f(s) + g(s)

(c f)(s) = c[f(s)].

To prove vector space, we need to hold eight properties of vector space.

1) Closed

 (f + g)(s) = f(s) + g(s) (given)

so, closed holds with respect to given defined operation.

2) Associative law

let defined three function, f,g,h in V,

(f+g)(s)+h(s)=f(s)+g(s)+h(s)

                   = f(s)+(g+h)(s)

hence associative law hold.

3) Existence of identity,

(f+o)(s)=f(s)

f(s)+o(s)=f(s)

Hence, O(s) is identity for this function,

4) Existences of Inverse,

(-f)(s) is inverse for f(s)

5) Commutative law,

(f+g)(s)=(g+h)(s)

6)( cf)(s)=c[f(s)]g

7) c(f+g)(s)=cf(s)+g(s)

8) c(fg)(s)=(cf(s))g(s)

Hence, the given function is defined Operations.

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Answer: When a polynomial is divided by x-1, the quotient is of the form of ax^2 + bx + c and the remainder is of the form of r. The quotient and the remainder are related by the formula:

f(x) = (x-1)(ax^2 + bx + c) + r

Given that the quotient is 1x^2+x-8 and the remainder is -2, we can substitute these into the above equation:

f(x) = (x-1)(1x^2+x-8) + (-2)

f(x) = x^3 - 9x^2 + (1-8)x +(-2)

f(x) = x^3 - 9x^2 - 7x - 2

Therefore the function f(x) is x^3 - 9x^2 - 7x - 2 in standard form.

Step-by-step explanation:

The simplification of the trigonometric function tan x csc x is;  sec x

The six trigonometric functions are defined as;

sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec).

Trigonometric Ratios are defined as the values of all the trigonometric functions based on the value of the ratio of sides in a right-angled triangle.

Now, we know that;

cot x = cos x/sin x

sec x = 1/cos x = (cos x)⁻¹

csc x = 1/sin x = (sin x)⁻¹

tan x = sin x/cos x

Thus;

tan x csc x = (sin x/cos x) * (1/sin x)

= 1/cos x = (cos x)⁻¹

= sec x

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The value of P(x < 1) is 0.5.

It is the chance of an event to occur from a total number of outcomes.

The formula for probability is given as:

Probability = Number of required events / Total number of outcomes.

We have,

Mean = 1

Standard deviation = 0.05

The P-value of Z when x = 1.

z = (x - mean) / standard deviation

z = (1 - 1)/0.05

z = 0

The P-value when z = 0 is 0.5.

So,

P(x < 1) = 0.5

Thus,

P(x < 1) is 0.5.

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A function is a rule that generates one unique output value for every specified input value. Therefore, the fourth statement is true.

In mathematics, the function is something that relates an input to an output. The output obtained will somehow be related to the input. We write the function as f(x). For example, consider f(x) = x². This means that the function f takes the value of x and squares this value. Like when we input x=4. Then the output is f(x)=4²=16.

So, here see the function assigns a single and a unique output value of 16 for the input value of 4. Therefore, option 4 is correct. That is function gives every input value a unique, separate output value.

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Answer: According to Mars, the makers of M&Ms, their factory produces 25% blue, 25% orange, and 12.5% for each of the remaining colors, which are yellow, green, brown, and red.

The observed values based on the entire lasso are 25 blue, 25 orange, 12.5 yellow, 12.5 green, 12.5 brown, and 12.5 red. These values match the expected values according to the manufacturer, which is 25% blue, 25% orange, and 12.5% for each of the remaining colors.

So, the observed values based on the entire lasso compare to the expected values from the manufacturer. The proportion of colors in the observed sample is the same as the proportion of colors that the factory produces, meaning that the factory is producing the expected ratio of colors.

Step-by-step explanation:

The associative property shows that the value of 8/9 / 7/5 x 3/16 is 5/24.

The associative property states that the way in which we group the numbers or variables in an arithmetic operation does not affect the result. In other words, we can change the grouping of the numbers or variables by using parentheses without changing the outcome of the calculation.

In this problem, we can use the associative property to group the fractions in a way that makes it easier to simplify.

= 8/9 ÷ 7/5 x 3/16

= (8/9) ÷ (7/5) x 3/16

= 8/9 x 5/7 x 3/16

= 8/9 x 15/112

= 120/1008

= 5 / 24

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Answer:

Step-by-step explanation:

1/3 is 2/6

2/6 + 1/6 = 3/6

3/6 = 1/2

answer 1/2 or half

Answer:

It is 1/2

Step-by-step explanation:

You make 1/3 the same denominator so it becomes 2/6.

2/6+1/6=3/6

Now when you simplify this you get 1/2.

The coordinates of X are (-6.4, 5.75)

From the question, we have the following parameters that can be used in our computation:

A = (-10,7)

B = (8,-5).

m : n = 1 : 4

The coordinate is then calculated as

X = 1/(m + n) * (mx2 + nx1, my2 + ny1)

So, we have

X = 1/5 * (1 * 8 + 4 * -10, 1 * -5 + 4 * 7)

Evaluate

X = (-6.4, 5.75)

Hence, the coordinate is (-6.4, 5.75)

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Complete question

given directed line segment ab determine point x on the graph such that x partitions the segment from a(-10,7) to b(8,-5) in the ratio 1:4 and give it's coordinates

The volume of the solid generated by revolving about the line y=-1 is 324π cubic units and the volume of the solid generated by revolving about the line y=-2 is 216π cubic units.

To find the volume of the solid generated by revolving the region bounded by the parabola y=-(x^2)/9 and the lines y=-1 about a line, we can use the method of cylindrical shells.

a) The line y=-1 is the axis of revolution, so the volume of the solid generated by revolving the region about this line is:

V = ∫[(-1-(x^2)/9)^2 - (-1)^2] * 2πx dx

where the integral is taken over the interval for which the parabola y=-(x^2)/9 is above the line y=-1. So the limits of the integral are from -3 to 3, the volume is:

V = ∫[(-1-(x^2)/9)^2 - (-1)^2] * 2πx dx from -3 to 3 = 324*π cubic units

b) The line y=-2 is the axis of revolution, so the volume of the solid generated by revolving the region about this line is:

V = ∫[(-2-(x^2)/9)^2 - (-2)^2] * 2πx dx

where the integral is taken over the interval for which the parabola y=-(x^2)/9 is above the line y=-2. So the limits of the integral are from -3 to -sqrt(92) = -3 to -3sqrt(2) and 3*sqrt(2) to 3 , the volume is:

V = ∫[(-2-(x^2)/9)^2 - (-2)^2] * 2πx dx from -3 to -3sqrt(2) + ∫[(-2-(x^2)/9)^2 - (-2)^2] * 2πx dx from 3sqrt(2) to 3 = 216*π cubic units

c) The line y=1 is not on the same side of the parabola y=-(x^2)/9 as the region bounded by the parabola and the lines y=-1 so there is no solid generated by revolving about this line.

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The company, under the new president, is most likely, earning a profit of $ 10, 000 per month.

When a person is said to have done a 180, it means that they are now doing the opposite of what they used to do. For example, a person who was eating only fast foods doing a 180 would end up eating only healthy food.

This means that the new president doing a 180, means that the company is doing the opposite of what they were doing and instead of making losses of $ 10, 000 monthly, they are making profits of $ 10, 000 every month.

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