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.
(Looking for an complete explanation on how to prove invertible, probably by showing the function is one-to-one and then determining the inverse. )

The expression for the simplified polynomial would be - x² - 10x - 24.

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

Given that the polynomial ( x + 5 )² is subtracted from 1. The simplified polynomial can be written as:-

Polynomial = 1 - ( x + 5 )²

Polynomial = 1 - ( x² + 10x + 25 )

Polynomial = 1 - x² - 10x - 25

Polynomial = -x² - 10x - 24

Hence, the polynomial can be written as -x² - 10x - 24.

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The decision rule given is x1 + x3 – 10 ≥ 0, which can be rearranged to x1 + x3 ≥ 10. This forms a line in the R2 space, and any point on or above this line will be classified as 1, while any point below this line will be classified as 0.

To draw the decision boundary, we can plot the line x1 + x3 = 10 on a graph with the x1 and x3 axes. The region above the line, where x1 + x3 ≥ 10, would be shaded to indicate that it is where the classifier predicts 1.

It is important to note that the classifier predicts 1 if x1 + x3 – 10 ≥ 0, but in the explanation above I have used x1 + x3 = 10 to draw the decision boundary, this is because x1 + x3 – 10 ≥ 0 is equivalent to x1 + x3 ≥ 10.

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The number of papers that Ms. Motley will grade in five days would be = 46 papers (approximately).

The number of papers Ms. Motley graded in first day = 9 papers

The number of papers Ms. Motley graded in the second day = 27 papers

The number of papers Ms. Motley graded in the third day = 31 papers

The average number of she graded for the three days is calculated as follows;

9+27+31 = 67/3 = 27.3 papers.

If in 3 days = 27.3 papers

5. days = X papers

Make X papers the subject of formula;

X papers = 5× 27.3/3

= 136.5/3

= 45.5 papers

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The probabilities for each type of blood are given as follows:

A probability is obtained as the division of the number of desired outcomes by the number of total outcomes.

The total number of people is given as follows:

22 + 20 + 6 + 2 = 50.

6 people have type B, hence the probability is given as follows:

6/50 = 0.12 = 12%.

2 people have type AB, hence the probability is given as follows:

2/50 = 0.04 = 4%.

8 people have either type B or type AB, hence the probability is given as follows:

8/50 = 0.16 = 16%.

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Answer:

-9x + -3

Step-by-step explanation:

1. Primeiro, vamos penteinar SO prazoos que possuem um mesmamã variável, ou seja, -5x e -4x:

-5x - 4x

2.  Agora, vamos penteinar SO prazoos que possuem o mesmo coeficiente, ou seja, 1 e -4:

-5x - 4x + 1 -

4 3.  Por fim, vamos penteinar SO prazoos que possuem o mesmo coeficiente, ou seja, -4 e 1:

-5x - 4x + -4 + 1

The formula the ratio of the displacement thickness to the boundary layer thickness is u = δ / 2

The term displacement thickness is defined as the distance by which the body surface should be shifted in order to compensate for the reduction in mass flow rate on account of boundary layer formation.

Here we need to find the way to determine the ratio of the displacement thickness to the boundary layer thickness.

As we all know that the boundary layer thickness. ideal fluid viscous force is neglected but in actual viscous stresses are prominent within the boundary layer.

AS we all also know that when fluid flows past a solid body the fluid particles adhere to the boundary and the condition of no-slip occurs.

Here the term non-slip conditions means the velocity of the fluid at body surface will be same as the velocity of body.

Then it can be calculated by using the following formula,

=> u = δ / 2

Complete Question:

Here by using the equation  u/U = y/δ and to determine the ratio of the displacement thickness to the boundary layer thickness.

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Answer:

= 53/42

Step-by-step explanation:

10/12 is equivalent to: 70/84

3/7 is equivalnt to: 36/84

Then:

10/12 + 3/7 = 70/84 + 36/84

= (70 + 36) / 84

= 106 / 84

= 53 / 42

The value of tan(Y) = 0.75.

What is trigonometry?

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles, and with the trigonometric functions that describe those relationships.

Since triangle XYZ is a right triangle, we can use the trigonometric ratios of sine, cosine, and tangent to find the value of tan(Y). We can use the Pythagorean theorem to find the lengths of the sides XZ and YZ.

XZ = √(X^2 + Z^2) = √(4^2 + 3^2) = 5

YZ = XZ * sin(Y) = 5 * sin(Y)

Since we know the value of XZ and YZ, we can use the tangent function to find the value of tan(Y):

tan(Y) = YZ / XZ = sin(Y) / cos(Y) = YZ / 5 = 3 / 4 = 0.75

So, the value of tan(Y) = 0.75.

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The best name for this shape is rectangle. There are two sets of parallel sides and two pairs of equal opposite sides in every rectangle.

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Answer:

Uma expressão está e sua forma mais simples quando não há mais nenhuma simplificação possível. Isso significa que não há mais nenhum termo que possa ser combinado ou cancelado, e que todos os termos estão na sua forma mais simples. Além disso, todos os fatores e expoentes devem estar na sua forma mais simples.

The probability that 1 battery in a pack of 16 is defective is approximately 0.3706 or 37.06%.

The binomial model can be used to calculate the probability of a specific number of successful outcomes (in this case, defective batteries) in a fixed number of trials (in this case, 16 batteries) with a constant probability of success (in this case, 5%) for each trial.

The probability of 1 defective battery in a pack of 16 can be calculated using the formula:

P(X = 1) = (¹⁶C₁) × (0.05)¹ × (0.95)⁽¹⁶⁻¹⁾

where (¹⁶C₁) is the binomial coefficient, which is equal to 16.

So;

P(X = 1) = (¹⁶C₁) × (0.05)¹ × (0.95)⁽¹⁶⁻¹⁾

P(X = 1) = 16 × 0.05 × 0.95¹⁵

P(X = 1) = 0.3706

Therefore, the probability that 1 battery in a pack of 16 is defective is approximately 0.3706 or 37.06%.

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Answer:

The value of z can be found using the Law of Sines, which states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle. In other words, if a, b, and c are the lengths of the sides of a triangle opposite angles A, B, and C, respectively, and A, B, and C are the angles opposite sides a, b, and c, respectively, then: a/sin(A) = b/sin(B) = c/sin(C). This can be used to solve for the unknown side length.

The required point that divides the line into 5:1 is given by (-5,0)

Let M(x, y) be the point which divides line segment PQ internally in the ratio m : n. PA, MN and QR are drawn perpendicular to x- axis. PS and MB are drawn parallel to x – axis. This is known as section formula.

Given here: A line segment with end points  (−10,−10) to ( − 4 , 2 )

Now the coordinates of the point that divides a line in the ratio m:n is given by

x= mx₂+nx₁ /m+n

y=my₂+ny₁/m+n

Thus the co-ordinates are x=5×-4+1×-10 / 5+1

                                             =-30/6

                                             = -5

And y= 5×2+1×-10 / 6

         =0

Hence, The required point that divides the line into 5:1 is given by (-5,0)

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9 pounds of Chamomile tea needed  to make a mixture that costs $14.74 per pound

What is the solution to an equation?

In order to make the equation's equality true, the unknown variables must be given values as a solution. In other words, the definition of a solution is a value or set of values (one for each unknown) that, when used as a replacement for the unknowns, transforms the equation into equality.

let:

x pounds of Chamomile tea needed

Therefore,

18.10X + 12.22(12) = 14.74(12 + x)

18.10X + 146.64 = 176.88 +  14.74x

3.36X = 30.24

X = 9 pounds

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From the given table the greatest frequency is 40,

midpoint of greatest frequency =$2100.

Relative frequency is also referred to as the empirical probability, which is calculated by the division of each frequency by the total number of frequency in the sample.

We can answer the question creating the following table:

Class            Frequency       Rel. Frequency

$600-$800             3                  3/120=0.025

$800-$1000            7                  7/120=0.059

$1000-$1200          11                  11/120=0.092

$1200-$1400          22                22/120=0.183

$1400-$1600          40                40/120=0.333

$1600-$1800          24                24/120=0.2

$1800-$2000         9                  9/120=0.075

$2000-$2200        4                  4/120=0.033

Total                       120                      1

From the above table the greatest frequency is 40,

midpoint of greatest frequency =$2100.

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Translations between points B and B' and points S and S' are result of using translation vector T(x, y) = (- 3, - 1).

In this problem we find the case of two points being translated by rigid transformation formula. Translations are defined by following equations:

P'(x, y) = P(x, y) + T(x, y)

Where:

First, derive translation vector from translation formula:

T(x, y) = B'(x, y) - B(x, y)

T(x, y) = (- 9, - 2) - (- 6, - 1)

T(x, y) = (- 3, - 1)

Second, check that point S' is the image of point S:

S'(x, y) = S(x, y) + T(x, y)

S'(x, y) = (5, 1) + (- 3, - 1)

S'(x, y) = (2, 0)

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The numbers that correctly complete this table are 53.83 years and 2.12%

The formula that can be used is:

Doubling time = 70 (constant)/ population growth

Doubling time =70 / 1.3

Doubling time = 53.84 years

The same formula can be applied:

33 = 70 (constant)/ population growth

population growth = 70 /33

population growth = 2.12 %

Based on this, the correct numbers to complete the table are 53.83 years and 2.12%.

Note: Here is the missing table:

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If two rooms in a tiny house share a wall the area of the living room is 54. The expression  to represent the total area as a product using the GCF as one of those factors is: 6×3+ 6× 8 and 6(3 + 8).

Living room area = 54 square feet

Area of the bedroom  = 42 square feet

Total area of the two rooms = 54 + 42

Total area of the two rooms = 66 square feet

So, the expressions that represent the total area of the two rooms  is: 6×3+ 6× 8 and 6(3 + 8).

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Answer:

54+42=6(9+7)

Step-by-step explanation:

Elementary row operations are always applied to an augmented matrix after the solution has been found. So the option D is correct.

Elementary row operations:

1. It is possible to convert a system of linear equations into an equivalent system, or into a new system of equations that has the same solutions as the original system, using these straightforward techniques.

2. In order to convert a matrix to row echelon form, Gaussian elimination is used.

3. Simple row operations on an augmented matrix never alter the underlying linear system's solution set.

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The perimeter of the rectangle WXYZ is 96 cm.

Here we see that

WXYZ is a rectangle (given)

Therefore,

∠WZY = 90 = ∠WZS

Since JS is parallel to PZ and JP is parallel to SZ, we get

JPSZ as a rectangle. Similarly, PKWQ, SYMR, and XQRL are rectangles.

JS = PZ

or, JS = PW + WZ

or, JS = KQ + WZ

or, WZ = JS - KQ

or, WZ = 52 - 25

or, WZ = 27

The side of the rhombus PS is a diagonal of rectangle JSPZ

Therefore length of the diagonal PS = √(JP² + JS²)

= √(52² + 39²)

= 65 cm

Hence for PWKQ,

PK = √(PQ² - KQ²)

= √(65² - 25²)

= 60 cm

For triangles ΔPWQ and ΔSYR,

the corresponding sides are parallel to each other. Hence we can say that

PW/YR = WQ/SY = PQ/RS

Therefore,

60/SY = 65/65            [Since they are the sides of a rhombus]

or, SY = 60 cm

Now, SY = SZ + ZY

or, SY = JP + ZY               [Opposite sides of a rectangle are equal]

or, ZY = 60 - 39

or, ZY = 21

Hence the perimeter of WXYZ

2(length + width)

= 2(27 + 21)

= 96 cm

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Complete Question

Rhombus PQRS is inscribed in a rectangle JKLM, as shown. Segments PZ and XR are parallel to JM. Segments QW and YS are parallel to JK. If JP = 39 cm, JS = 52 cm, and KQ = 25 cm, what is the perimeter of rectangle WXYZ?

Vector v and w have lengths of 11 and contain elements of 21, 10, 32, 2, -3, 4, 5, 6, 7, 4, -22 and -18, 72, 11, -9, 10, 2, 34, -5, 18, 9, 2 respectively. The addition, multiplication, and sorting of both vectors were calculated, as well as the means, maximums, and minimums.

a) The lengths of the vectors v and w are both 11.

b) Vector v contains the elements 21, 10, 32, 2, -3, 4, 5, 6, 7, 4, -22 and vector w contains the elements -18, 72, 11, -9, 10, 2, 34, -5, 18, 9, 2.

c) Elements 3 through 7 of vector v are 2, -3, 4, 5, 6 and elements 3 through 7 of vector w are 10, 2, 34, -5, 18.

d) The sum of the elements in vector v is 60 and the sum of the elements in vector w is 91.

e) The mean of vector v is 5.454545454545455 and the mean of vector w is 8.272727272727273.

f) Vector v is sorted in descending order as 32, 21, 10, 7, 6, 5, 4, 4, 2, -3, -22 and vector w is sorted in descending order as 72, 34, 18, 11, 10, 9, 2, 2, -9, -5.

g) The addition of vectors v and w is 53, 82, 28, 2, 7, 13, 6, 8, -7, 4, -20.

h) The multiplication of vectors v and w is -378, 720, 352, -18.

i) Elements in vector v that are greater than zero are 21, 10, 32, 2, 4, 5, 6, 7, 4.

j) Elements in vector w that are less than zero are -18, -9, -5.

k) Vector v multiplied by 6 is 126, 60, 192, 12, -18, 24, 30, 36, 42, 24, -132.

l) The maximum of vector v is 32 and the minimum of vector v is -22. The maximum of vector w is 72 and the minimum of vector w is -9.

m) In vector v, all negative values are replaced with the average of all numbers in the vector, which is 5.454545454545455. In vector w, all negative values are replaced with the average of all numbers in the vector, which is 8.272727272727273.

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Answer:

5/10 - 1/10 = 4/10

a = 4/10

4/10 + 1/10 = 5/10

Step-by-step explanation:

Hope it helps! =D

The length of RV is 14, SV is 7, TP is 15 and VP is 10. The solution has been obtained using centroid theorem.

What is centroid theorem?

According to the centroid theorem, the centroid of a triangle is located at a distance of 2/3 between the vertex and the midpoint of the sides.

In the given figure, V is the centroid of the triangle

The medians RS and PT intersect at V.

Now, since V divides each median in the ratio 2:1, therefore,

⇒VT = 1/3 of PT

⇒PT = 3*VT

Since, VT=5

⇒PT = 15 = TP

So, VP = PT - VT

⇒VP = 15-5

⇒VP = 10

Similarly,

⇒RV = 2/3 of RS

Since, RS = 21, therefore,

⇒RV = 2/3 of 21

⇒RV = 14

So, SV = RS-RV

⇒SV = 23-14

⇒SV = 7

Hence, the length of RV is 14, SV is 7, TP is 15 and VP is 10.

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The complete question is attached below

To find the slope of the line given the points (0,2) and (1,0), we can use the formula: m = (y2 - y1) / (x2 - x1).

m = (0 - 2) / (1 - 0) = -2

Using point-slope form, we can write the equation of the line as:

y - 2 = -2 (x - 0)

y = -2x + 2

To get the inequality we just need to change the equation to the inequality by replacing the equal sign with the inequality sign.

So the linear inequality for the points (0,2) and (1,0) is:

y >= -2x + 2

At that level of output and price, profit is maximized when marginal revenue and the marginal cost of production are equal:

MR = *TR/*Q

MC = *C/*Q

Eq.  = MR = MC

where, MR = Marginal Revenue

MC = Marginal Cost

* = Change in

TR = Total Revenue

Q = Quantity

C = Cost

The marginal revenue rule states that by selecting the amount at which marginal revenue equals marginal cost, we can maximize the net benefit of any action. The activity's net benefit is maximum at this amount.

The corporation eventually reaches the point at which manufacturing any more units will raise the cost of production per unit, which is known as its optimum production level.

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On solving the linear equation y = 3/2x - 29, the missing value for x is obtained as 14.66.

What is a linear equation?

A linear equation is one that has a degree of 1 as its maximum value. No variable in a linear equation, thus, has an exponent greater than 1. A linear equation's graph will always be a straight line.

The table shows the points (18,-2) and (16,-5).

The slope-intercept form of an equation/function is - y = mx + b

To find the slope m use the formula -

(y2 - y1)/(x2 - x1)

Substituting the values in the equation -

[-5 - (-2)]/[16 - 18]

(-5 + 2)/(-2)

-3/-2

3/2

So, the slope point is obtained as m = 3/2.

The equation becomes - y = 3/2x + b

To find the value of b substitute the values of x and y in the equation -

-2 = 3/2(18) + b

-2 = 27 + b

b = -2 - 27

b = -29

So, the value for b is -29.

Now, the equation becomes -

y = 3/2x - 29

To find the missing value substitute the value of  y = -7 in the equation -

-7 = 3/2x - 29

3/2x = -7 + 29

3/2x = 22

x = 22 × 2/3

x = 44/3

x = 14.66

Therefore, the value of x = 14.66.

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Answer: 7

Step-by-step explanation:

First, we need to prime factorize the number 189,

189 = 3 × 3 × 3 × 7 × m

From the factors we can see that there is no pair of same numbers for 7, so if we replace m with 7, we can get a perfect square.

Hence the answer is 7

The students expected to like soda are 324

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

150 students out of a total of 800 students on campus did not like soda.

This means that

650 students out of a total of 800 students on campus like soda.

i.e. 800 - 150 = 650

When half of the students are considered, we have

Soda = 650/2

Evaluate

Soda = 325

Hence, the number of students is 325

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Answer: 13:6

Ratio of length to width = 39:18

Divide both by 3 to simplify

13:6