Hello, is there any one to solve it please
Graph the function for the given domain, write the range. g(x) = 1/(x2+6)
Domain: {-6, -4, -2, 0, 2, 4, 6}​

Answer:

Answer: B. Symmetric.

Explanation:

In a symmetric distribution, the data is evenly distributed around the mean or median, creating a mirror image on both sides of the center. In this histogram, the median and mean are very close together at 55 and the bars on both sides of the center are roughly equal in height, indicating a fairly even distribution. Therefore, the histogram is symmetric.

Answer:

See explanation below

Step-by-step explanation:

1. 12x - 18 = 6(2x -3)

2. 15x + 25 = 5(3x + 5)

3. 14x + 21 = 7(2x + 3)

4. 5x - 5 = 5(x - 1)

5. 12x - 30 = 6(2x - 5)

6. 10x + 8 = 2(5x + 4)

7. 27x + 18 = 9(3x + 2)

8. 4x - 20 = 4(x - 5)

9. 20x + 30 = 10(2x + 3)

10. 4(x + 5) = 4x + 20

11. 3(x - 2) = 3x - 6

12. 5(2x + 4) = 10x + 20

13. 5(x - 1) = 5x - 5

14. 1/2(10x + 12) = 5x + 6

15. 4(2x + 4) = 8x + 16

16. 2(5x - 2) = 10x - 4

17. 2(x - 8) = 2x - 16

18. 4(2x + 1) = 8x + 4

two fractions with a common denominator of 8 can be expressed in the form of a/b and c/8, where a and c are integers. As long as a and c are not both multiples of 8  then these fractions would have a common denominator of 8.

A number that can be divided exactly by all of the denominators in a group of fractions is referred to as a common denominator. 2. A noun that counts. A trait or attitude that all members of a group share is known as a common denominator.

Since the two fractions have a common denominator of 8, they can be written in the form of a/b and c/8, where a and c are integers.

There are many possible combinations of integers that could satisfy this condition. Here are some examples:

1/8 and 3/8

2/8 (which simplifies to 1/4) and 6/8 (which simplifies to 3/4)

4/8 (which simplifies to 1/2) and 7/8

5/8 and 2/8 (which simplifies to 1/4)

3/8 and 4/8 (which simplifies to 1/2)

In general, any two fractions with a common denominator of 8 can be expressed in the form of a/b and c/8, where a and c are integers. As long as a and c are not both multiples of 8  then these fractions would have a common denominator of 8.

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The measure of the angle A is 49.99 degrees or 50 degrees if the length of AB = 14 and AC = 9.

Trigonometry is a branch of mathematics that deals with the relationship between sides and angles of a right-angle triangle.

We have a given a right angle triangle in the picture

It is required to find the measure of angle A

Applying cos ratio to find the measure of the angle A:

cosA = 9/14

cosA = 0.642

A = 49.99 ≈ 50 degree

Thus, the measure of the angle A is 49.99 degrees or 50 degrees if the length of AB = 14 and AC = 9.

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Answer:

A

Step-by-step explanation:

∠ o and ∠ z are alternate exterior angles and are congruent, that is

∠ z = ∠ o = 125°

The value of the height of the trapezium is 10.3 cm.

The area of a trapezium is given by the formula:

Area = (1/2) x (sum of parallel sides) x (distance between the parallel sides)

The parallel sides of the trapezium are given as 18 cm and 13 cm

The area of the trapezium is given as 160 cm²

The height of the trapezium = h

The value of the height of the trapezium is calculated as follows;

160 = ¹/₂ (18 + 13) h

2 x 160 = (31)h

320 = 31h

h = 320 / 31

h = 10.3 cm

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Answer:

2.5 hours

Step-by-step explanation:

To find the number of hours for each test,

Divide the number of hours by the number of tests

10 divided by 4

2.5 hours for each test

Using the length formula we know that the length of the given arc is 30.144 miles.

In mathematics, an arc is a portion of the boundary of a circle or curve. It is sometimes referred to as an open curve.

The measurement around a circle that determines its edge is called the circumference, often known as the perimeter.

So, the formula to find the length of the arc is:

Length of an Arc = θ × r

Now, insert values and calculate as follows:

Length of an Arc = θ × r

Length of an Arc = 4π/5 × 12

Length of an Arc = 4π/5 × 12

Length of an Arc = 12.56/5 * 12

Length of an Arc = 2.512 * 12

Length of an Arc =  30.144

Therefore, using the length formula we know that the length of the given arc is 30.144 miles.

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Answer:

Step-by-step explanation:

The slope of a line is the measure of the steepness and the direction of the line. Finding the slope of lines in a coordinate plane can help in predicting whether the lines are parallel, perpendicular, or none without actually using a compass.

The slope of any line can be calculated using any two distinct points lying on the line. The slope of a line formula calculates the ratio of the "vertical change" to the "horizontal change" between two distinct points on a line. In this article, we will understand the method to find the slope and its applications.

That is what Slope is.

Answer:

Step-by-step explanation:

Slope :( 1,1)

You start on the y-axis point which is (0,1) as you can see it is going up so I used the “up left” strategy. You go up 1 to the left 1 since the line intersects at point (1,2)

Answer:

x = [tex]\frac{7\pi }{6}[/tex], [tex]\frac{11\pi }{6}[/tex]  or x = 210°, 330°

Step-by-step explanation:

4sin(x) + 2 = 0

4sin(x) = -2

sin(x) = -1/2

x = [tex]\frac{7\pi }{6}[/tex], [tex]\frac{11\pi }{6}[/tex]

Once we have this data, we can substitute the values into the formula to find the empirical probability that a person prefers apple pie given that they prefer whipped cream. Therefore, the missing probability is 1/12, and we know this because it is the value that makes the sum of all probabilities equal to 1.

Probability is a measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 represents an event that is impossible, and 1 represents an event that is certain to occur. For example, if the probability of winning a coin toss is 1/2, this means that there is an equal chance of the coin landing heads or tails. Probability can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes. This is known as the classical probability approach. Another approach is empirical probability, where probabilities are calculated based on observed data or experiments. Lastly, subjective probability involves making an informed guess or estimate about the likelihood of an event occurring based on subjective factors such as experience, intuition, or expert opinion. Probability is a fundamental concept in statistics and is used in many application

Here,

1. The formula needed to calculate the empirical probability that a person prefers apple pie given that they prefer whipped cream is:

P(Apple Pie | Whipped Cream) = P(Apple Pie and Whipped Cream) / P(Whipped Cream)

where P(Apple Pie and Whipped Cream) is the probability that a person prefers both apple pie and whipped cream, and P(Whipped Cream) is the probability that a person prefers whipped cream.

This formula is used because it is a conditional probability, which is a measure of the probability of an event occurring given that another event has occurred. In this case, we want to find the probability that a person prefers apple pie given that they already prefer whipped cream.

To calculate P(Apple Pie and Whipped Cream), we would need to gather data on the number of people who prefer both apple pie and whipped cream. Similarly, to calculate P(Whipped Cream), we would need to gather data on the number of people who prefer whipped cream.

2. To find the missing probability, we need to use the fact that the sum of all probabilities in a probability distribution must be equal to 1. Therefore, we can set up an equation to solve for the missing probability:

1/6 + 1/3 + x + 5/12 = 1

Simplifying the equation by finding a common denominator gives:

2/12 + 4/12 + x + 5/12 = 1

Combining like terms gives:

11/12 + x = 1

Subtracting 11/12 from both sides gives:

x = 1 - 11/12

x = 1/12

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Answer:

4 pens.

Step-by-step explanation:

Now he has 2 pens.

Then his friend gives him two more.

As a sum, that's:

2 + 2 = 4 pens.

Answer:

He has 4 pens now.

Step-by-step explanation:

2+2=4

The cost to rent one shed of the rectangular prism shaped shed is $2772.

The size of a section on a surface is determined by its area. Surface area refers to the area of an open surface or the border of a three-dimensional object, whereas the area of a plane region or plane area refers to the area of a shape or planar lamina.

A rectangular prism is a polyhedron in geometry that has two parallel and congruent sides. It also goes by the name cuboid. Six faces, each with a rectangle form and twelve edges, make up a rectangular prism. It is referred to as a prism because of the extent of its cross-section.

Volume of prism= BH

where B= area of base and H= height

B= 11*7 = 77 feet²

H= 12 feet

Volume= 77*12=924 cubic feet

Cost =$3 per cubic foot

Total cost= 3*924= $2772

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Answer:

Step-by-step explanation:

Using a standard normal table, we can find the area under the curve between -2.11 and -0.85.

P(-2.11 < z < -0.85) = P(z < -0.85) - P(z < -2.11)

Using the table, we find:

P(z < -0.85) = 0.1977

P(z < -2.11) = 0.0174

Therefore,

P(-2.11 < z < -0.85) = 0.1977 - 0.0174 = 0.1803

So the probability that a standard normal random variable falls between -2.11 and -0.85 is 0.1803.

In response to the stated question, we may state that The equivalent trigonometry expression of 7/tan b + 7tan b is (7 - 7tan b cot b)/sin b.

The study of the connection between triangle side lengths and angles is known as trigonometry. The concept first originated in the Hellenistic era, during the third century BC, due to the application of geometry in astronomical investigations. The subject of mathematics known as exact techniques deals with certain trigonometric functions and their possible applications in calculations. There are six commonly used trigonometric functions in trigonometry. Sine, cosine, tangent, cotangent, secant, and cosecant are their separate names and acronyms (csc). The study of triangle characteristics, particularly those of right triangles, is known as trigonometry. As a result, geometry is the study of the properties of all geometric forms.

tan(A + B) = (tan A + tan B)/(1 - tan A tan B)

Set A = 90 degrees and B = b degrees:

tan(90 + b) = (tan 90 + tan b)/(1 - tan 90 tan b)

tan(90 + b) = (undefined + tan b)/(1 - undefined tan b)

tan(90 + b) = -cot b

7/tan b + 7tan b

= 7/(tan b) + 7(tan(90 + b) - 1)

= 7/(tan b) + 7(-cot b - 1)

= (7 - 7tan b cot b)/sin b

The equivalent expression of 7/tan b + 7tan b is (7 - 7tan b cot b)/sin b.

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the correct options are A), B), and E).

why it is and what is Gallon?

A) 2 quarts = 8 cups

B) 8 cm = 80mm

E) 96 inches = 8 feet

The number 8 cannot correctly fill in the blank for statements C and D.

C) 1 gallon = 16 cups, so 4 gallons = 64 cups, not 8 cups.

D) 1 hour = 60 minutes, so 96 minutes = 1 hour and 36 minutes, not 8 hours.

Therefore, the correct answers are A), B), and E).

A gallon is a unit of measurement for volume commonly used in the United States and some other countries. There are two different sizes of gallons: the US gallon and the imperial gallon.

The US gallon is defined as exactly 3.785411784 liters, and is used for measuring liquids such as gasoline, milk, and other beverages.

The imperial gallon, which is used in the United Kingdom and some other countries, is defined as exactly 4.54609 liters.

In both cases, a gallon is typically divided into smaller units such as quarts, pints, and fluid ounces for measuring smaller amounts of liquid.

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The result of the formula  [tex]12 - 3y2 + (v=4) / (2y - 2)[/tex] for y = 3 is  [tex]-29/2[/tex]  .

When [tex]y = 3[/tex] is used, the value of the expression [tex]12 - 3y2 + (v=4) / (2y - 2)[/tex]  has a value of  [tex]-29/2[/tex] .

To analyse an algebraic expression is to determine its value when a certain number is used in lieu of the variable. To evaluate the expression, we first replace the variable with the given number, then we use the order of operations to simplify the expression.

If  [tex]y = 3[/tex] , we can insert it into the expression & simplify as follows to evaluate   [tex]12 - 3y2 + (v=4) / (2y - 2)[/tex] for  [tex]y = 3[/tex]  .

[tex]12 - 3(3)^2 + (√4) / (2(3) - 2)[/tex] (y = 3 replacement)

[tex]12 - 27 + 2 / 4\s-15 + 1/2\s-29/2[/tex]

Therefore, The result of the formula  [tex]12 - 3y2 + (v=4) / (2y - 2)[/tex] for y = 3 is  [tex]-29/2[/tex]  .

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The solutions to the quadratic function 9x² - 15x - 6 = 0 are given as follows:

x = -1/3 and x = 2.

The quadratic equation for this problem is defined as follows:

9x² - 15x - 6 = 0.

The coefficients of the function are given as follows:

a = 9, b = -15 and c = -6.

The discriminant of the function is obtained as follows:

D = b² - 4ac

D = (-15)² - 4 x 9 x (-6)

D = 441.

Then the solutions to the quadratic function are obtained as follows:

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(a) 12 cm 40° : Area of shaded segments = 301.44 sq. cm.

(b) 58° 16 cm ​: Area of shaded segments = 777.04 sq. cm.

Two radii that meet at the center to form a sector define a circle. The sector is the portion of the circle created by these two radii. Knowing a circle's central angle calculation and radius measurement are both crucial for solving circle-related difficulties.

Area of sector of circle = Ф/360 * πr²

π = 3.14

r  is the radius

Ф is the angle subtended.

(a) 12 cm 40°

Area of shaded segments = 40/60 * 3.14* 12²

Area of shaded segments = 40/60 * 452.16

Area of shaded segments = 301.44 sq. cm.

(b) 58° 16 cm ​

Area of shaded segments = 58/60 * 3.14* 16²

Area of shaded segments = 58/60 * 803.84

Area of shaded segments = 777.04 sq. cm.

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The diagram for the question is attached.

The expression 1/30(ln(15/22))t = x when solved for t has a solution of t = -78.95x

Given the following expression

1/30(ln(15/22))t =

The above expression cannot be solved for t

This is so because the expression is not an equation or inequality

To do this, we must equate the expression to a value (say x)

So, we have

1/30(ln(15/22))t = x

Multiply through by 30

This gives

ln(15/22)t = 30x

Evaluate the natural logarithm expression

This gives

-0.38t = 30x

Divide both sides by -0.38

So, we have

t = -78.95x

Hence, the solution for t is t = -78.95x

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Toward the end of a game of Scrabble, you hold the letters D, O, G, and Q. You can choose 3 of these 4 letters and arrange them in order in 24 different ways.

To solve this problem, we need to use the concept of permutations. A permutation is an arrangement of objects in a specific order. In this case, we need to find the number of permutations that can be made from the letters D, O, G, and Q when we choose 3 of these 4 letters.

The formula for finding the number of permutations is:

n! / (n-r)!

where n is the total number of objects and r is the number of objects we choose.

Using this formula, we can calculate the number of permutations as follows:

4! / (4-3)!

= 4! / 1!

= 4 x 3 x 2 x 1 / 1

= 24

Therefore, we can arrange the chosen 3 letters in 24 different ways.

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Answer:

4-. c = ±26

Step-by-step explanation:

c² = a² + b²

c² = 10² + 24²

c² = 100 + 576

c² = 676

√c² = √676

c = ± 26

So,

120% is the same as 1.2 or 6/5.  80% is the same as 0.8 or 4/5.

We are given the value of a, which is 12.

6/5 of a, or 1.2 times a, is equal to 4/5 of b, or 4/5 times b.

[tex]\dfrac{6}{5} (12)=\dfrac{4}{5} b[/tex]

Simplify.

[tex]\dfrac{72}{5} =\dfrac{4}{5} b[/tex]

Multiply both sides by 5/4.

[tex]\dfrac{72}{4} \ \text{or} \ 18=b[/tex]

Therefore, the sum of a and b is 12 + 18 = 30.

The evaluation of the matrix multiplication can be presented as follows;

Question #1

The matrix M and A are;

[tex]M=\begin{bmatrix}1 &14 &24 &19 &14 &1&5 \\7 &20 &0 &0 &0 &14 &18\\5 &0 &9 &9 &4 &7 &0 \\\end{bmatrix}[/tex]

[tex]A = \begin{bmatrix} 1& 2 &3 \\3 & 1 & 2 \\ 2 & 3& 1 \\\end{bmatrix}[/tex]

The encoded message in matrix form is therefore;

[tex]A\times M=\begin{bmatrix}30 &54 &51 &46 &26 &50&41 \\20 &62 &90 &75 &50 &31&33 \\28 & 88 &57 &47 &32 &51&64 \\\end{bmatrix}[/tex]

The encoded message is;  30   20   28   54   62   88   51   90   57   46   75   47   26   50   31   51   41   33   64

Question #2

The matrices M and A × M are;

[tex]M=\begin{bmatrix}3 &16 &26 &21 &16 &3 &7\\9 & 22 & 0 &0 &0 &16&20 \\7 &0 &11 &11 &6 &9 &0\\\end{bmatrix}[/tex]

[tex]A\times M=\begin{bmatrix}42 &60 &59 &54 &34 &62&47 \\32 & 70 & 100 &85 &60 &43&41 \\40 & 98 & 63 &53 &38 &63 &74\\\end{bmatrix}[/tex]

The encoded message is; [tex]{}[/tex]42   32   40   60   70   98   59   100   63   54   85   53   34   60   38   62   43   63   47   41   74

The rule for matrix multiplication is that the number of columns in the first matrix should be the same as the number of rows in the second matrix

A m by n matrix can only multiply a n by p matrix to produce a m by p matrix

The matrix M for the message can be presented as follows;

The letters are replaced with the corresponding numbers to get;

[tex]{}[/tex]1  7  5   14   20  0  24  0  9    19   0   9   14   0   4   1   14   7   5   18   0

M = [tex]\begin{bmatrix}1 &14 &24 &19&14&1 &5 \\7 & 20 &0 &0 &0&14&18\\5 & 0& 9 & 9&4&7&0\\\end{bmatrix}[/tex]

The matrix A is presented as follows;

[tex]A=\begin{bmatrix}1 & 2 & 3 \\3 & 1 & 2 \\ 2& 3 &1 \\\end{bmatrix}[/tex]

Therefore;

[tex]A \times M = \begin{bmatrix}30 &54 &51 &46&26&50 &41 \\20 & 62 &90 &75 &50&31&33\\28 & 88& 57 & 47&32&51&64\\\end{bmatrix}[/tex]

The encoded message is therefore;

Question #2

The 2-letter circular shift indicates that the message becomes;

AGENT X IS IN DANGER

The above message becomes;

CIGPV Z KU KO FCOIGT

The numbers comes;

[tex]{}[/tex]3  9  7  16   22  0  26   0   11  21  0  11   15  0  6  3   15   9   7   20

The Matrix Mis therefore;

[tex]M = \begin{bmatrix}3 &16 &26 &21&16&3 &7 \\9 & 22 &0 &0 &0&16&20\\7 & 0& 11 & 11&6&9&0\\\end{bmatrix}[/tex]

The message becomes;

[tex]A \times M = \begin{bmatrix}42 &60 &59 &54&34&62 &47 \\32 & 70 &100 &85 &60&43&41\\40 & 98& 63 & 53&38&63&74\\\end{bmatrix}[/tex]

The encoded message is; [tex]{}[/tex] 42  32   40   60   70   98   59   100   63   54   85   53   34   60   38   62   43   63   47   41   74

The information in the attached file related to the question includes;

Two agents are to send a confidential message which is to be encoded using a 3 × 3 invertible matrix A that consists of natural number elements.  The coding strategy consists of replacing each letter by the number representing the letter in the alphabet, after which the numbers representing the letters are placed in columns of 3 elements, completing the missing elements in the last column and blanks with zero in the matrix M. See above.

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The value of the probability is 0.3125 and this is proved by the calulations below

The probability of getting exactly 3 heads in 5 coin tosses can be calculated by multiplying the probability of one specific combination of 3 heads and 2 tails by the number of possible combinations.

The probability of one specific combination, for example HHTTT, is (1/2)^5 = 1/32, because each toss has a 1/2 chance of being a head or a tail.

There are 5C3 = 10 possible combinations of 3 heads and 2 tails in 5 tosses.

For example: HHTTT, HTHTT, HTTHT, HTHHT, TTHHH, etc.

Therefore, the probability of getting exactly 3 heads is:

Probability = 10 * (1/32)

Probability = 10/32

Probability = 0.3125.

Hence, the value of the probability is 0.3125.

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The car must drive at a constant speed of approximately 112.89 km/hr to cover the same distance in 2 hours 35 minutes.

The formula for time is: time = distance / speed

where "distance" is the distance traveled by an object, and "speed" is the rate at which the object is moving.This formula can be used to calculate the time taken by an object to travel a certain distance at a constant speed, or to calculate the speed or distance if the other two variables are known.

The formula for speed is: speed = distance / time

where "distance" is the distance traveled by an object and "time" is the duration of travel.

This formula can be used to calculate the speed of an object if the distance it has traveled and the time it took to travel that distance are known. It can also be used to calculate the distance traveled by an object if its speed and the time it traveled at that speed are known.

In the given question,

Let's first calculate the distance traveled in 2 hours 45 minutes (2.75 hours) at an average speed of 106 km/hr.

distance = speed × time

distance = 106 × 2.75

distance = 291.5 km

Now, we need to find at which constant speed the car must drive to cover the same distance in 2 hours 35 minutes (2.5833 hours). Let's call this speed "x".

distance = speed × time

291.5 = x × 2.5833

x = 291.5 / 2.5833

x ≈ 112.89 km/hr

Therefore, the car must drive at a constant speed of approximately 112.89 km/hr to cover the same distance in 2 hours 35 minutes.

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Answer:

Devan's age = 2 years.

Deepa's age = 6 years.

Step-by-step explanation:

Present age:      

 Let the present age of Devan = x

             Present age of Deepa = 3x

After 2 years:

                     Age of Devan = x + 2

                     Age of Deepa = 3x + 2

     Deepa's age = 2* Devan's age

          3x + 2        = 2 *(x + 2)

                3x + 2  = 2x + 2*2    {Use distributive property}

               3x + 2   = 2x + 4

  Subtract '2' from both sides,

                           3x = 2x + 4 - 2

                           3x = 2x + 2

Subtract '2x' from both sides,

                   3x  - 2x = 2

                             x = 2

Devan's age = 2 years.

Deepa's age = 3*2

                      = 6 years  

Probability of winning at home is 0.67, Probability of winning away is 0.58.

To calculate the probability of home and away wins, we need to divide the number of wins by the total number of games played at home and away respectively.

Probability of winning at home = (Number of wins at home) / (Total number of games played at home)

= 8 / 12

= 0.67

Probability of winning away = (Number of wins away) / (Total number of games played away)

= 7 / 12

= 0.58

Based on these results, Sasha's team has a higher probability of winning at home (0.67) compared to away games (0.58).

Therefore, her belief that her team plays better away is not supported by the data.

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(a) T is one-to-one if and only if T carries linearly independent subsets of V onto linearly independent subsets of W.

(b) If T is one-to-one, then S is linearly independent if and only if T(S) is linearly independent.

(c) If β is a basis for V and T is one-to-one and onto, then T(β) is a basis for W.

(a) Assume T is one-to-one. Let S be a linearly independent subset of V, and suppose T(S) is linearly dependent. Then there exist distinct vectors s1, s2, ..., sn in S such that T(s1), T(s2), ..., T(sn) are linearly dependent. This means that there exist scalars c1, c2, ..., cn, not all zero, such that c1T(s1) + c2T(s2) + ... + cnT(sn) = 0. Since T is linear, we have T(c1s1 + c2s2 + ... + cnsn) = 0. But since T is one-to-one, this implies that c1s1 + c2s2 + ... + cnsn = 0, contradicting the assumption that S is linearly independent. Hence, T(S) must be linearly independent.

Conversely, assume that T carries linearly independent subsets of V onto linearly independent subsets of W. Let v1 and v2 be distinct vectors in V, and suppose T(v1) = T(v2). Then {v1, v2} is linearly dependent, which implies that there exist scalars c1 and c2, not both zero, such that c1v1 + c2v2 = 0. Applying T to both sides yields c1T(v1) + c2T(v2) = 0, which implies that T(v1) and T(v2) are linearly dependent. This contradicts the assumption that T carries linearly independent subsets of V onto linearly independent subsets of W. Hence, T must be one-to-one.

(b) Assume T is one-to-one and let S be a subset of V. Suppose S is linearly independent and that T(S) is linearly dependent. Then there exist distinct vectors s1, s2, ..., sn in S such that T(s1), T(s2), ..., T(sn) are linearly dependent. This means that there exist scalars c1, c2, ..., cn, not all zero, such that c1T(s1) + c2T(s2) + ... + cnT(sn) = 0. Since T is linear, we have T(c1s1 + c2s2 + ... + cnsn) = 0. But since T is one-to-one, this implies that c1s1 + c2s2 + ... + cnsn = 0, contradicting the assumption that S is linearly independent. Hence, T(S) must be linearly independent.

Conversely, assume that T(S) is linearly independent whenever S is a linearly independent subset of V. Let v1 and v2 be distinct vectors in V, and suppose T(v1) = T(v2). Then {v1, v2} is linearly dependent, which implies that there exist scalars c1 and c2, not both zero, such that c1v1 + c2v2 = 0. Since {v1, v2} is linearly dependent, we have either v1 = 0 or v2 = 0. Without loss of generality, assume v1 = 0. Then T(v1) = 0 = T(v2), and hence T({v1, v2}) = {0} is linearly dependent. This contradicts the assumption that T carries linearly independent subsets of V onto linearly independent subsets of W. Hence, S must be linearly independent.

(c) First, we will show that T(β) spans W. Let w be an arbitrary vector in W. Since T is onto, there exists some vector v in V such that T(v) = w. Since β is a basis for V, there exist scalars c1, c2, ..., cn such that v = c1v1 + c2v2 + ... + cnvn. Applying T to both sides, we have w = T(v) = T(c1v1 + c2v2 + ... + cnvn) = c1T(v1) + c2T(v2) + ... + cnT(vn), which implies that T(β) spans W.

Next, we will show that T(β) is linearly independent. Suppose there exist scalars c1, c2, ..., cn such that c1T(v1) + c2T(v2) + ... + cnT(vn) = 0. Applying T to both sides, we have T(c1v1 + c2v2 + ... + cnvn) = 0. But since T is one-to-one, this implies that c1v1 + c2v2 + ... + cnvn = 0, which implies that c1 = c2 = ... = cn = 0, since β is a basis for V. Hence, T(β) is linearly independent.

Since T(β) spans W and is linearly independent, it is a basis for W. Therefore, if β is a basis for V and T is one-to-one and onto, then T(β) is a basis for W.

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