Part A: Graph the piecewise function g(x) and determine the domain.

Part B: Determine the x-intercepts of g(x). Show all necessary calculations.

Part C: Describe the interval(s) in which the graph of g(x) is positive.

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:  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]

For any k ∈ (-∞, o), function f is strictly decreasing on the interval (-1, 1).

In mathematics, a function is a relationship or mapping between two sets of elements, known as the domain and the codomain, that assigns each element from the domain to a unique element in the codomain. It describes how inputs from the domain are transformed or related to corresponding outputs in the codomain.

The given function is f = [tex]\frac{kx}{x^2+1}[/tex] ...(1),

Differentiate (1) w.r.t. x,

[tex]\dfrac{d}{dx}f = \dfrac{d}{dx}\frac{kx}{x^2+1}[/tex]

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

Since f is decreasing on (-1, 1),

therefore

f' < 0 on (-1, 1),

[tex]k\frac{1-x^2}{(x^2 + 1)^2}[/tex] < 0 on (-1, 1).

As [tex]\frac{1-x^2}{(x^2 + 1)^2}[/tex] > 0 on (-1, 1),

k < 0 for f' < 0.

Therefore, k ∈ (-∞, o) for f to be strictly decreasing on (-1, 1).

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The complete question is as follows:

Let f be the function given by [tex]\frac{kx}{x^2+1}[/tex], where k is a constant. For what values of k, if any, is f strictly decreasing on the interval (-1, 1)?

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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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 measure of the length HL will be 18.5 units for the given figure.

One of the earliest areas of mathematics is geometry, along with arithmetic. It is concerned with spatial characteristics like the separation, shape, size, and relative placement of objects.

Given that in the quadrilateral the two sides are Jk = 12 and GF = 25. The value of HL can be calculated by the theorem,

HL = ( 1 / 2 ) x GF x JK

HL = ( 1 / 2 ) x 25 x 12

HL = 18.5

Therefore, the length of section HL will be 18.5 units.

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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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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:

m=21/8

Step-by-step explanation:

5 /38+m=8
Move the constant to the right.
43/8+m=8

m=8-43/8

m=21/8

Answer: 1

Step-by-step explanation:

1 + 0 = 1

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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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:

Answer:

10/10/10

Step-by-step explanation:

The area and heights  and bases of the figures are: 8.9, 54 in², 35cm²,20.25m², 168 ft², 6cm, 37.8in² and 5mm respectively

The figures include triangles and trapezoids.  The areas are the floor spaces the figures occupy

1) Area of a triangle = 1/2 *b*h

b = 14.3*2 = 28.6

h=8.9

Area = 1/2 *28.6 * 8.9

Area = 127.27 cm²

2) Area of a trapezoid = 1/2(a+b)h

Area = 1/2 (12+6)*6

Area = 1/2*18*6

= 54in²

3) Area of a trapezoid = 1/2(a+b)h

Area = 1/2 (8+6)*5

= (14*5)/2 = 35cm²

4) Area of a triangle = 1/2 *b*h

Area = 1/2 * 9 * 4.5

=20.25m²

5) Area of a trapezoid = 1/2(a+b)h

Area = 1/2 (18+24)*8

= 168ft²

6) Area of a triangle = 1/2 *b*h

38.4 = 1/2*b*12.8

Making b the subject of the relation we have

76.8 = 12.8b

b=6cm

7) Area of a triangle = 1/2 *b*h

Area = 1/2*8.4*9

=37.8in²

8) Area of a trapezoid = 1/2(a+b)h

35=1/2(6+8)*h

70=14h

Making h the subject we have

h= 5 mm

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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:

Answer:

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

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 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 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 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

The area of the circle is πr², where r is the radius. The area of the shaded region is half the area of the circle.

The area of the circle is πr², where r is the radius. The area of the shaded region is half the area of the circle.

Therefore, Area of the shaded region = πr²/2

Given that minor arc AD measures 2π, the radius of the circle is 2.

Therefore, Area of the shaded region = π(2)²/2 = 4π = 72

The area of the shaded region is half the area of the circle, which is πr²/2. The minor arc AD measures 2π, so the radius of the circle is 2, and the area of the shaded region is 4π, or 72.

The area of the circle is πr², where r is the radius. The area of the shaded region is half the area of the circle.

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Answer:

Option A) [tex]132ft[/tex]

Step-by-step explanation:

It is given that,

Radius of a circle = 21ft

We know that π = [tex]\frac{22}{7}[/tex]

So the circumference of a circle = 2πr

Substituting the values,

= [tex]2X\frac{22}{7} X 21[/tex]

So we get,

[tex]= 2X22X3[/tex]

[tex]= 132ft[/tex]

The required circumference of the circle is 132 feet.  Option A is correct.

To find the circumference of a circle, we use the formula C = πd, where C is the circumference, π is the mathematical constant pi (approximately equal to 3.14), and d is the diameter of the circle.

Here,
The circumference of a circle with a radius of 21 feet can be found using the formula:

C = 2πr

where "C" is the circumference, "r" is the radius, and "π" is the mathematical constant pi, which is approximately equal to 22/7.

Substituting the values given, we have:

C = 2 x (22/7) x 21

C = 132 feet

Therefore, the circumference of the circle is 132 feet. Answer: 132 feet.

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The estimate of the total number of computers that needed repairing last year in Point City is given as follows:

Between 132,000 and 328,000 computers.

The estimate is obtained applying the proportions in the context of this problem.

The confidence interval is given as follows:

1.15 plus/minus 0.49.

Hence the bounds of the interval are given as follows:

Hence, out of 200,000 households, the lower bound of the estimate is given as follows:

0.66 x 200,000 = 132,000.

The upper bound is of:

1.64 x 200,000 = 328,000.

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Answer:

0.125 pounds of skittles

Step-by-step explanation:

Lauren and Shayla have a total of ½ pound of skittles and they want to share them with 2 friends, including themselves, so the total number of people is 2+2 = 4

If they want to divide the ½ pound of skittles equally among 4 people, each person will receive ½ pound / 4 = 0.125 pounds of skittles.

Answer:

0.43, 43%

Step-by-step explanation:

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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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: 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

A multiplication equation to show an equivalent fraction of 15 using fifteenths would gives 10/75

A fraction consisting of a quotient and remainder is a mixed fraction. we can convert the mixed fraction to improper fraction by first dividing the numerator by denominator and then taking the quotient as whole number and remainder as the numerator of proper fraction keeping the denominator same.

We have to write an equivalent fraction of 15 using fifteenths.

2/15

So let multiply by 5 .

2 / 15 x  5/ 5 = 10/75

Thus, 10/75 is equivalent expression.

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