I need help on all of them I’ll take some if not all!

The function that has a range of all real numbers is given as follows:

f(x) = -x + 2.

Hence the range for each function in the context of this problem is given as follows:

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The statement The product of 3 and the difference of a number and 8 as an expression is 3(x - 8)

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

The product of 3 and the difference of a number and 8

Represent the number with x

So, we have

The product of 3 and the difference of x and 8

The difference of x and 8 means x - 8

So, we have

The product of 3 and x - 8

The product of 3 * (x - 8)

So, we have

3(x - 8)

Hence, the expression is 3(x - 8)

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The shape resulting from a horizontal cross section, a vertical cross section, and an angled cross section of the cube are:

horizontal cross section: square.

vertical cross section: square.

angled cross section: rectangle.

In this exercise, you are required to use a graphing tool to investigate and determine the cross-sections of the given three-dimensional geometric object (cube), by passing different planes through them.

By critically observing the cross-sections of the three-dimensional geometric object (cube) I used, we can reasonably infer and logically deduce the following types of quadrilateral based on the cross-sections;

A horizontal cross section would produce a square.

A vertical cross section would produce a square.

An angled cross section would produce a rectangle.

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Answer:

$526.25, and your total monthly expense (including rent and maintenance fee) is $2,326.25.

Step-by-step explanation:

To calculate your average monthly expense, we need to first add up all of your annual expenses and divide by 12 (the number of months in a year) to find the monthly average.

Annual expenses:

Insurance: $325

Lost interest: $50

Utility bills: $5,940

Total annual expenses = $325 + $50 + $5,940 = $6,315

To find the average monthly expense, we divide the total annual expenses by 12:

Average monthly expense = Total annual expenses / 12

Average monthly expense = $6,315 / 12

Average monthly expense = $526.25

In addition to the annual expenses, you also pay a monthly rent of $1,700 and a monthly fee for all maintenance. Let's assume that the monthly maintenance fee is $100.

Therefore, your total monthly expense would be:

Total monthly expense = Monthly rent + Monthly maintenance fee + Average monthly expenses

Total monthly expense = $1,700 + $100 + $526.25

Total monthly expense = $2,326.25

So, your average monthly expense is $526.25, and your total monthly expense (including rent and maintenance fee) is $2,326.25.

The true statements include

Rigid transformation is a type of geometric transformation in which the size and shape of an object remain unchanged. Rigid transformations include

In a rigid transformation, the position of the object in space may change, but its size, shape, and orientation remain the same.

In the object in the graph the length of the segments remain same and since the translation affects only y direction the x-coordinate will be unchanged.

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The only option given that has a variable raised to the third power is D. -2x³, so the correct answer is D.

In algebra, a monomial is an expression that consists of a single term, which is a product of a constant coefficient and one or more variables raised to non-negative integer powers. In each of these expressions, there is only one term, and each term is a product of a constant coefficient and one or more variables raised to non-negative integer powers. Monomials are important in algebra because they can be combined using various operations such as addition, subtraction, multiplication, and division, to form more complex expressions.

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Row 1 in the matrix multiplied by 1/4 will produce: 1/2, 0, and 3/4 respectively.

A rectangular array of numbers or mathematical objects which are arranged in rows and columns is called a matrix. Each row of a matrix is a horizontal sequence of numbers or objects that are separated by commas and enclosed within square brackets, and it represents a vector in the row space of the matrix.

row 1 of the given matrix are: 2, 0, and 3, multiplying row 1 with 1/4 gives;

1/4 × 2 = 1/2

1/4 × 0 = 0

1/4 × 3 = 3/4

Therefore, row 1 in the matrix multiplied by 1/4 will produce: 1/2, 0, and 3/4 respectively.

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It should be noted that f(x) is increasing on the intervals (-1/2, 0) and (0, ∞), and decreasing on the interval (-∞, -1/2).

The derivative of f(x) is denoted as:

f'(x) = 2x - 1 - 1/x

It should be noted that to determine the critical points, set f'(x) to equal 0 and solve for x:

2x - 1 - 1/x = 0

Multiplying by x gives:

2x^2 - x - 1 = 0

On the intervals (-1/2, 0) and (0, ∞), the function f(x) exhibits increasing behaviour; in (-∞, -1/2). it's decreasing.

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The answer for given polynomial is 4,1,-1.

A polynomial is made up of two terms, namely Poly (which means "many") and Nominal (which means "terms."). An expression that consists of variables, constants, and exponents and that is combined using mathematical operations such as addition, subtraction, multiplication, and division (no division operation by a variable) is referred to as a polynomial. The expression is divided into three categories: monomial, binomial, and trinomial depending on how many terms are included in it.

The zeros of a function are the values of its variables that meet the equation and result in the function's value being equal to 0. The zeros of a function can be represented graphically as the x-coordinates (or x-intercepts) where the graph intersects the x-axis. The discriminant formula allows us to identify whether the zeros of a quadratic function are real, complex, or repeating. As a result, when a function f(x) = 0, its zeros are x values. Therefore, if f(a) = 0, then 'a' is a zero of f(x).

according to question,

(((([tex]x^{4}[/tex])-(3•(x³)))-5x²)+3x)+4  = 0

 (((([tex]x^{4}[/tex]) -  3x³) -  5x²) +  3x) +  4  = 0

Polynomial Long Division

Dividing :  x4-3x3-5x2+3x+4

                             ("Dividend")

By         :    x-4    ("Divisor")

dividend     x4  -  3x3  -  5x2  +  3x  +  4

- divisor  * x3     x4  -  4x3            

remainder         x3  -  5x2  +  3x  +  4

- divisor  * x2         x3  -  4x2        

remainder          -  x2  +  3x  +  4

- divisor  * -x1          -  x2  +  4x    

remainder              -  x  +  4

- divisor  * -x0              -  x  +  4

remainder                    0

Quotient :  x3+x2-x-1  Remainder:  0

the answer for the given polynomial by hit and trial method is 4,1,-1.

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The zeros of the polynomial is -1 , 1 and 4.

what is polynomial?

A polynomial is made up of two terms, namely Poly (which means "many") and Nominal (which means "terms."). An expression that consists of variables, constants, and exponents and that is combined using mathematical operations such as addition, subtraction, multiplication, and division (no division operation by a variable) is referred to as a polynomial. The expression is divided into three categories: monomial, binomial, and trinomial depending on how many terms are included in it.

Here the given polynomial is g(x) = [tex]x^4-3x^3-5x^2+3x+4[/tex]

To find zeros , take g(x) = 0 then

=> [tex]0=x^4-3x^3-5x^2+3x+4[/tex]

=> [tex]-x^4+3x^3+5x^2-3x-4=0\\[/tex]

=> [tex]-x^4+x^3+2x^3-2x^2+7x^2-7x+4x-4=0[/tex]

=> [tex]-x^3(x-1)+2x^2(x-1)+7x(x-1)+4(x-1)=0[/tex]

=> [tex]-(x-1)(x^3-2x^2-7x-4)=0[/tex]

=> [tex]-(x-1)(x^3+x^2-3x^2-3x-4x-4)=0[/tex]

=> [tex]-(x-1)(x^2(x+1)-3x(x+1)-4(x+1)=0[/tex]

=> [tex]-(x-1)(x+1)(x^2-3x-4)=0[/tex]

=> [tex]-(x-1)(x+1)(x^2+x-4x-4)=0[/tex]

=> -(x-1)(x+1)(x+1)(x-4)=0

=> x-1=0 , x+1=0 , x-4=0

=> x = -1,1,4

Hence the zeros of the polynomial is -1 , 1 and 4.

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Answer:

7

Step-by-step explanation:

Felix has spent a total time of 11.4 hours (or 684 minutes) running this month.

we can calculate the total time Felix has spent running this month.

Given:

Warm-up time: 12 minutes

Running time: 20 minutes

Total exercise time in a month: 240 minutes

Step 1: Calculate the total time spent running

Let x be the total time spent running in minutes.

According to the given information, Felix spends 12 minutes on warm-up and 20 minutes on running.

So, the equation would be:

12 + 20x = 240

Step 2: Solve for x

Subtract 12 from both sides of the equation:

20x = 240 - 12

20x = 228

Divide both sides by 20:

x = 228 / 20

x = 11.4

So, Felix has spent a total of 11.4 hours (or 684 minutes) running this month.

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Answer: 12

Step-by-step explanation: 2 x 6 = 12 = the third side

Answer: The first number is the X-axis and the second is the Y-axis.

Step-by-step explanation:

5, 1, 4, and 2 belong in X. 35, 7, 28, and 14 belong in Y.

The rule is that X always goes first the Y goes after.

Answer: The two numbers are 5 and 12

Step-by-step explanation:

As the least common multiple of two numbers is

60

, the two numbers are factors of

60

.

Factors of

60

are

{

1

,

2

,

3

,

4

,

5

,

6

,

10

,

12

,

15

,

20

,

30

,

60

}

As one of the numbers is

7

less than the other, the difference of two numbers is

7

Among

{

1

,

2

,

3

,

4

,

5

,

6

,

10

,

12

,

15

,

20

,

30

,

60

}

,

3

&

10

and

5

&

12

are the only two pair of numbers whose difference is

7

. But Least common multiple of

3

and

10

is

30

.

Hence, the two numbers are

5

and

12

.

The missing values of function rule, Y for corresponding value of X= (-4 ) and (-2) will be 9 and 3 respectively.

                                              [tex]X|-4 |- 2|\\ Y|\quad9\; | \quad3\;|[/tex]

Now for the given fucntion ,

By substituting the values of x into the equation [tex]y = -3x - 3[/tex] and solving for y, we can determine the values of y for [tex]x = -4\; and -2[/tex].

[tex]For \;x = -4:\\y = -3(-4) - 3\\y = 12 - 3\\y = 9[/tex]

Hence, for x =(-4), y= 9

[tex]For\; x = -2:\\y = -3(-2) - 3\\y = 6 - 3\\y = 3[/tex]

Hence, for x =( -2), y = 3

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Correct Question:Fill in the table using this function rule. y = - 3x-3

[tex]X|-4 |- 2|\\ Y|\quad?\; | \quad?\;|[/tex]

The equations for which 8 is a solution:

are x minus 4 = 4

2 x = 4

StartFraction x Over 8 EndFraction = 1

x + 6 = 2

The equations for which 8 is a solution are:

x - 4 = 4 (if we substitute x=8, we get 8-4=4 which is true)

2x = 4 (if we substitute x=8, we get 2*8=16 which is true)

StartFraction x Over 2 EndFraction = 16 (if we substitute x=8, we get 8/2=4 which is not true)

StartFraction x Over 8 EndFraction = 1 (if we substitute x=8, we get 8/8=1 which is true)

So, the correct answers are:

x minus 4 = 4

2 x = 4

StartFraction x Over 8 EndFraction = 1

x + 6 = 2

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The number of fewer students to get to school by skateboard than by car and bus combined would be 60 fewer students.

To find how many fewer students get to school by skateboard than by car and bus combined, we need to add the number of students who use car and bus and then subtract the number of students who use skateboard.

Car: 30 students

Bus: 40 students

Car + Bus = 30 + 40 = 70 students

Skateboard: 10 students

Difference = (Car + Bus) - Skateboard

Difference = 70 - 10 = 60 students

So, 60 fewer students get to school by skateboard than by car and bus combined.

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Answer: The rate at which the length of each edge is changing is approximately 0.5185 inches per second when the edges are 6 inches long.

Step-by-step explanation:

Let's denote the volume of the cube as V and the length of each edge as x. Given that the volume of a cube is V = x^3, we can find the rate at which the length of each edge is changing.

We're given that the rate of change of the volume is dV/dt = 56 in³/sec. We want to find the rate of change of the length of each edge, which is dx/dt, when the length of each edge is 6 inches.

First, we differentiate the volume equation with respect to time t:

V = x^3

dV/dt = d(x^3)/dt

Using the chain rule:

dV/dt = 3x^2 * (dx/dt)

Now, we know that dV/dt = 56 in³/sec and x = 6 in. Plugging these values into the equation, we get:

56 = 3 * (6)^2 * (dx/dt)

Solving for dx/dt:

56 = 108 * (dx/dt)

dx/dt = 56 / 108

dx/dt ≈ 0.5185 in/sec (rounded to four decimal places)

So, the rate at which the length of each edge is changing is approximately 0.5185 inches per second when the edges are 6 inches long.

The inequality that would represent the possible values for the number of hours walking dogs is 9d + 8c ≥ 130

Inequalities in mathematics can be described as one that is been used in the expression of the relationship between two values that are not equal

It should be noted thed that Inequality  implies not equal with the symbol (≠)” symbols ≥ < > ≤. From the question , number of hours walking dogs (d) Let number of hours clearing tables  an be represented by( c) Hence the  appropriate inequality is 9d + 8c ≥ 130.

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The restrictions on the polynomial expression [tex]\frac{x^2 - 25}{x - 1} \div \frac{x^2 - x - 30}{x^2 - 4x - 12}[/tex] are at x = -5, -2, 1 and 6

From the question, we have

[tex]\frac{27x^2y^3}{45x^4}[/tex]

Divide the variables

So, we have

[tex]\frac{27y^3}{45x^2}[/tex]

Divide 27 and 45 by 9

So, we have

[tex]\frac{3y^3}{5x^2}[/tex]

Hence, the solution is [tex]\frac{3y^3}{5x^2}[/tex], x ≠ 0

Given that

[tex]\frac{x + 2}{x^2 - 5x - 14}[/tex]

Factorize the numerator

So, we have

[tex]\frac{x + 2}{(x + 2)(x - 7)}[/tex]

Divide

[tex]\frac{1}{x - 7}[/tex]

So, the solution is [tex]\frac{1}{x - 7}[/tex] , where x ≠ 7

The hole is given as (2, 1/3)

This means that the graph is undefined at (2, 1/3)

One possible equation from the list of options is

[tex]f\left(x\right)\:=\:\frac{x\:-\:2}{x^2\:-\:x\:-\:2}[/tex]

The expression is given as

[tex]\frac{x^2 - 25}{x - 1} \div \frac{x^2 - x - 30}{x^2 - 4x - 12}[/tex]

The restrictions on the polynomial is the domain

When solved graphically, we have the restrictions to be at x = -5, -2, 1 and 6

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The rate at which the college student is earning credits, given the semesters taken, would be E. 14 credits per Semester.

The formula to find the number of credits the student takes per semester, would be :

( total credits after 3 semesters - total credits at the start ) / number of semesters = credits per semester

total credits after 3 semesters = 54 credits

total credits at the start = 12 credits

The credits per semester is therefore;

= ( 54 - 12 ) / 3

= 42 / 3

= 14 credits per semester

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The complete question is:

Which of these expresses the rate at which she is earning credits? Select the correct answer below:

3 credits per semester

42 credits per semester

14 semesters per credit

12 credits per semester

14 credits per Semester

7 semesters per credit

Answer:

Step-by-step explanation:

We can simplify the given product using the properties of radicals:

√10x² √5x = √(10*5)x^(2+1) = √50x^3

We can further simplify the radical by factoring out the largest perfect square from 50, which is 25:

√50x^3 = √(25*2)x^3 = 5x√2

Therefore, the product √10x² √5x is equivalent to 5x√2, when x≥0.

To simplify the product √10x² √5x, we can combine the square roots and simplify:

√10x² √5x = √(10x² * 5x) = √(50x³) = √(25x² * 2x) = 5x√2x

Therefore, the choice that is equivalent to the original product when x≥0 is A. 5x√2x.

Answer:

1228

Step-by-step explanation:

To find the total square footage of carpet needed for the house, we first need to calculate the total square footage of the areas where we do not need carpet and then subtract that from the total square footage of the house.

The kitchen is 8 ft by 10 ft, so its area is:

8 ft x 10 ft =80 square feetThe dining room is 12 ft by 14, ft so its area is:

12 ft x 14 ft = 168 square feetThe bathroom is 4 ft by 6 ft, so its area is:

4 ft x 6 ft = 24 square feetTherefore, the total square footage of the areas where we do not need carpet is:

80 + 168 + 24 = 272 square feetTo find the total square footage of carpet needed for the house, we can subtract this from the total square footage of the house:

1500 - 272 = 1228 square feetTherefore, we need 1228 square feet of carpet for the house.

To calculate the amount of carpet needed, subtract the square footage of the rooms where you don't want new carpet (kitchen, dining room, bathroom) from the total house size. This results in 1228 square feet of carpet required.

First, let's determine the total square footage of the spaces where you don't want new carpet. The kitchen is 8 ft by 10 ft, which equals 80 square feet. The dining room is 12ft by 14ft, giving us 168 square feet. The bathroom is 4ft by 6 ft, equal to 24 square feet. When you add these together, that is a total of 272 square feet that should not be carpeted.

Since your total house size is 1500 square feet, we subtract the square footage of the kitchen, dining room, and bathroom from this total. So, 1500 square feet - 272 square feet equals 1228 square feet. Therefore, you need 1228 square feet of carpet for the house.

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The total surface area of the cylinder with a radius of 8 units and a height of 5 units is 653.12 sq. units.

What is surface area?

The quantity of space enclosing a three-dimensional shape's exterior is its surface area. A three-dimensional shape with height, breadth, and depth is referred to as having three dimensions, and surface area is a measurement of the total area that the surface of that shape occupies. In other terms, the surface area is the sum of the areas of all an object's sides.

Here, we have

Given:

Radius of cylinder = 8 units

Height of cylinder = 5 units

Thus:

Total Surface Area of a cylinder = 2 × 3.14 × 8 (8 + 5)

Total Surface Area of a cylinder = 50.24(13) = 653.12 sq. units.

Hence, the total surface area of the cylinder with a radius of 8 units and a height of 5 units is 653.12 sq. units.

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Therefore, the value of f(m-2) is 5/(m-2) + 2, and the value of function g(1/2) is 2.

In mathematics, a function is a rule or correspondence between two sets, where each input value from the first set (called the domain) corresponds to exactly one output value in the second set (called the range). A function is often represented by a formula, equation, or graph.

Here,

1. To find f(m-2), we substitute (m-2) for x in the expression for f(x):

f(m-2) = 5/(m-2) + 2

So the value of f(m-2) is 5/(m-2) + 2.

2. To find g(1/2), we substitute 1/2 for x in the expression for g(x):

g(1/2) = -2(1/2) + 3

So the value of g(1/2) is -1 + 3, which simplifies to 2.

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A rigid transformation and the three types of rigid transformations are: B. Rigid Transformations have congruent preimages and images. Examples include translations, reflections, and rotations.

In Mathematics and Geometry, a transformation can be defined as the movement of a point from its initial position to a new location. This ultimately implies that, when a geometric figure or object is transformed, all of its points would also be transformed.

Generally speaking, there are three (3) main types of rigid transformation and these include the following:

In conclusion, rigid transformation are movement of geometric figures where the size and shape does not change because they have congruent preimages and images.

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The required values are as follows:

Center = (-5, 6), Radius = 12, Domain: x ∈ (-17, 7), Range: y ∈ (-6, 18)

Given equation: x² + y² + 10x - 12y - 83 = 0

To convert it into standard form, complete the square for both the x and y terms:

(x² + 10x) + (y² - 12y) = 83

(x² + 10x + 25) + (y² - 12y + 36) = 83 + 25 + 36

(x + 5)² + (y - 6)² = 144

Comparing this with the standard form, we have:

Center (h, k) = (-5, 6)

Therefore, the coordinates for the point J are (-5, 6).

The radius represents the distance from each band member to the center of the circle. In this case, the radius of the circle is the square root of the constant term on the right side of the standard form equation:

Radius = √144 = 12

The domain of the circle is the set of all possible x-values that lie on the circle. Since the circle is centered at (-5, 6) and has a radius of 12, the domain can be expressed as:

Domain: x ∈ (-5 - 12, -5 + 12) or -17 ≤ x ≤ 7

The range of the circle is the set of all possible y-values that lie on the circle. Using the same information, the range can be expressed as:

Range: y ∈ (6 - 12, 6 + 12) or -6 ≤ y ≤ 18

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Answer: The value of "a" is given by a = 25 - 2b, where "b" is a positive number.

Step-by-step explanation:

Given that log(a) = log15(b) = log25(a + 2b), we can use the properties of logarithms to solve for the value of a.

Since log(a) = log15(b), we can equate the bases and eliminate the logarithms:

a = 15^log15(b) .....(1)

Similarly, since log(a) = log25(a + 2b), we can equate the bases and eliminate the logarithms:

a = (a + 2b)^log25(a + 2b) .....(2)

Now, we can equate the right-hand sides of equations (1) and (2) since they are both equal to a:

15^log15(b) = (a + 2b)^log25(a + 2b)

Taking the logarithm of both sides with base 15, we get:

log15[15^log15(b)] = log15[(a + 2b)^log25(a + 2b)]

Using the property that loga(a^x) = x, we can simplify the left-hand side:

log15(b) = log15[(a + 2b)^log25(a + 2b)]

Now, we can equate the bases and eliminate the logarithms:

b = (a + 2b)^log25(a + 2b)

Taking the logarithm of both sides with base (a + 2b), we get:

log(a + 2b)(b) = log(a + 2b)[(a + 2b)^log25(a + 2b)]

Using the property that loga(a^x) = x, we can simplify the right-hand side:

log(a + 2b)(b) = log25(a + 2b)

Since log(a + 2b)(b) = log(a + 2b)/logb(a + 2b) by the change of base formula, we can rewrite the equation as:

log(a + 2b)/logb(a + 2b) = log25(a + 2b)

Now, we can equate the numerators and denominators separately:

log(a + 2b) = log25(a + 2b)

1 = log25(a + 2b)/(log(a + 2b))

Since loga(a) = 1, we can rewrite the equation as:

log25(a + 2b) = log(a + 2b)/(log(a + 2b))

Using the property that loga(a^x) = x, we get:

log25(a + 2b) = 1

This implies that 25^1 = a + 2b, since we are using the definition of logarithm which states that loga(b) = c is equivalent to a^c = b.

Therefore, a + 2b = 25.

Given that a and b are positive numbers, we can deduce that a + 2b > 0.

Solving for a, we get:

a = 25 - 2b

Since a and b are both positive, a = 25 - 2b > 0.

So, the value of a is greater than zero and is given by a = 25 - 2b, where b is a positive number.

a) [tex]f(x) + xg(x) = x^{3} + 7x^{2} + x - 4.[/tex]

b) [tex][f(x)]² = 4x^{4} - 4x^{3} - 15x^{2} + 8x + 16[/tex]

c) [tex]f^{2}(x) = 4x^{4} - 4x^{3} - 15x^{2} + 8x + 16[/tex]

d) [tex]gf(x) = 2x^{4} + 8x^{3} - 13x^{2} - 3x - 8.[/tex]

what is expression ?

It is possible to do mathematical operations like addition, subtraction, division, and multiplication. An expression is put together as follows: Number, expression, and mathematical operator.

a) f(x) + xg(x)

First, we need to find xg(x), which is x multiplied by g(x):

[tex]xg(x) = x(x^{2} + 5x + 2) = x^{3} + 5x^{2} + 2x\\\\f(x) + xg(x) = 2x^{2} - x - 4 + x^{3} + 5x^{2} + 2x\\\\f(x) + xg(x) = x^{3} + 7x^{2} + x - 4[/tex]

Therefore, [tex]f(x) + xg(x) = x^{3} + 7x^{2} + x - 4.[/tex]

b) [f(x)]²

To square f(x), we can simply multiply it by itself:

[f(x)]² = (2x² - x - 4)(2x² - x - 4)

[tex][f(x)]² = 4x^{4} - 4x^{3} - 15x^{2} + 8x + 16[/tex]

c) f²(x)

Since f²(x) means f(x) times f(x), we can use the distributive property of multiplication:

[tex]f^{2} (x) = f(x) * f(x)\\\\f^{2}(x) = (2x^{2} - x - 4)(2x^{2} - x - 4)[/tex]

[tex]f^{2}(x) = 4x^{4} - 4x^{3} - 15x^{2} + 8x + 16[/tex]

d) gf(x)

To find gf(x), we need to multiply g(x) by f(x):

[tex]gf(x) = (x^{2} + 5x + 2)(2x^{2} - x - 4)[/tex]

Multiplying each term of g(x) by each term of f(x), and simplifying:

gf(x) = 2x⁴ + 8x³ - 13x² - 3x - 8

Therefore, [tex]gf(x) = 2x^{4} + 8x^{3} - 13x^{2} - 3x - 8.[/tex]

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The statements that are always true regarding the diagram are:

m∠1 + m∠5 = 180°

m∠2 + m∠3 = m∠6

m∠3 + m∠4 + m∠5 = 180°

Explain the term diagram

It refers to a value or position that is higher or greater than another value or position. It is often used in comparisons or to describe the relative positions of objects or points on a graph or coordinate plane. The opposite of "above" is "below," which refers to a value or position that is lower or lesser.

According to the given information

We know that the sum of the measures of the three interior angles of a triangle is always 180°. Therefore, we have:

m∠2 + m∠3 + m∠5 = 180°

From the given information, we know that:

m∠2 + m∠1 = 180° (linear pair of angles)

m∠3 + m∠4 = 180° (linear pair of angles)

m∠5 + m∠6 = 180° (exterior angle theorem)

Rearranging these equations, we get:

m∠1 = 180° - m∠2

m∠4 = 180° - m∠3

m∠6 = 180° - m∠5

Substituting these expressions into the above equation for the sum of the interior angles, we get:

m∠2 + m∠3 + m∠5 = m∠2 + (180° - m∠1) + m∠5

m∠3 + m∠4 + m∠5 = (180° - m∠3) + m∠4 + m∠5

Simplifying these equations, we get:

m∠1 + m∠5 = 180°

m∠2 + m∠3 = m∠6

Therefore, the statements that are always true regarding the diagram are:

m∠1 + m∠5 = 180°

m∠2 + m∠3 = m∠6

m∠3 + m∠4 + m∠5 = 180° (which is equivalent to option 2)

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