Question 6 of 25
f(x) = 5x + 3. Find the inverse of f(x).
O A. f-¹(x)=3-5x
OB. f-¹(x) = x-5/3
OC. f-¹(x) = 5x - 3
OD. f-¹ (x) = x-3/5

The equation of a sphere if one of its diameters has endpoint (2, 1, 4) and (4, 3, 10) is (x+1)²+(y-1)²+(z+3)²= 11

Using the distance equation to find the distance of the diameter:

d² = (xf-xi)² + (yf-yi)² +(zf-zi)²

d² = (2-4)²+(1-3)²+(4-10)²

d²=4+4+36=44

d=sqrt(44)

The radius is half the diameter,  which is: r=d/2

r=sqrt(44)/2

To find the middle of the diameter we are using the center equation:

(xf-xi)/2, (yf-yi)/2, (zf-zi)/2

(2-4)/2, (1-3)/2, (4-10)/2

(-1,1,-3) Center

Now, using the equation for a sphere:

(x-xc)²+(y-yc)²+(z-zc)²=r²

(x+1)²+(y-1)²+(z+3)²=(sqrt(44)/2)²

(x+1)²+(y-1)²+(z+3)²=44/4

(x+1)²+(y-1)²+(z+3)²= 11

Thus, the equation of a sphere if one of its diameters has endpoint (2, 1, 4) and (4, 3, 10) is (x+1)²+(y-1)²+(z+3)²= 11

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d² = (xf-xi)² + (yf-yi)² +(zf-zi)²

d² = (2-4)²+(1-3)²+(4-10)²

d²=4+4+36=44

d=sqrt(44)

The radius is half the diameter,  which is: r=d/2

r=sqrt(44)/2

(xf-xi)/2, (yf-yi)/2, (zf-zi)/2

(2-4)/2, (1-3)/2, (4-10)/2

(-1,1,-3) Center

Answer:

(1, -9)

Step-by-step explanation:

The slope-intercept formula of a line is: y = mx + b

m = slope

b = y-intercept

We know one point already.

3 = 3(5) + b

3 = 15 + b

3 - 15 = 15 - 15 + b

-12 = b

So now we know the formula: y = 3x - 12

We can plug in 1 as x to find another point

y = 3(1) - 12

y = 3 - 12

y = -9

Answer:

I'm unable to plot graph, but I can give you the steps to graph the equation y = -²- 10x - 16 on the given set of axes.

Plot the vertex of the parabola: The vertex of the parabola is the point where the parabola changes direction. To find the vertex, you can use the formula (-b/2a, f(-b/2a)), where a, b, and c are the coefficients of the equation y = ax² + bx + c. In this case, a = -1, b = -10, and c = -16, so the vertex is (-10/2(-1), -(-10/2(-1))² - 10(-10/2(-1)) - 16) = (5,16).

Plot the roots: The roots of the equation are the values of x that make y = 0. To find the roots, you can set y = 0 and solve for x. In this case, y = 0 = -x² - 10x - 16. This equation can be solved by factoring or by using the quadratic formula. The roots are x = 4 and x = -2.

Plot five points on the parabola, including the vertex and the roots. To do this, you can substitute different values of x into the equation y = -x² - 10x - 16 and find the corresponding y-values. For example, you could plot the point (4,0), which is one of the roots, and (5,16), which is the vertex.

Connect the points to form a parabola.

Keep

Step-by-step explanation:

The two data sets for which a linear regression model is reasonable include the following: a) Aaliyah’s and Julio’s data sets.

In Mathematics, a positive correlation is used to described a scenario in which two variables move in the same direction and are in tandem.

This ultimately implies that, a positive correlation exist when two variables have a linear relationship or are in direct proportion. This ultimately implies that, when one variable increases, the other increases, as well.

In order to determine if a data set has a good linear relationship, we would have to interpret the correlation coefficient (r), which typically ranges between -1 to 1 as depicted by both Aaliyah’s and Julio’s data sets.

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Complete Question:

Aaliyah, Julio, Dan, and Miriam are each studying sets of data. Aaliyah is working with 14 data pairs with a correlation coefficient of 0.9. Julio is working with 13 data pairs with a correlation coefficient of –0.8. Dan is working with 10 data pairs with a correlation coefficient of 0.1. Miriam is working with 15 data pairs with a correlation coefficient of –0.2. For which two data sets is a linear regression model reasonable?

a)Aaliyah’s and Julio’s data sets

b)Julio’s and Miriam’s data sets

c)Dan’s and Miriam’s data sets

d)Miriam’s and Aaliyah’s data sets

Answer:The American colonies declared independence from British rule in 1776 because they were seeking to establish themselves as a sovereign nation, separate from British control and governance.

The colonies had grown increasingly discontent with British rule and policies, particularly those related to taxation and trade. The colonists felt that they were not being fairly represented in British government and that the laws and taxes imposed on them by Britain were unjust and burdensome.

In 1765, the British Parliament passed the Stamp Act, which placed a tax on printed materials such as newspapers, legal documents, and books. The colonists saw this as a violation of their rights as British citizens, as they had no representation in Parliament and felt that they were being taxed without their consent. This led to widespread protests and resistance to the tax, known as the Stamp Act riots.

In 1773, the British government passed the Tea Act, which granted a monopoly on the sale of tea to the British East India Company and imposed taxes on tea imported to the colonies. This led to the famous Boston Tea Party, in which colonists dumped 342 chests of tea into the Boston Harbor as a protest against the tax.

These and other events, such as the Quartering Acts and the Intolerable Acts, which were designed to punish the colonies for their resistance, further fueled the colonists' anger and resentment towards British rule.

In 1774, the First Continental Congress was held, in which representatives from the colonies met to discuss a unified response to British policies. The Congress adopted a Declaration of Rights and Grievances, in which they demanded that their rights as British citizens be respected and that they be given fair representation in Parliament.

In 1775, the American Revolutionary War broke out between the colonies and Great Britain. On July 4, 1776, the Second Continental Congress adopted the Declaration of Independence, which stated that the colonies were no longer subject to British rule

Step-by-step explanation:

Answer: The colonists joined together to fight Britain and gain independence.

Step-by-step explanation: Many colonists were angry because no one represented their needs in the British government. Colonists believed they did not have self-government. The British forced colonists to allow British soldiers to sleep and eat in their homes.

Therefore, if the borrower had paid the loan just at the end of the second year, he would have owed R 167250. The additional interest due would be R 167250 - 150000 = R 17250.

a) To calculate the amount of money that the person would have to pay at the end of the second year to clear his debt, you can use the formula for compound interest:l,

A = P(1 + r/n)^(nt)

Where A is the final amount, P is the initial principal, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the number of years.

In this case, the person took a loan of R 150000 for 2 years at an annual interest rate of 10%. He paid R 85000 at the end of the first year. So, the new principal at the beginning of the second year is P = 150000 - 85000 = 65000.

So, A = P(1 + r/n)^(nt)

A = 65000(1 + 0.1)^2

A = 65000*1.1^2

A = 73,650

So, the person should have to pay R 73650 at the end of the second year to clear his debt.

b) If the person had paid the loan only at the end of the second year, the interest would have been calculated on the original principal of R 150000 for 2 years. So,

A = P(1 + r/n)^(nt)

A = 150000(1 + 0.1)^2

A = 150000*1.1^2

A = 167250

So, the person would have had to pay R 167250 if he had paid the loan only at the end of the second year. Therefore, he would have to pay more interest of R 167250 - 150000 = R 17250.

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Answer:

You start your first timer, and let it run out, then either let the second timer run for 2 minutes, then take the eggs out.

Step-by-step explanation:

The maximum value of log(x)/ x is 1/e.

The function given is f(x) = log(x)/ x

As base is not mentioned assuming that the base is e.

To find maximum value, let us find the critical point of this function equating its first derivative to 0.

d/dx (f(x) = log(x)/ x) = ( 1 - log(x))/ x²

So d/dx (f(x)) = 0 when 1 - log(x) = 0

That is log(x) = 1

So x = e

f''(x) = (2log(x) - 3)/ x³

f''(e) = (2log(e) - 3)/ e³ = -1/e³ < 0

So at x = e, f(x) is maximum.

Thus maximum value of f(x) is at x = e

f(e) = log(e)/ e = 1/e

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The reduced row-echelon form of the augmented matrix for the system of equations is: 1 0 0 1 | 8, 0 1 0 -2 | -2, 0 0 1 0 | 0. This means that the first equation is reduced to x_1 + x_4 = 8, the second equation is reduced to x_2 - 2x_4 = -2, and the third equation is reduced to x_3 = 0.

Step 1: Subtract 2 times row 1 from row 2 to get row 2 in the form [0 1 0 -2 | -2].

Step 2: Divide row 2 by -2 to get row 2 in the form [0 1 0 1 | 8].

Step 3: Subtract 3 times row 2 from row 1 to get row 1 in the form [1 0 0 1 | 8].

Step 4: Subtract 4 times row 2 from row 3 to get row 3 in the form [0 0 1 0 | 0].

Step 5: The reduced row-echelon form of the augmented matrix for the system of equations is now: [1 0 0 1 | 8], [0 1 0 -2 | -2], [0 0 1 0 | 0]. This means that the first equation is reduced to x_1 + x_4 = 8, the second equation is reduced to x_2 - 2x_4 = -2, and the third equation is reduced to x_3 = 0.

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The zeros of function p(x) = 5 - x² are x = √5 and x = -√5

Consider given polynomial function.

p(x) = 5 - x²

We know that the zeros of a function are the values of the variable of the function that makes the function value equal to 0.

To find the zeros of function p(x), we need to equate p(x) to zero then solve it for x.

⇒ p(x) = 0

⇒ 5 - x² = 0

⇒ 5 - x² - 5 = 0 - 5          ....(subtract 5 from each side of equation)

⇒ -x² = -5

⇒ x² = 5                            ......(multiply both the sides by -1)

⇒ x = ±√5                         .........(taking square-root of both the sides)

Thus x = √5 and x = -√5 are zeros.              

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The complete question is:

What are the zeros of p(x) = 5 - x²?.

Answer: C. 145 degrees

Step-by-step explanation:

Using the information given above, we are now supposed to figure out the measure of angle C. Considering the two angles are supplementary, or two angles with the sum of 180°, we can finally assume that the answer is C.

Angle B + Angle C = 180°

Angle B = 35°

180° - 35° = 145°

Answer: C. 145 degrees

The scale factor used in the scale drawing which has the scale of 6 centimeters = 7 meters is 116.667

Scale factors are used to increase or decrease image. The situation of increment is usually called magnifying.

For the scale drawing with a scale of scale of 6 centimeters = 7 meters

the units are converted to be same, using the conversion unit of

100 cm = 1 m

x cm  = 7 meters'

solving for x

x = 700 cm

hence the scale of the drawing is 6 cm = 700 cm, we solve for the scale factor

6 cm = 700 cm

1 cm = scale factor

solving for the scale factor

scale factor = 700 / 6

= 350 / 3

the factor is 116.667

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A parabola is a u shaped curve. It is a plane curve whose all points are at a fixed distance form a point called focus.

A parabola is formed by cutting a right circular cone in half along a plane perpendicular to the cone's generator. It is a locus of a moving point whose separation from the focus is equal to its separation from a stationary line (directrix)

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The cosine function (also known as the cos function) in triangles is the ratio between the neighbouring side and the hypotenuse.

The complement of sine and one of the three primary trigonometric functions, cosine is one of the three basic trigonometric functions.

The cosine of a right triangle is the ratio between the lengths of the adjacent side and the longest side, or hypotenuse.

Adjacent Side/Hypotenuse = cos theta

If the ratio between the side and hypotenuse is known, the cos inverse function can be used to calculate the angle of any right-angled triangle. The symbol for the cos inverse is [tex]cosx^{-1}[/tex].

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The cosine function (also known as the cos function) in triangles is the ratio between the neighbouring side and the hypotenuse.

The complement of sine and one of the three primary trigonometric functions, cosine is one of the three basic trigonometric functions.

The cosine of a right triangle is the ratio between the lengths of the adjacent side and the longest side, or hypotenuse.

Adjacent Side/Hypotenuse = cos theta

If the ratio between the side and hypotenuse is known, the cos inverse function can be used to calculate the angle of any right-angled triangle. The symbol for the cos inverse is .

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Based on the constructed difference table, the next term of this sequence 6, −2, −3, 7, 32, 76, . . . is 143.

In Mathematics, a sequence can be defined as a series of real and natural numbers in which each term or list of elements are ordered in a particular order and repetitions are allowed.

In Mathematics, a difference table can be defined as a type of table that is constructed by subtracting adjacent elements (entries) in a sequence, and then repeating the process with next those elements (entries).

Number            Difference from previous             Difference of differences

6

-2                               -8                                          

-3                               -1                                                7            

7                                 10                                              11            4

32                               25                                             15            4

76                              44                                               19            4

143                            67                                                23           4

First column: -2 - 6 = -8; -3 - (-2) = 1; ....... 143 - 76 = 67.

Second column: -1 - (-8) = 7; 10 - (-1) = 11; .... 67 - 44 = 23.

Third column: 11 - 7 = 4; 15 - 11 = 4 .... 23 - 19 = 4.

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To find log7 to the base 2 on a calculator, you will need to use the change of base formula that I mentioned earlier: log_b(x) = log_c(x) / log_c(b).

If you want to find log7 to the base 2 on a calculator, you can use the following steps:

Press the log button on your calculator to change to the logarithm mode (usually represented by "log" or "ln").

Enter the number 7

Press the "log" or "ln" button again to switch to the base 2 logarithm mode.

Enter the number you want to find the logarithm of.

Press the division button (/)

Press the log or ln button again and enter the number 7

Press the equals button (=)

The result will be the logarithm of the number with respect to base 2.

Alternatively, you can find the logarithm to base 2 of 7 by dividing the logarithm to base 10 of 7 by logarithm to base 10 of 2 (or natural logarithm of 7 by natural logarithm of 2)

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The correct pairs pf points and axis of symmetry  are  (-1,-8) & y=-8 respectively.

The vertex of a parabola is the point at the intersection of the parabola and its line of symmetry. For a parabola whose equation is given in standard form , the vertex will be the minimum (lowest point) of the graph if and the maximum (highest point) of the graph if .

Given here; A parabola with upwards curve.

Now we know, The vertex is the highest point if the parabola opens downward and the lowest point if the parabola opens upward. The axis of symmetry is the line that cuts the parabola into 2 matching halves and the vertex lies on the axis of symmetry.

Thus we can clearly observe from the graph that (-1,-8) is the vertex of the parabola and we know the parabola is symmetric around its vertex.

Therefore the axis of symmetry of the parabola is y=-8

Hence, The correct pairs pf points and axis of symmetry  are  (-1,-8) & y=-8 respectively.

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The equation of the degree 6 polynomial with a root at 3, a double root at 2, and a triple root at -1, and y-intercept at y = 5 is given as follows:

y = -5/12(x - 3)(x - 2)²(x + 1)³.

The equation of the function is obtained considering the Factor Theorem, as a product of the linear factors of the function.

The zeros of the function, along with their multiplicities, are given as follows:

Then the linear factors of the function are given as follows:

The function is then defined as:

y = a(x - 3)(x - 2)²(x + 1)³.

In which a is the leading coefficient.

When x = 0, y = 5, due to the y-intercept, hence the leading coefficient a is obtained as follows:

5 = -12a

a = -5/12

Hence the polynomial is:

y = -5/12(x - 3)(x - 2)²(x + 1)³.

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The vertex of the graph of  y = 3x² + 2x + 1 is (-0.33, 1.33)

We know that the general form of a parabola is given by an equation y = ax² + bx + c.

the x coordinate of the vertex of the quadratic equation is given by the formula -b/(2a)

To find y-coordinate we substitute value of x in given quadratic equation and solve it for x.

Consider given quadratic equation y = 3x² + 2x + 1

Comparing given quadratic equation y = 3x² + 2x + 1 with y = ax² + bx + c we get, a = 3, b = 2 and c = 1

The x-coordinate of vertex would be,

-b/(2a)

= -2/(2 × 3)

= -2/6

= -1/3

= - 0.33

substitute x = -1/3 in given quadrtic equation in order to find the y-coordinate.

 y = 3(-1/3)² + 2(-1/3) + 1

y = (-1)² - 2/3 + 1

y = 2 - 2/3

y = 4/3

y = 1.33

So, (-1/3, 4/3) is the vertex.

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This equation can be solved using the addition and subtraction property of equality, x + 7 = 13.

An equation is an expression containing an equal sign that states the equality of two mathematical expressions. It is a statement that two expressions have the same value. Equations often involve variables, which are placeholders for unknown numbers. An equation contains at least one operation, such as addition, subtraction, multiplication, or division. Equations are used in a variety of mathematical applications, from basic algebra to more advanced topics such as calculus.

Subtract 7 from both sides to isolate x.

x + 7 - 7 = 13 - 7

x = 6

Therefore, the two step equation that equals 13 is x + 7 = 13.

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A 360-degree angle can be visualized as a full rotation or circle, and it is often used to measure angles in a circular context.

A 360-degree angle is a complete rotation and is equivalent to one full circle. It is often represented as a circle with a line that goes from the center of the circle to the outer edge, marking the angle.

When looking at a 360-degree angle, it can be divided into four quadrants: the top right, top left, bottom left, and bottom right. Each quadrant is 90 degrees or one-fourth of the total angle. This division is useful when analyzing or measuring angles in different sections of a circle.

In terms of visual representation, imagine a clock face with the numbers 1-12. The clock face is a 360-degree angle, with each number representing a 30 degree increment. The hour hand of the clock would rotate around the face, completing a full rotation and reaching the starting point at the 12 o'clock position, completing a 360 degree angle.

In other words, a 360 degree angle can be visualized as a full rotation or circle, and it is often used to measure angles in a circular context. It can also be divided into smaller sections for analysis or measurement purposes.

Therefore, A 360 degree angle can be visualized as a full rotation or circle, and it is often used to measure angles in a circular context.

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Yes. 144 is a perfect square. It is a perfect square of 12. The integer that can be expressed as the square of another integer is called as a perfect square.

A number is said to be a perfect square if its square root is an integer. It  means that a perfect square is an integer's product with itself. It is also defined as the second exponent of an integer. A number squaring a whole number is a perfect square. A perfect square can be determined by the prime factorization method. Prime factorization is defined as the  process of writing all numbers as a product of prime. Through prime factorization we can determine the square root. Every integer has its prime factorization.

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Answer:

QP = 2

QR = 2√3

Step-by-step explanation:

This is a special right triangle with angle measures as follows:

30° - 60° - 90°

And side lengths represented with:

x - x√3 - 2x respectively to angle measures.

The side length that sees angle measure 90, hypotenuse, is given as 4 so the legs would be:

2 and 2√3

The process of solving systems by elimination with multiplication for the equation is explained below.

In math the term equation is known as a mathematical statement that is made up of two expressions connected by an equal sign.

Here we need to write down the steps to solve systems by elimination with multiplication.

Here the first steps id to out the equations in Standard Form.

And then here we need to determine which variable to eliminate.

Finally we have to Multiply the equations and solve.

And this process that allows us to multiply both sides of an equation by the same constant while still having an equivalent expression.

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The distance between two points in a two-dimensional coordinate system represents the length of the shortest path between those two points. This length is also called the Euclidean distance, it is a measure of the straight-line distance between two points.

In other words, it is the length of the line segment connecting the two points. It is calculated using the distance formula which is derived from the Pythagorean theorem, it is calculated as the square root of the sum of the squares of the differences of the coordinates of the two points.

The distance between two points is always a positive value and it is a scalar quantity, it does not have any direction associated with it. It is a measure of the separation between two points, regardless of the direction of the line connecting them.

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Answer: The prime factorization of 630 is 2 x 3 x 5 x 7.

The prime factorization of 1248 is 2^4 x 3 x 7.

So, 630 can be written as 2 x 3 x 5 x 7 and 1248 can be written as 2^4 x 3 x 7

In general, to find the prime factorization of a number, you can divide the number by the smallest prime number (2) and continue dividing by prime numbers until the quotient is a prime number. Then divide that quotient by the next smallest prime number, and so on, until the quotient is 1.

Step-by-step explanation:

The average expenditure which I did by the maximum number of the manual worker is 1847.826

We note that the maximum frequency for the class 1500–2000 is 40. As a result, it belongs to the modal class.

The class interval with the most data points, or what we may refer to as the highest frequency, inside a collection of data is called the modal class.

The class that will include the mode is known as the modal class. The modal class may be determined using the formula I0 I 0 + (f1−f02f1−f0−f2)h ( f 1 − f 0 2 f 1 − f 0 − f 2 ) h

l=1500,h=500,f=40,f

1=24, and f 2=33

∴Mode=(l+ 2f−f1/2f−f2f−f 1 )×h

Mode=1500+ 40−24 / 80−24−33×500

⇒Mode=1500+ 16/23 ×500

=1847.826

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Answer:

Step-by-step explanation:

y= 1/4x + 2.5

Answer: 1) 1,500     2) 36     3) 15     4) 8,400

Step-by-step explanation: Hope it helps!

An expression of more than two algebraic terms is called Polynomial.

Polynomials are made up of two terms: poly (meaning "many") and nominal (meaning "terms"). A polynomial is defined as an expression consisting of variables, constants, and exponents combined by mathematical operations such as addition, subtraction, multiplication, and division (not division by a variable). Based on the number of terms in the expression, it is classified as monomial, binomial, and ternary. Examples of constants, variables, and exponents are:

Polynomials can be classified by degree and power. Based on the number of terms, there are three main types of polynomials:

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