Given the function defined in the table below, find the average rate of change, in simplest form, of the function over the interval 15\le x \le 3015≤x≤30.

The cosecant of ∠G is 1/sin(∠G) = 37/35

The cosecant of an angle is defined as the reciprocal of the sine of the angle. To find the sine of ∠G, we can use the Pythagorean theorem on the right triangle GHI:

The Pythagorean theorem is a mathematical principle that relates the lengths of the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

So, [tex]H^2 = side^2 + side^2[/tex]

[tex]35^2 + 12^2 = GI^2 + IH^2 = HG^2[/tex]

1225 + 144 = 1225 + 144 = [tex]37^2[/tex]

1369 = 1369 = 1369

So, HG = 37.

Hence, the sine of ∠G is 35/37.

Therefore, the cosecant of ∠G is 1/sin(∠G) = 37/35.

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The sum of all primes below 2 million is 142,913,828,922.

The sum of all the primes below two million refers to the total of all prime numbers that are less than two million.

Prime numbers are positive integers that are only divisible by 1 and itself, making them an important aspect of mathematics.

To find the sum of all the primes below two million, we need to first identify all the prime numbers and then add them up.

To identify prime numbers, we use the process of trial division, where we divide the number by all positive integers that are less than or equal to the square root of the number. If no positive integer divides the number, it is considered a prime number.

This can be done using a program that loops through all the numbers and performs the trial division to identify the prime numbers.

The sum of all the primes below two million can also be calculated using a mathematical formula, such as the Prime Number Theorem is written as,

=> 142,913,828,922.

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The solutions to the equation is 4 ± 2√3

The equation x^2 - 8x + 4 = 0 is a quadratic equation.

To find the solutions, we can use the quadratic formula or factor the equation.

Using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

where a = 1, b = -8, c = 4

Plugging in the values:

x = (-(-8) ± √((-8)^2 - 4 * 1 * 4)) / 2 * 1

x = (8 ± √(64 - 16)) / 2

x = (8 ± √(48)) / 2

x = (8 ± 4√(3)) / 2

Divide

x = 4 ± 2√3

So the solutions to the equation are 4 ± 2√3

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So, given the plane curve r(t) = -ti - ln(cos t)j, we obtain T, N, and kappa:

T(t) = -i + tan(t)j

N(t) = sec^2(t) j

kappa(t) = sec^2(t)

A vector is a matrix that has only one row or one column. A row vector is a matrix with only one row. Example Because it only contains one row, the matrix is a row vector. A column vector is a matrix with only one column. If all of the scalars in a matrix are real-valued, it is indicated using uppercase boldface letters, such as A R m n. That is, the matrix exists in a real-valued vector space with m n dimensions. As a result, matrices are just vectors expressed in a two-dimensional table-like format.

Here,

The position vector of the curve is given by:

r(t) = -ti - ln(cos t)j, -π/2 < t < π/2

Differentiating the position vector, we get the tangent vector:

T(t) = dr/dt = -i + tan(t)j

The normal vector is given by:

N(t) = T'(t) = sec^2(t) j

Finally, the curvature is given by:

kappa(t) = ||T'(t)|| = sec^2(t)

So, we have T, N, and kappa for the plane curve r(t) = -ti - ln(cos t)j:

T(t) = -i + tan(t)j

N(t) = sec^2(t) j

kappa(t) = sec^2(t)

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A boxplot consists of several components including a box that shows the interquartile range (IQR) of the data, a line in the box representing the median, and "whiskers" that extend out from the box to show the range of the data.

If the box part of boxplot A is shorter in length than the box part of boxplot B, it suggests that the data in the first dataset (A) has a smaller interquartile range (IQR) compared to the data in the second dataset (B).

This means that the middle 50% of the data in dataset A is more tightly clustered together compared to the middle 50% of the data in dataset B. This can be due to the presence of outliers in the second dataset.

Boxplots are commonly used in statistical analysis and data visualization, but it is not used on a daily basis in every field. They are particularly useful in fields such as finance, biology, and engineering where large amounts of data need to be analyzed and visualized to draw insights and make decisions.

In these fields, they may be used daily or several times a week, depending on the specific use case and the data that needs to be analyzed. However, in other fields such as the arts, humanities, or some service industries, boxplots may not be used as frequently.

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As per the definition of  Laplace equation, none of the functions satisfy the Laplace equation.

The equation ∇²z/∇x² + ∇²z/∇y² = 0 is known as the Laplace equation, which describes the behavior of a scalar field that is in a state of equilibrium, meaning it is not changing over time.

When examining the two functions, z = ey (y cos x - x sin y) and z = e (y cos x - x sin x), it is important to determine if they satisfy the Laplace equation.

For the first function, z = ey (y cos x - x sin y), the partial derivatives can be calculated as follows:

∇²z/∇x² = e^y (cos x + x cos y - y sin x)

∇²z/∇y² = e^y (x sin x - sin y - y cos y)

By adding these two partial derivatives together, we can see that the first function does not satisfy the Laplace equation, as the result is not equal to zero.

For the second function, z = e (y cos x - x sin x), the partial derivatives can be calculated as follows:

∇²z/∇x² = -e (cos x + x cos x - y sin x)

∇²z/∇y² = -e (x sin x - sin y - y cos x)

Once again, by adding these two partial derivatives together, we can see that the second function also does not satisfy the Laplace equation, as the result is not equal to zero.

Therefore, none of the given functions satisfy the Laplace equation.

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0.125 liter of water this jug contain it is between 150ml and 100ml.

The conversion of 1 jug (beer jug) into a volume and capacity measure equals 1.14 liters, according to its commonly used equivalent volume and capacity unit type measure.

Three dimensions need to be taken into account in order to determine the round water tank's liter capacity. The length, breadth, and height make up those three dimensions. Let B, W, and H stand for length, width, and height, respectively. As a result, 240 x 28.31 = 6794.4 liters will be the tank's total volume in liters.

That given jug contain some water

Quantity of water is 125ML

1000ml = 1Liter

125ml = 0.125Liters.

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Two numbers, 4, and -2, satisfy the equation such that =. The expression is 2(x+2)(x-2) and can be stretched to 2x2 - 8 by writing it that way. By putting the expression's value equal to 0, we may get the two solutions for x.

Expanding the formula 2(x+2)(x-2) yields 2x2 - 8. Thus, the two numbers that make the expression equal to 0 are the two solutions for x. The expression is 2(x+2)(x-2) and can be stretched to 2x2 - 8 by writing it that way You must set the equation equal to 0 and solve for x in order to find these values. This will allow you to identify the two x values that satisfy the equation: 4 and -2. With the expression on one side and 0 on the other, this can be compared to a balance act. The two solutions for x are the two values of x that make the equation balanced.

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I followed you since I could not find the awnser I hope thats okay

The slope and y-intercept are 2 and 1 respectively.

In Mathematics, the slope of any straight line can be determined by using this mathematical equation;

Slope (m) = (Change in y-axis, Δy)/(Change in x-axis, Δx)

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Note: Each interval on this graph represent 1 unit.

Based on the information provided, we can logically deduce the following slope and data points on the line:

Substituting the given data points into the slope formula, we have the following;

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope (m) = (3 - 1)/(2 - 1)

Slope (m) = 2/1

Slope (m) = 2.

In conclusion, the y-intercept of this line is at (0, 1).

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Answer:

y=1x+1

Step-by-step explanation:

This triangle is not a right triangle because the sum of the squares of the two smaller sides is not equal to the square of the largest side.

To determine if a triangle is a right triangle, you can use the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two smaller sides is equal to the square of the largest side.

In this case, we can compare the squares of the sides with lengths of 33 km and 72 km, and the square of the side with length 78 km:

33^2 + 72^2 = 5184 + 5184 = 10368

78^2 = 6084

Since 10368 is not equal to 6084, we can conclude that this triangle is not a right triangle.

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1.0, 3.5, 5, 6.5, 10 are the 5 number summary of the given table.

How do you find the 5 number summary?

When conducting descriptive analyses or conducting an initial analysis of a sizable data set, a five-number summary is particularly helpful. The maximum and minimum values in the data set, the lower and upper quartiles, and the median make up a summary's five values. Finding the smallest data value (minimum), the 25th percentile (Q1 - the first quartile), the median (25th percentile, Q2, the second quartile), the 75th percentile (Q3 - the third quartile), and the highest data value will yield the data set's five-number summary (maximum). The minimum, lower quartile (Q1), median (Q2), upper quartile (Q3), and maximum make up the five-number summary. The median is the middle value; Q1 is the median of the first half of the data, and Q3 is the median of the second half.

Smallest value = 1.0

Median of the first half data Q1 = 3.5

Median Q2= 5.0

Median of the second half data Q3 = 6.5

Largest number = 10.0

The lowest is the smallest number, the maximum is the highest, the median is the intermediate value.

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The fully simplified expression for the derivative of f(x)=(x2 6)(2x−5) is f'(x)=6x2−26x+30.

The derivative of f(x)=(x2 6)(2x−5) can be found by first expanding the polynomials. When we expand the polynomials, we get f(x)=2x3−13x2+30x−30. The derivative of this expression is f'(x)=6x2−26x+30.

To calculate the derivative of f(x), we can use the power rule of derivatives. The power rule states that the derivative of a term with a power of n is n times the coefficient of the term, multiplied by the term with a power of n-1. Applying this rule to the terms in f(x) gives us the derivative of f(x).

For the first term, 2x3, the derivative is 6x2, which is the coefficient of the term, 2, multiplied by the term with a power of n-1, x2. For the second term, -13x2, the derivative is -26x, which is the coefficient of the term, -13, multiplied by the term with a power of n-1, x. For the third term, 30x, the derivative is 30, which is the coefficient of the term, 30, multiplied by the term with a power of n-1, 1.

Therefore, the fully simplified expression for the derivative of f(x)=(x2 6)(2x−5) is f'(x)=6x2−26x+30.

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Answer:

Step-by-step explanation:

Maybe just add or multiply the number by something to get the answer

Answer:

500 kilometers

Basically, just find out what is 25% of 2000.

First, you find what 25% is as a decimal and then multiply it by 2000.

25% as a decimal is 0.25

0.25 * 2000 = 500

25% of 2000 is 500.

Step-by-step explanation:

Hope it helps! =D

a, b, d, e are arranged from lowest to highest priority when assigning r/s configuration.

Three chemists, R.S. Cahn, C. Ingold, and V. Prelog, developed the approach for clearly identifying the handedness of molecules; as a result, it is sometimes referred to as the Cahn-Ingold-Prelog rules. There are two methods for figuring out an enantiomer's absolute configuration empirically in addition to the Cahn-Ingold system:

1- analysis of X-ray diffraction. Be aware that the structure of a specific enantiomer has nothing to do with the rotation's sign.

2- Chemical correlation with a molecule whose X-ray diffraction structure has previously been established.

Thus, a, b, d, e are arranged from lowest to highest priority when assigning r/s configuration.

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Therefore , the solution of the given problem of polynomial comes out to be  quotient comes out to be (p³ + 2p² + 6p -10) .

A polynomial is a mathematical statement with coefficients or uncertainty that uses only additions, erasures, multiplications, and powers of positive integer variables. There is just one indeterminate x polynomial identified by the formula x2 4x + 7. The term "polynomial" refers to a phrase in mathematics that consists of variables (sometimes referred to as "indeterminates") and coefficients that can be added, deducted, multiplied, then raised to negative number powers of non-variables.

Here,

Given :

=> (p⁴ -7p³ - 12p² - 64p + 90 )  / (p-9)

=> (p-9) * (p³ + 2p² + 6p -10)  / (p-9)

=>  (p³ + 2p² + 6p -10)

Thus , quotient comes out to be (p³ + 2p² + 6p -10) .

Therefore , the solution of the given problem of polynomial comes out to be  quotient comes out to be (p³ + 2p² + 6p -10) .

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S is linearly dependent and does not span R^2, making it not a basis for R^2.

The set S = {(1, 5), (1, 0), (0, 1)} is not a basis for R^2, as it is linearly dependent and does not span R^2.

A set is considered linearly dependent if one of the vectors in the set can be expressed as a linear combination of the other vectors. In the set S, the vector (1,0) can be expressed as a linear combination of (1,5) and (0,1), meaning that the set is linearly dependent.

A set is considered to span R^2 if every vector in R^2 can be expressed as a linear combination of the vectors in the set. In the set S, the vector (2,5) cannot be expressed as a linear combination of the vectors in S, meaning that S does not span R^2.

Therefore, S is linearly dependent and does not span R^2, making it not a basis for R^2.

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Using three equal subintervals and a right riemann sum, left riemann sum, and a midpoint sum is -6.4, 3.8, -1.0.

1) Integral with left end points:

         A-left = -3.4*2 + (-0.6)*2 + 0.8*2 = -6.4

2) Integral with right end points:

        A-right = -0.6*2 + 0.8*2 + 1.7*2 = 3.8

3) Integral with midpoints:

         A-mid = -2.2*2 + 0.2*2 + 1.5*2 = -1.0

A specific type of approximation of an integral by a finite sum in mathematics is known as a Riemann sum. It bears the name of the German mathematician Bernhard Riemann from the nineteenth century. Approximating the area of functions or lines on a graph, as well as the length of curves and other approximations, is a highly typical use.

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Answer: C is the answer

Step-by-step explanation:

If the matrix is said to be in reduced echelon form then it should satisfy the following conditions:

1. The pivot in any non-zero row is the only non-zero entry in its column.

2. The pivot in any non-zero row is positive.

What is an echelon form?

A matrix in reduced echelon form is also sometimes referred to as a row-reduced echelon form or simply a reduced row echelon form (RREF). To check if a matrix is in RREF, you need to verify if the matrix satisfies both the conditions for echelon form and the additional conditions for RREF.

1. A matrix is said to be in echelon form if it satisfies the following conditions:

2. All zero rows (if any) are at the bottom of the matrix.

3. The leading entry (the first non-zero entry) of each non-zero row occurs to the right of the leading entry of the previous row.

4. The leading entry in any non-zero row is 1, called a pivot.

A matrix is said to be in reduced echelon form if it is in echelon form and additionally satisfies the following conditions:

1. The pivot in any non-zero row is the only non-zero entry in its column.

2. The pivot in any non-zero row is positive.

Hence, if the matrix is said to be in reduced echelon form then it should satisfy the following conditions:

1. The pivot in any non-zero row is the only non-zero entry in its column.

2. The pivot in any non-zero row is positive.

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Answer:

The new ratio of the history to the maths to the science books would be 22:43:55. With the addition of 20 history, 40 maths, and 10 science books, the total number of books in the library is now 570. The new ratio of the history to the maths to the science books is therefore 22:43:55.

Step-by-step explanation:

The volume of the solid is 1789.33.

Volume is defined as a capacity occupied by a three-dimensional solid shape. In any shape, it is hard to visualize but can be compared between shapes. For example, the volume of a compass box is greater than the volume of an eraser placed inside it. For calculating the area of any two-dimensional shape, we divide the portion into equal square units. Similarly, while calculating the volume of solid shapes we will divide it into equal cubical units. The S.I. unit of volume is cubic meter (m3) since volume is a quantity of the three-dimensional space occupied by a shape or surface. However, the most commonly used unit for volume is liter. Apart from this, large and small volumes are measured in other units like milliliter (ml), pints, gallons, and others.

For a square base parallel to the x-axis, you can use the formula:

[tex]V = \int\limits^a_b {f(y)}^{2} \, dy[/tex]

The equation for a circle is x² + y² = r²

⇒ x = √(11² - y²) = √(121 - y²)

Then, if we consider that for a height y, the length x is double, we have that the length of each cross-section is given by:

f(y) = 2√(81 -y²)

With this, we can propose the following integral to obtain the volume that they are asking us:

[tex]V = \int\limits^{11}_0 {f(y)^{2} } \, dx \\V = \int\limits^{11}_0 {(2\sqrt{121-y^{2} } )^{2}} \, dx \\V = 4(121y - \frac{y^{3} }{3} )[/tex]

We get that the volume is V=1789.33.

Thus, the volume of the solid is 1789.33.

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The solid has a volume of 1789.33.

The capacity occupied by a three-dimensional solid shape is known as volume. It is difficult to visualize in any shape, yet it may be compared among shapes. A compass box, for instance, has a bigger volume than an eraser inserted inside of it. We split the area into equal square units to determine the area of any two-dimensional form. Similarly to this, when computing the volume of solid things, we will divide it into equal cubical units. The cubic meter is the SI unit for volume because it measures how much three-dimensional space is occupied by a form or surface (m3). However, the liter is the unit of volume that is most frequently used. Other units used to measure large and small amounts include milliliters (ml), pints, gallons, and others.

The formula is: for a square base parallel to the x-axis.

[tex]V=\int\limits^a_b {f}(y)^2 \, dy[/tex]

The circle's equation is x2 + y2 = r2.

⇒ x = √(11² - y²) = √(121 - y²)

The length of each cross-section is thus determined by: if we assume that at a height y, the length x is twofold.

f(y) = 2√(81 -y²)

As a result, we can offer the following integral to acquire the requested volume:

[tex]V=\int\limits^1_0 {f}(y)^2 \, dx[/tex]

[tex]V= \int\limits^1_0 {(2\sqrt{(121-y^2)^2} } \, dx[/tex]

[tex]V=4(121y-\frac{y^3}{3} )[/tex]

The volume is determined to be V=1789.33.

As a result, the solid has a volume of 1789.33.

The complete question is:-

Use the general slicing method to find the volume of the following solid.

The solid with a semicircular base of radius 11 whose cross sections perpendicular to the base and parallel to the diameter are squares. Place the semicircle on the xy-plane so that its diameter is on the x-axis and it is centered on the y-axis. Set up the integral that gives the volume of the solid. Use increasing limits of integration.

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The Parametric equation is x = 2 + 3t y = -8.

Using the two points as a starting point, we can get the vector equation of the line by first locating the direction vector and then adding it to one of the points to obtain the general equation.

Subtracting the two points yields the direction vector:

b - a = (5, -8) - (2, -8) = (3, 0) (3, 0)

The direction vector is then added to one of the points, and t is used as the parameter to find the general equation of the line through point a parallel to point b:

x = 2 + 3t

y = -8

Therefore, x = 2 + 3t y = -8 is the parametric equation of the line through point a parallel to point b.

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Step-by-step explanation:

1. true

5 is a constant (term).

2. false

7y is not a coefficient. 7 is the coefficient for that y-term.

3. true

7y + 2x + 5 is the sum of 3 terms.

4. false

7y is not a factor. it would have to be multiplied to something else to be a factor for that something else.

5. false

they would be like terms, if both were constants, or both contained y.

6. false

the length of side a'b' will be greater than the length of side ab. The length of side a'b' is 12 units.

Rectangle abcd is a 4-sided shape with sides of length a, b, c, and d. When a rectangle is dilated, it is enlarged or scaled up. The length of the sides of the new rectangle, a', b', c', and d', will be greater than the original sides of the rectangle abcd. Therefore, the length of side a'b' will be greater than the length of side ab. The length of side a'b' is 12 units.

The length of side ab of the new rectangle (a'b'c'd) is 12 units, since it was dilated from 4 units and 6 units (abcd).

The complete question is :

Rectangle abcd has side lengths of 4 units and 6 units. It was dilated to create rectangle a'b'c'd. What is the length of side ab of the new rectangle?

6 units

7. 6 units

9. 5 units

12 units.

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A, b, c, and d-length sides make up the four-sided rectangle abcd. A rectangle is expanded or scaled up when it is dilated. The new rectangle's sides, a', b', c', and d', will all be longer than the rectangle's initial sides, abcd. As a result, side a'b' will have a length that is longer than side ab. Side a'b' measures 12 units in length.

6 units

7. 6 units

9. 5 units

12 units.

The answer to 5.6 to the power of 6 is 95,788.67.

According to the question

The result of multiplying 5.6 by itself 6 times is expressed as 5.6 to the power of 6. In this situation, the number 5.6 is being exponentiated, which is a mathematical process that elevates a number to a specified power, which is the power of six. This formula yields an actual number, which is the result of multiplying 5.6 by 6 times. Both a calculator and manual multiplication may be used to figure this out. The outcome of this computation is represented numerically as 95,788.67, which represents the value that is achieved when the number 5.6 is multiplied by itself six times.

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The answer to 5.6 to the power of 6 is 95,788.67.

According to the question

The result of multiplying 5.6 by itself 6 times is expressed as 5.6 to the power of 6. In this situation, the number 5.6 is being exponentiated, which is a mathematical process that elevates a number to a specified power, which is the power of six. This formula yields an actual number, which is the result of multiplying 5.6 by 6 times. Both a calculator and manual multiplication may be used to figure this out. The outcome of this computation is represented numerically as 95,788.67, which represents the value that is achieved when the number 5.6 is multiplied by itself six times.

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Answer: 5.5

Step-by-step explanation:

0.72÷[tex]0.6^{2*2}[/tex]
Simplify the 2*2 to 4
0.72÷[tex]0.6^{4}[/tex]
[tex]0.6^{4}[/tex]=0.1296
0.72÷0.1296=5.555555556
5.555555556 can be rounded to 5.5

0.72÷[tex]0.6^{2}[/tex] x 2 on the other hand is 4 -- sorry I can't really see if it's both 2's as exponents or not
0.6^2=0.36
0.72÷0.36×2=4

On solving the provided question, we can say that, by solving the equations  that amount is $36.95

A mathematical equation is a formula that joins two statements and uses the equal symbol (=) to indicate equality. A mathematical statement that establishes the equality of two mathematical expressions is known as an equation in algebra. For instance, in the equation 3x + 5 = 14, the equal sign places the variables 3x + 5 and 14 apart. The relationship between the two sentences on either side of a letter is described by a mathematical formula. Often, there is only one variable, which also serves as the symbol. for instance, 2x – 4 = 2.

Suppose x be the number of minutes of local calls when cost of plans are the same

Cost of Plan 1is $27.95+.12x

Cost of Plan 2 is $12.95+.32x

[tex]$27.95+0.12x=12.95+.32x \\[/tex]

[tex]$27.95-$12.95+.12x-.12x=$12.95-$12.95+.32x-.12x\\$15.00=.20x[/tex]

x=75 minutes

[tex]27.95+.12(75)=12.95+.32(75)\\27.95+9.00=12.95+24.00\\36.95=36.95[/tex]

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The required formula for given transformation is  2√(x - 4) - 3.

In mathematics, a graph is a visual representation or diagram that shows facts or values in an ordered way. The relationships between two or more items are frequently represented by the points on a graph.

The formula for the function g(x) can be found by applying the transformations to f(x):

Shift down 3 units: g(x) = f(x) - 3

Shift left 4 units: g(x - 4) = f(x) - 3

So, the formula for g(x) is:

g(x) = f(x - 4) - 3

= 2√(x - 4) - 3.

Thus, required formula is 2√(x - 4) - 3.

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