Recursive Rules from Word Problems

1) Sally started a new job. After depositing her 1st paycheck into her bank account she had
$225. Every week she was paid the same amount. After her 8th paycheck she had $1275 in her
bank account.

Complete the table

The cosine of an angle in a right triangle is defined as the ratio of the length of the leg adjacent to the angle to the length of the hypotenuse.

[tex]tanL = 15/8[/tex][tex]sinL = 15/17[/tex][tex]cosL = 8/17[/tex]

In a right triangle, the side opposite the right angle is called the hypotenuse (in this case, it's the side with length 17).

The other two sides are called the legs. We can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs:

hypotenuse^2 [tex]= leg1^2 + leg2^2[/tex]

For this triangle, we have:

[tex]17^2 = 8^2 + 15^2[/tex]

Simplifying this equation, we get:

[tex]289 = 64 + 225[/tex]

Therefore, the equation is true, and we have verified that this is a right triangle.

Now, we can use the trigonometric ratios to find the values of tanL, sinL, and cosL, where L is the angle opposite the leg with length 8.

The tangent of an angle in a right triangle is defined as the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. Therefore, we have:

tanL = opposite/adjacent [tex]= 15/8[/tex]

The sine of an angle in a right triangle is defined as the ratio of the length of the leg opposite the angle to the length of the hypotenuse. Therefore, we have:

sinL = opposite/hypotenuse [tex]= 15/17[/tex]

The cosine of an angle in a right triangle is defined as the ratio of the length of the leg adjacent to the angle to the length of the hypotenuse. Therefore, we have:

cosL = adjacent/hypotenuse = 8/17

Therefore, , the values we found are:

[tex]tanL = 15/8[/tex]

[tex]sinL = 15/17[/tex]

[tex]cosL = 8/17[/tex]

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The solution of the exponential equation 3^(3x-2) = 9^(4x-1) is x = 0.

We can solve this exponential equation for x by using logarithms. We can take the logarithm of both sides of the equation, using any base that we prefer. For instance, we can use the natural logarithm, ln:

ln(3^(3x - 2)) = ln(9^(4x - 1))

Now, we can use the properties of logarithms to simplify both sides of the equation. First, recall that ln(a^b) = b ln(a), for any positive value of a and any real value of b. Therefore, we have:

(3x - 2) ln(3) = (4x - 1) ln(9)

Next, we can use another property of logarithms, namely ln(a^b) = b ln(a) = ln(c) → a^b = c, to eliminate the natural logarithms from both sides of the equation. Specifically, we can rewrite ln(9) as ln(3^2), and then use the power rule for logarithms, ln(a^b) = b ln(a), to get:

(3x - 2) ln(3) = (4x - 1) ln(3^2) = 2 (4x - 1) ln(3)

Now, we can simplify the equation by multiplying out the coefficients of ln(3) on the left-hand side:

3x ln(3) - 2 ln(3) = 8x ln(3) - 2 ln(3)

Then, we can collect like terms:

3x ln(3) - 8x ln(3) = -2 ln(3) + 2 ln(3)

Finally, we can solve for x by factoring out ln(3) and dividing both sides by the resulting factor:

(3 ln(3) - 8 ln(3)) x = 0

-5 ln(3) x = 0

x = 0

Therefore, the solution of the exponential equation 3^(3x-2) = 9^(4x-1) is x = 0.

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The pair of ratios that does not form a proportion is 103/24 and 5/40.

To check if two ratios form a proportion, we need to simplify them to their simplest form and compare them. If the two ratios are equal after simplification, then they form a proportion.

In this question, we are given five ratios: 103/24, 5/40, -30/15, 10/5, and 3/S.

To simplify the first ratio, we can divide both the numerator and denominator by their greatest common factor, which is 1. Therefore, the simplified form of 103/24 is 4.29 (rounded to two decimal places).

To simplify the second ratio, we can also divide both the numerator and denominator by their greatest common factor, which is 5. Therefore, the simplified form of 5/40 is 0.125.

When we compare these two simplified ratios, we can see that they are not equal. Therefore, the pair of ratios that does not form a proportion is 103/24 and 5/40.

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Which pair of ratios does NOT form a proportion?

103 24

5 40

-30 15

10 5

3 S

The average annual rate of change during the time period 2007-2012 is approximately 10,797 thousand subscribers per year.

The rate of change function is defined as the rate at which one quantity is changing with respect to another quantity. In simple terms, in the rate of change, the amount of change in one item is divided by the corresponding amount of change in another.

To find the average annual rate of change for this time period, we need to determine the total change in the number of cellular telephone subscribers from 2007 to 2012, and then divide by the number of years in the time period.

The total change in the number of subscribers from 2007 to 2012 is:

335,244 - 270,461 = 64,783

The number of years in the time period is:

2012 - 2007 + 1 = 6

So the average annual rate of change is:

64,783 / 6 = 10,797.17 (rounded to two decimal places)

If we round the average annual rate of change to the nearest unit, it would be 10,797 subscribers per year.

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Answer:

a mode of negative number again gives the positive value

d. -(-61) =61 by multiplication sign rule

e. opposite of -61 =61

Answer: e

Step-by-step explanation:

the opposite is 61

The standard form of the expression is 2m² + m.

The standard form is referred to as the general way of representing any type of notation. The standard form formula represents the standard form of an equation which is the commonly accepted form of an equation. For example - The standard form of a polynomial is to write the terms with a higher degree first (descending order of degree) and its coefficients must be in integral form.

Combining like terms, we get:

(m² - m) + (2m + m²) = m² + m² - m + 2m

Simplifying further, we get:

2m² + m

Therefore, the sum of the given expressions is 2m² + m.

To write the answer in standard form, we arrange the terms in descending order of degree of the variable:

2m² + m = 2m² + 1m¹ + 0m⁰

So the standard form of the expression is 2m² + m.

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To calculate the NPV of the project, we need to find the cash inflows and outflows for each year and discount them back to their present value using the corporate discount rate of 15%.

Year 0:

Initial investment = 1,500 million VND

Working capital = -300 million VND

Year 1:

Revenue = 860 million VND

Variable costs = -344 million VND (40% of net sales)

Fixed costs = -250 million VND

Depreciation = -250 million VND (1,500 million VND / 6)

Operating income before taxes = 16 million VND (860 - 344 - 250 - 250)

Taxes = -3.2 million VND (20% of operating income before taxes)

Operating income after taxes = 12.8 million VND (16 - 3.2)

Add back depreciation = 250 million VND

Net cash flow = 262.8 million VND

Discounted cash flow = 228 million VND (262.8 / 1.15)

Year 2:

Revenue = 920 million VND

Variable costs = -368 million VND (40% of net sales)

Fixed costs = -250 million VND

Depreciation = -250 million VND (1,500 million VND / 6)

Operating income before taxes = 52 million VND (920 - 368 - 250 - 250)

Taxes = -10.4 million VND (20% of operating income before taxes)

Operating income after taxes = 41.6 million VND (52 - 10.4)

Add back depreciation = 250 million VND

Net cash flow = 291.6 million VND

Discounted cash flow = 231.9 million VND (291.6 / 1.15^2)

Year 3:

Revenue = 1,050 million VND

Variable costs = -420 million VND (40% of net sales)

Fixed costs = -250 million VND

Depreciation = -250 million VND (1,500 million VND / 6)

Operating income before taxes = 130 million VND (1,050 - 420 - 250 - 250)

Taxes = -26 million VND (20% of operating income before taxes)

Operating income after taxes = 104 million VND (130 - 26)

Add back depreciation = 250 million VND

Net cash flow = 354 million VND

Discounted cash flow = 255.3 million VND (354 / 1.15^3)

Year 4:

Revenue = 1,200 million VND

Variable costs = -480 million VND (40% of net sales)

Fixed costs = -250 million VND

Depreciation = -250 million VND (1,500 million VND / 6)

Operating income before taxes = 220 million VND (1,200 - 480 - 250 - 250)

Taxes = -44 million VND (20% of operating income before taxes)

Operating income after taxes = 176 million VND (220 - 44)

Add back depreciation = 250 million VND

Net cash flow = 426 million VND

Discounted cash flow = 286.6 million VND (426 / 1.15^4)

Year 5:

Revenue = 880 million VND

Variable costs = -352 million VND (40% of net sales)

Fixed costs = -250 million VND

Depreciation = -250 million VND (1,500 million VND / 6)

Operating income before taxes = 28 million VND

By setting up the equation and solving for the unknown variable, we find that the number in question is -15. The answer provides a step-by-step method for solving an equation that represents a word problem.

Let's start by translating the given problem into an equation.

"The product of a number and -6" can be written as "-6x", where "x" is the unknown number. "Five times the sum of that number and 33" can be written as "5(x+33)".

Putting these together, we get:

-6x = 5(x+33)

Now we can solve for "x":

-6x = 5x + 165

-11x = 165

x = -15

Therefore, the number we're looking for is -15.

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Answer:

B

Step-by-step explanation:

Bus A

1. 9:55

2. 10:20

3.10:45

4.11:10

5.11:35

6.12:00

7.12:25

8.12:50

Bus B

1. 10:10

2.10:50

311:30

4. 12:10

5. 12:50

(a) The area of the family room is 146 square feet .

(b) 21.6 tiles of the 9 inches by 9 inches tiles will take to cover the floor.

(c) total number of boxes that Mr. Dieter will buy for the room is 1.8 boxes.

The area of ​​the rectangle is on its side. Basically, the formula for the area is equal to the product of the length and width of a rectangle. And when we talk about the perimeter of a rectangle, it is equal to all four of its sides.

(a) the area of the family room can be determined by calculating the area of each of the shapes and adding the 3 areas together

area of a rectangle = length x breadth

⇒ 16 x 7 = 112 ft²

Area of a triangle = 1/2 x base x height

Area of the smaller triangle = (1/2) x 4 x 3 = 6 ft²

Area of the bigger triangle = (1/2) x 8 x 7 = 28 ft²

Some of the areas = 112 + 6 + 28 = 146 ft²

(b)

1. First convert the area of the room to inches

⇒ 1 ft = 12 in

⇒ 146 x 12 = 1752 in²

2. the next step is to determine the area of the tile

area of a square = length²

⇒ 9² = 81 in²

3. Divide the area of the room by the area of the tile

⇒ 1752 / 81 = 21.6 tiles

(c)

total number of boxes that would be bought = 21.6 /12 = 1.8 boxes.

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Answer:

X-intercept(s): (-3,0), (-8,0)

Y-intercept(s): (0,48)

Vertex: (-11/2,-25/2)

Step-by-step explanation:

X-intercepts: When you are finding the x-intercepts, there are two ways to find your x-intercepts like you can find in the Quadratic Formula or plug the equation into your calculator and see it on the graph/ table. If you like the quadratic formula, you need plug into the a, b, and c and it will look like x= -22±√(22^2)-4(2)(48))/2(2). It will be the same answer. on the another hand, if you like the calculator way, you get your calculator that need be TI-84 plus CE or TI-84 then go on y= then put your equation like 2x^2+22x+48 after that you click on graph. if you cant find the x-intercepts on the graph, you can do 2nd then above the graph buttom you can see it in the table. When you looking at the table, look for the 0s in the y table.

Y-intercept: When you are finding the y-intercepts, there are two ways to find your y-intercepts like you need plug in 0 for x or plug the equation into your calculator and see it on the graph/ table. If you like the quadratic formula, you need plug in 0 for x and it will look like y=2(0)^2 +22(0)+48. it would the get the same answer. if you like the calculator way, you get your calculator that need be TI-84 plus CE or TI-84 then go on y= then put your equation like 2x^2+22x+48 after that you click on graph. if you cant find the x-intercepts on the graph, you can do 2nd then above the graph buttom you can see it in the table. When you looking at the table, look for the 0s in the x table.

Vertex:

1. Get your equation in the form like this y=ax^2+bx+c

2. Calculate -b/2a. This is the x-coordinate of the vertex.

3. To find the y- coordinate of the vertex, simply plug the x to the quadratic equation and solve for y.

Comment if i am right or wrong

Option D is the correct answer, As a result, LM is 16.1 centimeters long.

The mid-segment of a triangle is aligned to the base and is half the length of the triangle.

MQ = therefore 1/2 LN.

LN = 16.1 centimeters because we know MQ = 8 cm.

MQ is parallel to side LM and half of its length because it is also the midpoint of the triangular LMN.

A line segment linking the midpoints of two of the triangle's sides is known as a mid-segment. It is half the length of the triangle's third edge and parallel to it. In other words, the mid-segment linking the midpoints of sides a and b has length c/2 and is parallel to side c if a triangle has sides of lengths a, b, and c.

Because of this, LM = 2MQ = 16.1 centimeters.

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As the calculations show, account B will yield a bigger total sum after two  interest  years than account A. As a result, we should deposit our funds in account B.

In mathematics, interest is the amount of money earned or payable on an original investment or loan. You can use either simple or compound interest. Simple interest is calculated as a percentage of the initial amount, whereas compound interest is calculated on the principal amount plus any previously earned interest. If you invest $100 at a 5% annual simple interest rate, you will get $5 in interest every year for three years, for a total of $15.

Account A: Semi-annual interest rate = 10% divided by two equals 5%.

2 years multiplied by 2 equals 4 compounding periods.

Total interest earned Equals Rs. 30,000 multiplied by (1 + 5%/2).

4 - Rs. 30,000 = Rs. 6,380.

13 After two years, the total sum is Rs. 30,000 plus Rs. 6,380.

13 = Rs. 36,380.

13

About Account B:

12% annual interest rate

The number of compounding periods is equal to two.

Total interest earned = Rs. 30,000 multiplied by (1 + 12%)

2 - Rs. 30,000 = Rs. 7,440

After two years, the total payment is Rs. 30,000 + Rs. 7,440 = Rs. 37,440.

As the calculations show, account B will yield a bigger total sum after two years than account A. As a result, we should deposit our funds in account B.

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Answer: 1,797

Step-by-step explanation:

Step 1: Round Numbers Up:

500+600+700 = 1,800

Step 2: Subtract 3 to get answer (-1 for every number rounded up)

1,800-3 = 1,797

Addition using compensation method : 1800

Subtraction using compensation method :  1797

Given,

499+599+699

Here,

Compensation : Round off to the nearest number so that the calculation of addition and subtraction becomes more easy .

So,

Firstly addition,

499 + 599 + 699

Add 3 adding one to each, we get,

500 + 600 + 700 = 1800

Now,

Subtracting 3, 1800 - 3 = 1797

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Using the LIFO perpetual inventory method, the cost of the 18 units sold is $420.

The perpetual inventory strategy known as LIFO (Last In, First Out) is predicated on the idea that the most recent inventory purchases are sold first.

In order to account for the number of units sold, we use this method to count backward from the most recent inventory acquisition.

The business sold 18 units on November 8, which is more than its most recent purchase of 6 units on November 6. Therefore, starting with a total of 18 units, we first use the 10 units from the November 2 purchase and the 8 units from the November 6 buy.

10 units were bought on November 2 for a total of $220, or $22 each unit. The 8 pieces that were bought on November 6 cost $25 apiece, for a total of $200. Hence, $220 plus $200 equals $420 for the 18 units that were sold.

The cost of the 18 sold units, calculated using the LIFO perpetual inventory approach, is $420.

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Probability that a randomly selected August day has a high temperature that exceeds 70°F is approximately 0.7727, or 77.27%.

Assuming that the range of high temperatures in a specific city throughout the month of August is uniform, the probability density function (pdf) of the distribution is as follows:

[tex]f(x) = 1 / (b - a) = 1 / (87 - 65) = 1 / 22[/tex]

where the temperature interval's lower and higher bounds are set at 65 and 87, respectively.

We must determine the area under the pdf curve to the right of x = 70°F in order to determine the likelihood that a randomly chosen August day will have a high temperature that is higher than 70°F. This can be done by integrating the pdf from 70 to 87:

[tex]P(x > 70) = ∫(70 to 87) (70 to 87) f(x) dx = ∫(70 to 87) (70 to 87) 1/22 dx = [1/22 * (x)] (70 to 87) = (1/22) * (87 - 70) = 0.7727[/tex]

As a result, there is a 77.27% probability that an August day chosen at random would have a high temperature above 70 degrees Fahrenheit. This suggests that out of 100 randomly selected August days, we may expect roughly 77 days to have a high temperature surpassing 70°F.

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We can approach this problem by breaking down the two displacements of the ship into their respective x- and y-components and then adding them together to find the net displacement.

For the first displacement, the ship traveled 25° South of West for 250 miles. This can be broken down into an x-component and a y-component as follows:

x = 250 cos(25°) (to the west) y = -250 sin(25°) (to the south)

For the second displacement, the ship changed direction to 70° East of South and traveled 45 miles further. This can also be broken down into an x-component and a y-component:

x = 45 cos(70°) (to the east) y = -45 sin(70°) (to the south)

To find the net displacement, we can add the x-components and y-components separately:

total x = 250 cos(25°) + 45 cos(70°) total y = -250 sin(25°) - 45 sin(70°)

We can use these values to find the distance of the ship from its starting point by using the Pythagorean theorem:

distance = sqrt((total x)^2 + (total y)^2)

Substituting the values from above and evaluating:

distance = sqrt((250 cos(25°) + 45 cos(70°))^2 + (-250 sin(25°) - 45 sin(70°))^2)

distance ≈ 272.8 miles

To find the direction of the ship from its starting point, we can use the inverse tangent function to find the angle:

angle = atan(total y / total x)

Substituting the values from above and evaluating:

angle ≈ -65.1°

Since the angle is negative, we know that the direction is to the west of south. Therefore, the ship is approximately 272.8 miles away from its starting point in a direction that is 65.1° west of south.

Answer:

Step-by-step explanation:

[tex]\frac{e^2\times e^3}{e^6}=\frac{e^{2+3}}{e^6}[/tex]

         [tex]=\frac{e^5}{e^6}[/tex]

         [tex]=e^{5-6}[/tex]

         [tex]=e^{-1}[/tex]

Solution:   (C)

162,356 doctors were registered that year.

Explanation:

The questions says, "34.8% of all registered doctors are females"

(Consider, total number of registered doctors as [tex]x[/tex])

that's, [tex]34.8\%[/tex] of [tex]x[/tex] are female.

and [tex]34.8\%[/tex] of [tex]x[/tex] is 56,500.

Mathematically,

[tex]\dfrac{34.8}{100} \times x=56500[/tex]

[tex]x=56500\times\dfrac{100}{34.8}[/tex]

[tex]x=162356[/tex] doctors were registered that year.

The relative frequency of music downloads that were regional, either tropic or urban, were not urban are 42%,16% and 89% respectively.

Relative frequency is a statistical concept that refers to the proportion or percentage of times a particular event or category occurs in relation to the total number of events or categories. It is calculated by dividing the frequency of the event or category by the total number of events or categories

(a) The relative frequency of music downloads that were regional is:

230/550 = 0.42 or 42%

(b) The relative frequency of music downloads that were either tropical or urban is:

(80+10)/550 = 0.16 or 16%

(c) The relative frequency of music downloads that were not urban is:

(230+170+80)/550 = 0.89 or 89%

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Answer:

The volume of the prism is 800(sqrt(3)) cubic inches.

Step-by-step explanation:

The volume of a triangular prism can be calculated by multiplying the area of the base (which is an equilateral triangle in this case) by the height of the prism.The area of an equilateral triangle can be calculated using the formula:A = (sqrt(3)/4) x s^2where A is the area and s is the length of one side of the triangle.In this case, the length of one side of the equilateral triangle is 20 inches, so we can substitute that into the formula:A = (sqrt(3)/4) x 20^2

A = (sqrt(3)/4) x 400

A = 100(sqrt(3)) square inchesNow that we have the area of the base, we can calculate the volume of the prism:V = A x h

V = 100(sqrt(3)) x 8

V = 800(sqrt(3)) cubic inches

To find the volume of a triangular prism, one must calculate the area of the base (in this case an equilateral triangle), and then multiply this by the height of the prism. Using the provided dimensions, the volume is calculated to be 800√3 cubic inches.

The question is asking us to find the volume of a triangular prism with an equilateral triangle as the base and a given height. The formula for the volume of a prism is Volume = Base Area * Height. Since the base is an equilateral triangle, its area can be calculated using the formula: Area = (sqrt(3) / 4) * side². By substituting the given side length of 20 inches, we find that the area of the base is sqrt(3) * 100 square inches. We then multiply this by the height of the prism, which is 8 inches, to find the total volume. So, the volume of the prism is 800√3 cubic inches.

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La magnitud física que determina que un trabajador haga el trabajo en un tiempo menor que otro puede ser la RAPIDEZ.

La rapidez o también velocidad (si hablamos de vector) define porque un trabajador realiza un trabajo en 20 segundos y el otro en 12 segundos, significa que el primero es menos rápido que el segundo.  

De cierta manera pudieran estar otras magnitudes relacionadas como la potencia.

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Answer:

B) 3

Step-by-step explanation:

Divide 12 and 4 then you will get 3 as your answer.

Answer:

You can Buy 3 oranges from 1 dollar.

Given- A Store sells 12 Oranges for 4 dollars

let dollar = x (equation 1)

Now,

      12 oranges = 4x (Equation 1)

Taking 4 to the other side of equals sign, it divides 12 : -

       12/4 oranges = x

       3 oranges = x

Thus we get the value of x which is 3 oranges. As we know that x = dollar, we get to know that each dollar will get us 3 oranges.

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Answer:

distance = 180 km

Step-by-step explanation:

To find the total distance, we use this formula:

distance = speed * time

where speed is 48 km/hour and time is 3 [tex]\frac{3}{4}[/tex] hours.

inserting the value we get

distance = 48 km/hour × 3.75 hours

distance = 180 km

You need to pay mortgages of  $6370 monthly.

A mοrtgage is an agreement between yοu and a lender that gives the lender the right tο take yοur prοperty if yοu fail tο repay the mοney yοu've bοrrοwed plus interest. Mοrtgage lοans are used tο buy a hοme οr tο bοrrοw mοney against the value οf a hοme yοu already οwn

In this case,

P = $600000

r = 13.10 % = 0.131      [Credit card rate fοr Octοber 2015]

t = 15years

n = 12  [Cοmpοunded mοnthly]

Monthly payment = [tex]$ \rm \frac{r \times P}{n} \times \left[1 - (1 +\frac{r}{n}) - nt \right][/tex]

= [tex]$ \rm \frac{0.131 \times 600000}{12} \times \left[1 - (1 +\frac{0.131}{12}) - 12 \times 15 \right][/tex]

= 6550 - 180

= 6550 - 180

= 6370

Thus, You need to pay mortgages of  $6370 monthly.

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Answer:

The answer to your problem is, g(3) = 18

Step-by-step explanation:

Given that the function, g, has a domain of -1 ≤ x ≤ 4 and a range of                       - 0 ≤ g(x) ≤ 18 and that g(-1) = 2 and g(2) = 8

Listed in order 1 - 4

A. The value of x=5. This contradicts property 1 stated above. Therefore, it is not true.

B. The value of g(x)=-2. This contradicts property 2 stated above. Therefore, it is not true.

C. The value of g(2)=4. However by property 4 stated above, g(2)=9. Therefore, it is not true.

D. This statement can be true as its domain is in between -1 and 4 and its range is in between 0 and 18.

Thus the answer to your problem is, D. g(3) = 18

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