Derive the equation for the ellipse centered at the origin with foci (0, ±2c) and major vertices (0, ±2a).

The volume of the cone in terms of π is 168π in³

A cone is a three-dimensional shape in geometry that narrows smoothly from a flat base (usually circular base) to a point(which forms an axis to the centre of base) called the apex or vertex.

The volume of a cone is one- third of a volume of a cylinder. Therefore;

The volume of a cone is expressed as;

V = 1/3 πr²h

where r is the radius and h is the height

r = 6 in

h = 14 in

V = 1/3 × π × 6² × 14

V = 504π/3

V = 168π in³

Therefore the volume of the cone in terms of π is 168π in³

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Answer:

the mean is 8 for x

8 = x + 2x / 2

16 = 3x

x = 16/3

x = 5.3

therefore the coordinate of x are ( 5.3,10.6)

for the y coordinate

-10= y +2y /2

-20 = 3y

y = -6.6

The coordinate of y are ( -6.6, - 13.3 )

If we add 2.75 to the product of 4.0 and 0.25 then the value id 3.8

We have to find the value by adding  2.75 to the product of 4.0 and 0.25

The product of 4.0 and 0.25 is 1.0

4.0 × 0.25 = 1.0

Add 2.75 to the product of 4.0 and 0.25

2.75+1=3.75

Which is equal to 3.8

Therefore, if we add 2.75 to the product of 4.0 and 0.25 then the value id 3.8

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The slope of the line is 3.

Given are the coordinates of points on a line, we need to find the slope of the line,

Considering the points (2, 7) and (4, 13).

We know that the slope = y₂-y₁ / x₂-x₁

slope = 13-7 / 4-2 = 6/2

Slope = 3

Hence the slope of the line is 3.

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Answer:

---------------------

First, find the common ratio:

Next, use the formula for the nth term of a geometric sequence:

Plugging in the values we have:

Main Answer:The volume of the solid enclosed by the paraboloid and the planes is 18.67 cubic units.

Supporting Question and Answer:

How do we calculate the volume of a solid bounded by surfaces using triple integration?

To calculate the volume of a solid bounded by surfaces using triple integration, we set up a triple integral with the integrand equal to 1, representing the infinitesimal volume element. The bounds of integration are determined by the equations defining the surfaces that enclose the solid. By evaluating the triple integral over the specified region, we can find the volume of the solid.

Body of the Solution: To find the volume of the solid enclosed by the paraboloid z = 2 + x^2 + (y - 2)^2 and the planes z = 1, x = -2, x = 2, y = 0, and y = 3, we can set up a triple integral in the given region.

To find the  volume,using the triple integral:

V = ∫∫∫ R (1) dz dy dx

where R is the region bounded by the given planes and the paraboloid.

The bounds of integration for x are -2 to 2, for y are 0 to 3, and for z are the lower bound function z = 1 and the upper bound function z = 2 + x^2 + (y - 2)^2.

Setting up the triple integral:

V = ∫ from x = -2 to 2 ∫ from y = 0 to 3 ∫ from z = 1 to 2 + x^2 + (y - 2)^2 (1) dz dy dx

Integrating the innermost integral with respect to z:

V = ∫ from x = -2 to 2 ∫ from y = 0 to 3 [(2 + x^2 + (y - 2)^2) - 1] dy dx

Simplifying the expression inside the integral:

V = ∫ from x = -2 to 2 ∫ from y = 0 to 3 [x^2 + (y - 2)^2 + 1] dy dx

Integrating the inner integral with respect to y:

V = ∫ from x = -2 to 2 [x^2(y) + ((y - 2)^3)/3 + y] evaluated from y = 0 to 3 dx

Substituting the limits of integration for y:

V = ∫ from x = -2 to 2 [x^2(3) + (3 - 2)^3/3 + 3 - (x^2(0) + (0 - 2)^3/3 + 0)] dx

Simplifying further:

V = ∫ from x = -2 to 2 [3x^2 +2/3] dx

Integrating the final integral with respect to x:

V = [(x^3) + (2/3)x] evaluated from x = -2 to 2

Evaluating the expression at the limits:

V = [(2^3) +(2/3) 2] - [((-2)^3) + (2/3)(-2)]

V = (8 +4/3) - (-8 - 4/3)

V = 16+8/3

V =56/3

Final Answer:Therefore, the volume of the solid enclosed by the paraboloid and the given planes is 56/3 cubic units.

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The volume of the solid enclosed by the paraboloid and the planes is 18.67 cubic units.

To calculate the volume of a solid bounded by surfaces using triple integration, we set up a triple integral with the integrand equal to 1, representing the infinitesimal volume element. The bounds of integration are determined by the equations defining the surfaces that enclose the solid. By evaluating the triple integral over the specified region, we can find the volume of the solid.

Body of the Solution: To find the volume of the solid enclosed by the paraboloid z = 2 + x^2 + (y - 2)^2 and the planes z = 1, x = -2, x = 2, y = 0, and y = 3, we can set up a triple integral in the given region.

To find the  volume,using the triple integral:

V = ∫∫∫ R (1) dz dy dx

where R is the region bounded by the given planes and the paraboloid.

The bounds of integration for x are -2 to 2, for y are 0 to 3, and for z are the lower bound function z = 1 and the upper bound function z = 2 + x^2 + (y - 2)^2.

Setting up the triple integral:

V = ∫ from x = -2 to 2 ∫ from y = 0 to 3 ∫ from z = 1 to 2 + x^2 + (y - 2)^2 (1) dz dy dx

Integrating the innermost integral with respect to z:

V = ∫ from x = -2 to 2 ∫ from y = 0 to 3 [(2 + x^2 + (y - 2)^2) - 1] dy dx

Simplifying the expression inside the integral:

V = ∫ from x = -2 to 2 ∫ from y = 0 to 3 [x^2 + (y - 2)^2 + 1] dy dx

Integrating the inner integral with respect to y:

V = ∫ from x = -2 to 2 [x^2(y) + ((y - 2)^3)/3 + y] evaluated from y = 0 to 3 dx

Substituting the limits of integration for y:

V = ∫ from x = -2 to 2 [x^2(3) + (3 - 2)^3/3 + 3 - (x^2(0) + (0 - 2)^3/3 + 0)] dx

Simplifying further:

V = ∫ from x = -2 to 2 [3x^2 +2/3] dx

Integrating the final integral with respect to x:

V = [(x^3) + (2/3)x] evaluated from x = -2 to 2

Evaluating the expression at the limits:

V = [(2^3) +(2/3) 2] - [((-2)^3) + (2/3)(-2)]

V = (8 +4/3) - (-8 - 4/3)

V = 16+8/3

V =56/3

Therefore, the volume of the solid enclosed by the paraboloid and the given planes is 56/3 cubic units.

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The Expansion of  (√2x + 2y - √3z)² is 2x + 4y + 3z - 2√6xy - 2√6xz - 4√3yz.

To expand the expression (√2x + 2y - √3z)², we can use the following identity:

(a + b + c)² = a² + b² + c² + 2ab + 2ac + 2bc

In this case, we can identify a = √2x, b = 2y, and c = -√3z. So we have:

(√2x + 2y - √3z)²

= (√2x)² + (2y)² + (-√3z)² + 2(√2x)(2y) + 2(√2x)(-√3z) + 2(2y)(-√3z)

Simplifying, we get:

(√2x + 2y - √3z)² = 2x + 4y + 3z - 2√6xy - 2√6xz - 4√3yz

Therefore, (√2x + 2y - √3z)² = 2x + 4y + 3z - 2√6xy - 2√6xz - 4√3yz.

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Answer: 0.175325

Step-by-step explanation:

Step-by-step explanation:

b = 37  degrees due to  opposite angles of crossing lines

c = a          37 + c = 180 degrees   so     c = a = 143 degrees

It will take Alice 65 minutes to finish mowing the lawn.

To mow a 6 x 32 m strip of lawn, Alice covered an area of:

= 6 x 32

= 192 square meters.

As it take 30 minutes to mow this area, her mowing rate is:

= 192/30

= 6.4 square meters per minute.

The area of remaining lawn is:

= 13 x 32

= 416 square meters.

The minute to finish mowing is:

= 416 square meters / 6.4 square meters per minute.

= 65 minutes

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Step-by-step explanation:

To find how many of the same cards the farmer can form at most, we need to find the greatest common factor (GCF) of 78 and 90. One way to do this is to list the factors of each number and find the largest one they have in common:

78: 1, 2, 3, 6, 13, 26, 39, 78

90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

The largest factor they have in common is 6, so the farmer can form at most 6 baskets of the same size.

To find how much the farmer gets from the sale of 10 baskets, we need to first find the total number of artichokes and courgettes he has:

78 + 90 = 168

Next, we can find the total amount of money he gets from selling the artichokes and courgettes:

Money from artichokes = 78 × $0.70 = $54.60

Money from courgettes = 90 × $0.40 = $36.00

Total money = $54.60 + $36.00 = $90.60

If the farmer sells 10 baskets, then each basket would have 168/10 = 16.8 artichokes and courgettes. Since he can only form baskets of the same size, he would sell 6 baskets with 16 artichokes and 24 courgettes in each basket. The total amount of money he would get from selling these 6 baskets is:

Money from artichokes = 6 × 16 × $0.70 = $67.20

Money from courgettes = 6 × 24 × $0.40 = $57.60

Total money = $67.20 + $57.60 = $124.80

Therefore, the farmer would get $124.80 from the sale of 10 baskets.

Answer:

64cm^2 is the correct answer

The angle formed at the gas station is 130.3°. The correct option is (D).

Recall that, the area of a triangle is:

area = 1/2 * base * height

In this case, the base of the triangle is the distance between the two cars after 2 hours, and the height of the triangle is the distance from the gas station to the midpoint of the base.

Let's first find the distance traveled by each car in 2 hours:

distance traveled by car 1 = 45 mph * 2 hours = 90 miles

distance traveled by car 2 = 60 mph * 2 hours = 120 miles

The total distance between the two cars is the sum of their distances, which is:

total distance = 90 miles + 120 miles = 210 miles

The midpoint of the base is located at a distance of half the total distance from the gas station:

distance to midpoint = 1/2 * 210 miles = 105 miles

Now we can use the formula for the height of a triangle, which is:

height = 2 * area / base

Plugging in the values we have, we get:

height = 2 * 5190.713 mi²/ 210 miles = 123.824 mi

Finally, we can use the inverse tangent function to find the angle formed at the gas station, which is:

angle = tan⁻¹(height / distance to midpoint)

angle = tan⁻¹(123.824 mi / 105 miles)

angle = tan⁻¹(1.1793)

angle = 49.7°

However, this angle is the internal angle of the triangle formed by the two cars and the gas station. To find the angle formed at the gas station, we need to subtract this angle from 180°, which gives:

angle at gas station = 180° - 49.7° = 130.3°

Therefore, the answer is not one of the choices provided.

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The volume of the solid enclosed by the two paraboloids is approximately 1365.34π cubic units.

Here, we have to find the volume of the solid enclosed by the two paraboloids [tex]y = x^2 + z^2[/tex] and [tex]y = 32 - x^2 - z^2[/tex],

we need to set up a triple integral in cylindrical coordinates.

In cylindrical coordinates, we have:

x = r cos(θ)

z = r sin(θ)

y = y

The limits of integration for r, θ, and y will depend on the region of integration.

First, let's find the intersection points of the two paraboloids:

[tex]x^2 + z^2 = 32 - x^2 - z^2\\2x^2 + 2z^2 = 32\\x^2 + z^2 = 16[/tex]

This represents a circular region with radius r = 4 in the x-z plane.

Now, let's find the limits of integration for r, θ, and y:

For r:

Since the circular region has a radius of 4, the limits of r will be from 0 to 4.

0 ≤ r ≤ 4

For θ:

The intersection points form a complete circle in the x-z plane, so the limits of θ will be from 0 to 2π.

0 ≤ θ ≤ 2π

For y:

The lower paraboloid is given by [tex]y = x^2 + z^2[/tex], and the upper paraboloid is given by [tex]y = 32 - x^2 - z^2[/tex].

The limits of y will be from the lower paraboloid to the upper paraboloid.

[tex]x^2 + z^2 \leq y \leq 32 - x^2 - z^2[/tex]

Now, we can set up the triple integral to find the volume V:

V = ∫∫∫ (y) dy dθ dr

V = ∫∫∫ [tex](r^2 cos^2(\theta) + r^2 sin^2(\theta)) dy d\theta dr[/tex]

V = ∫∫∫ [tex](r^2) dy d\theta dr[/tex]

V = ∫∫ [tex](r^2) y|_{(x^2+z^2)}^{(32-x^2-z^2)} d\theta dr[/tex]

V = ∫∫ [tex](r^2) (32 - 2x^2 - 2z^2) d\theta dr[/tex]

V = ∫ [tex](32r^2 - 2r^2(x^2 + z^2)) d\theta dr[/tex]

V = ∫ [tex](32r^2 - 2r^2r^2) d\theta dr[/tex]

V = ∫ [tex](32r^2 - 2r^4) d\theta dr[/tex]

Now, integrate with respect to θ from 0 to 2π:

V = ∫(0 to 2π) ∫(0 to 4) [tex](32r^2 - 2r^4)[/tex] dr dθ

Now, integrate with respect to r from 0 to 4:

V = ∫(0 to 2π) [tex][(32/3)r^3 - (1/3)r^5] |_0 ^4[/tex] dθ

V = ∫(0 to 2π) [tex][(32/3)(4^3) - (1/3)(4^5)][/tex] dθ

V = ∫(0 to 2π) [(32/3)(64) - (1/3)(1024)] dθ

V = ∫(0 to 2π) [682.67] dθ

V = 682.67 * (2π) - 682.67 * (0)

V = 1365.34π cubic units

Therefore, the volume of the solid enclosed by the two paraboloids is approximately 1365.34π cubic units.

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Therefore, the probability that Sam gets more heads than Max is 0.5.

Let X be the number of heads Max gets and Y be the number of heads Sam gets. Both X and Y follow a binomial distribution with parameters k and 0.5.

We want to find P(Y>X), which can be expressed as:

P(Y>X) = P(Y-X>0)

Let Z = Y - X. Then Z follows a binomial distribution with parameters k and p = 0.5 - 0.5 = 0.

The mean of Z is E(Z) = E(Y-X) = E(Y) - E(X) = k/2 - k/2 = 0.

The variance of Z is Var(Z) = Var(Y-X) = Var(Y) + Var(X) = k/4 + k/4 = k/2.

Using the normal approximation to the binomial distribution, we can approximate Z as a normal distribution with mean 0 and variance k/2, for large enough values of k.

Therefore, P(Y>X) = P(Z>0) can be approximated using the standard normal distribution:

P(Z>0) = P((Z-0)/sqrt(k/2) > (0-0)/sqrt(k/2)) = P(Z/sqrt(k/2) > 0)

Using a standard normal table or calculator, we find that P(Z/sqrt(k/2) > 0) = 0.5.

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The library is 6.3 miles away from the park. the library and football fields are 11.8 miles away  

The Airplane travel 1750 miles in 3.5 hours.

The equation that models the number of miles flown by the airplane given x hours is:

y = 500x

where y is the dependent variable, representing the number of miles flown, and x is the independent variable, representing the number of hours flown.

The dependent variable is y because it depends on the independent variable x. The number of miles flown by the airplane is determined by the number of hours it has flown. The longer the airplane flies, the more miles it will cover. Therefore, the number of miles flown is a function of the number of hours flown, and it is the dependent variable.

To find how many miles the airplane will travel in 3.5 hours, we substitute x = 3.5 into the equation:

y = 500x

y = 500(3.5)

y = 1750

Therefore, the airplane will travel 1750 miles in 3.5 hours.

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The area to the left of the z-score corresponds to the cumulative probability up to that point. The correct answer is b. z=0.84.

To find the z-score that separates the top 70% from the bottom 30%, we need to look up the corresponding percentile in a standard normal distribution table.

The area to the left of the z-score corresponds to the cumulative probability up to that point. For example, if we look up a z-score of 0.84 in the table, we find that the area to the left of that value is 0.7995, or 79.95%.

Since we want to find the value that separates the top 70% from the bottom 30%, we subtract 0.30 (or 30%) from 1.00 to get 0.70 (or 70%). We then find the z-score that corresponds to that area, which is approximately 0.84.

Therefore, the answer is b. z=0.84.

To find the Z-score value that separates the top 70% of a normal distribution from the bottom 30%, you need to look for the Z-score corresponding to the 70th percentile. In this case, the correct answer is:

b. z=0.52

Here's a step-by-step explanation:

1. Determine the percentile you're looking for, which is the 70th percentile (separating the top 70% from the bottom 30%).
2. Consult a Z-score table or use a calculator to find the Z-score corresponding to the 70th percentile.
3. The Z-score for the 70th percentile is approximately 0.52.

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Birth weights of full-term babies in a certain area are normally distributed with a mean of 7.13 pounds and a standard deviation of 1.29 pounds. a newborn weighing 5.5 pounds or less is a low-weight baby. There is a 10.3 % probability that a randomly selected newborn is low-weight. The correct answer is option B.

Given that the birth weights of full-term babies in a certain area are normally distributed with a mean of 7.13 pounds and a standard deviation of 1.29 pounds. A newborn weighing 5.5 pounds or less is a low-weight baby. We need to find the probability that a randomly selected newborn is low-weight.

Probability of newborn weighing 5.5 pounds or less = P(x ≤ 5.5)

We need to convert x to standard normal variable z.z = (x - μ) / σz = (5.5 - 7.13) / 1.29z = -1.26

Now, we need to find the area under the normal curve to the left of z = -1.26, using normal tables or calculators we get 0.1038A

Approximately, the probability that a randomly selected newborn is low-weight is 10.3%. Hence, option (B) is the correct answer.

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Assuming that the regular price for each item is $4 and there are 40 items in total, the deal of buy 4 get 1 free would save $32 compared to the deal of buy 9 get 2 free.

To calculate the amount saved with each deal, we first need to determine how many free items we would receive with each deal. With the buy 4 get 1 free deal, for every 5 items purchased, 1 item is free. This means that we would receive a total of 8 free items (40/5 = 8). With the buy 9 get 2 free deal, for every 11 items purchased, 2 items are free. This means that we would receive a total of 7.27 free items (40/11*2 = 7.27).

With the buy 4 get 1 free deal, we would pay for a total of 32 items (40 - 8 free items). This would cost a total of $128 (32 x $4). With the buy 9 get 2 free deal, we would pay for a total of 32.73 items (40 - 7.27 free items). This would cost a total of $130.92 (32.73 x $4). Therefore, the buy 4 get 1 free deal would save us $2.92 compared to the buy 9 get 2 free deal.

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You have $2.30 and would like to buy some rice flour buy 9.2 ounces of rice flour with $2.30.

To arrive at this answer, we need to first convert the price per pound to price per ounce. There are 16 ounces in a pound, so the price per ounce is $4/16 = $0.25 per ounce.

Next, we divide the amount of money you have ($2.30) by the price per ounce ($0.25).

$2.30/$0.25 = 9.2 ounces.

Therefore, the conclusion is that you can buy 9.2 ounces of rice flour with $2.30.
Hi! I'm happy to help you with your question.


You can buy 9.2 ounces of rice flour.


1. First, we need to find out how many dollars you have per ounce of rice flour: $4 per pound / 16 ounces per pound = $0.25 per ounce.
2. Next, we'll determine how many ounces of rice flour you can buy with $2.30: $2.30 / $0.25 per ounce = 9.2 ounces.


With $2.30, you can buy 9.2 ounces of rice flour at the given price of $4 per pound.

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Answer:

1231.5 units^2

Step-by-step explanation:

The formula for volume of a cone is

V = 1/3πr^2, where

We see from the diagram that the height is 24 cm and the radius is 7 cm.  Now, we can plug these values in for h and r respectively in the volume formula:

V = 1/3π(7)^2(24)

V = 1/3π(49)(24)

V = 1/3π*1176

V = 392π

V = 1231.50432

V = 1231.5 cm^3

Answer:17

Step-by-step explanation:

you have to multiply both sides by 2 so A+5= 22 then subtract 5 over so A=17

Answer:

615.75cm

Step-by-step explanation:

The circumference of a circle is 2r×pi

If the circumference is 87.92cm, we can work out the radius

2×r×pi=87.92cm

r=87.92÷(2×pi)

r=14

Now we can work out the area using the formula r²×pi

A=r²×pi

A=14²×3.14

A=196×3.14

A=615.75

The area of the shaded region is given as follows:

85.245 cm².

The area of a rectangle is given by the multiplication of it's dimensions, hence it is given as follows:

Ar = 13.8 x 7.6

Ar = 104.88 cm².

The area of a circle of radius r is given by the equation presented as follows:

A = πr²

The radius for this problem is of r = 2.5 cm, hence the area is given as follows:

A = π x 2.5²

A = 19.635 cm²

Hence the area of the shaded region is given as follows:

104.88 - 19.635 = 85.245 cm².

The diagram is given by the image presented at the end of the answer.

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The probability of the first two songs played being slow songs is approximately 0.5454.

The probability that the first two songs played are slow songs can be calculated by dividing the number of favorable outcomes (two slow songs) by the total number of possible outcomes.

Since there are 9 slow songs out of 12 total songs, the probability of selecting a slow song as the first song is 9/12. After the first slow song is played, there are 8 slow songs left out of the remaining 11 songs. Therefore, the probability of selecting a slow song as the second song, given that the first song was slow, is 8/11.

To find the probability of both events occurring (selecting a slow song first and then selecting a slow song second), we multiply the probabilities of each event:

P(First song slow) * P(Second song slow | First song slow) = (9/12) * (8/11) = 72/132 = 0.5454 (rounded to four decimal places).

The probability that the first two songs played are slow songs is 0.5454.

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The volume of the given cone is 193.65 in³.

Given is a cone of height 5.9 in and radius of 5.6 in, we need to find the volume of the cone,

V(cone) = π × radius² × height / 3

= 3.14 × 5.6² × 5.9 / 3

= 193.65 in³

Hence the volume of the given cone is 193.65 in³.

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Answer:

502.65yd^2

Step-by-step explanation:

To find the surface area of a cylinder, the surface area is 2(pi)(r)(h)+2(pi)(r^2)

Plug in all values into the formula.

We will use pi as 3.14.

Because the diameter is 10, the radius will be 5.

2(3.14)(5)(11)+2(3.14)(25)

Therefore, the surface area of the cylinder is about 502.65 yd^2.

The transformations applied to f(x) to obtain the graph of g(x) can be summarized as a vertical stretch by a factor of 2, followed by a vertical shift upward by 12 units, and a horizontal shift to the left by 20 units.

Assuming that the expression 2 2 refers to 2 raised to the power of 2, the transformations applied to f(x) to change it to the graph of g(x) can be identified as follows:

Vertical stretch: The coefficient 2 in front of the function f(x) indicates that the graph of f(x) is stretched vertically by a factor of 2 to obtain the graph of 2f(x).

Vertical shift: The term +12 added to the function 2f(x) shifts the graph vertically upward by 12 units.

Horizontal shift: The term +20 added to the function 2f(x) shifts the graph horizontally to the left by 20 units.

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Note The full question is :

"What are the specific transformations applied to the function f(x) to obtain the graph of g(x)?"