6. Find the HCF and LCM of: (b) (a) 4(a²-4), 6(a²-a-2) and 12(a² + 3a-10) 2x²-3xy-2y², 6x² + xy - y² and 3x² - 7xy + 2y² (c) a(c + a)-b(b + c), b(a + b)-c(c + a) and c(b + c)-a(a + b) (d) p² +q²+2pq-1, q²-p² + 2q + 1 and p² - q² + 2p + 1 (e) 6x²-5x-4, 8x² + 2x - 15 and 12x²-43x +35​

The equation Square root of 5 square root of 5 = x can be simplified as follows:

√5 ·√5 = x

√(5·5) = x

√25 = x

x = 5

Therefore, the value of x that will make the equation true is 5.

We can use the power series expansion of the exponential function e^(-x) to evaluate the sum of the series:

e^(-x) = ∑(n=0 to infinity) (-1)^n (x^n) / n!

Setting x = 3/2, we get:

e^(-3/2) = ∑(n=0 to infinity) (-1)^n (3/2)^n / n!

Multiplying both sides by (3/2)^2 and simplifying, we get:

(9/4) e^(-3/2) = ∑(n=0 to infinity) (-1)^n (3/2)^(n+2) / (n+2)!

Comparing this with the given series, we can see that they differ only by a factor of (-1) and a shift in the index of summation. Therefore, we can write:

∑(n=0 to infinity) (-1)^n (2n) (3/2)^(2n) / (2n)!

= (-1) ∑(n=0 to infinity) (-1)^n (3/2)^(n+2) / (n+2)!

= (-1) ((9/4) e^(-3/2))

= - (9/4) e^(-3/2)

Hence, the sum of the series is - (9/4) e^(-3/2).

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a. There are 648 integers from 1 through 999 that do not have any repeated digits.

b. There are 351 integers from 1 through 999 that have at least one repeated digit.

c. The probability that an integer chosen at random from 1 through 999 has at least one repeated digit is approximately 0.351.

Learn more about the count of integers without repeated digits from 1 to 999.In a range from 1 through 999, there are 900 integers in total. To determine the number of integers without repeated digits, we need to consider the possible combinations. For the hundreds place, there are 9 options (1-9) since zero cannot be used as the first digit. For the tens place, there are 9 options again (0-9 excluding the digit already used in the hundreds place). Similarly, for the units place, there are 8 options available (0-9 excluding the two digits already used in the hundreds and tens places). Multiplying these options together, we get 9 * 9 * 8 = 648 integers without repeated digits.To calculate the number of integers with at least one repeated digit, we can subtract the count of integers without repeated digits from the total count of integers in the range. Therefore, 900 - 648 = 252 integers have at least one repeated digit.

To find the probability, we divide the count of integers with at least one repeated digit by the total count of integers in the range, resulting in 252 / 900 ≈ 0.351. Therefore, the probability that a randomly chosen integer from 1 through 999 has at least one repeated digit is approximately 0.351.

Out of the three-digit integers from 100 to 999, there are 351 integers that have at least one repeated digit. To determine this count, we subtract the number of unique-digit integers (648) from the total count of three-digit integers (900). Hence, 900 - 648 = 252 integers have at least one digit repeated.

If a three-digit integer is randomly selected from the range 100 to 999, the probability that it will have at least one repeated digit is approximately 0.39 or 39%. This probability is calculated by dividing the count of integers with repeated digits (351) by the total count of three-digit integers (900). Therefore, the probability is 351 / 900 ≈ 0.39 or 39%.

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The radius, r, of the larger cone is equal to 24 mm.

In Mathematics and Geometry, the volume of a cone can be calculated by using this formula:

Volume of cone, V = 1/3 × πr²h

Where:

Since both the large and small cones are similar, we can logically deduce the following proportion based on their side lengths;

19,008/704 = (r/8)³

19,008/704 = r³/512

r³ = 19,008/704 × 512

Radius of larger cone = 24 mm.

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

11 yards of fencing left.

Given that Edgar decided to add a second gate. He removes 2 yards of fencing from his section of 13 yards.

Therefore, the total length of the fencing was 13 yards.We have to remove 2 yards of fencing from the section.Therefore, the total fencing remaining will be=

Total fencing - Fencing Removed Fencing Removed = 2 yardsTotal fencing = 13 yards We can substitute the values in the above equation.Fencing remaining= 13 - 2 = 11 yards  In total, 11 yards of fencing are left.

Edgar had 13 yards of fencing. He had to remove 2 yards of fencing from it. Thus, he could not use the removed fencing for the gate. We need to calculate the remaining length of the fencing.Edgar had to remove 2 yards of fencing to add a second gate.

Therefore, the total fencing remaining will be= Total fencing - Fencing RemovedFencing Removed = 2 yardsTotal fencing = 13 yardsWe can substitute the values in the above equation.

Fencing remaining= 13 - 2 = 11 yards

Thus, Edgar has only 11 yards of fencing left to use. This will be less fencing available to Edgar to use for his purpose. With a smaller area to work with, Edgar will have to ensure that the fencing is placed appropriately.

Edgar had a total of 13 yards of fencing before removing 2 yards of fencing to add a second gate. Therefore, he had only 11 yards of fencing left.

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The given series ∑=3[infinity]710 is a geometric series with the first term a=3 and the common ratio r=7/10. Therefore, the sum of the given geometric series is 10, and the series is convergent.

To determine whether the series converges or diverges, we can apply the formula for the sum of an infinite geometric series, which is S = a / (1 - r). Plugging in the values for a and r, we get:

S = 3 / (1 - 7/10) = 3 / (3/10) = 10

Therefore, the sum of the infinite geometric series is 10. This means that as we add up more and more terms of the series, the sum gets closer and closer to 10. In other words, the series converges to a finite value of 10.

In conclusion, the sum of the given geometric series is 10, and the series is convergent.

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To find out whether Mia has enough fabric to make the dress, you need to add the amount of fabric required for the top and skirt. Then compare it with the amount of fabric she has.

So, let's do that.To make the dress, we need 11/8 yards of fabric for the top2 5/8 yards of fabric for the skirt Total fabric required

= 1 1/8 + 2 5/8

= 3 3/4 yards

Mia has 3 1/2 yards of fabric

So, Mia does not have enough fabric to make the dress because she needs 3 3/4 yards of fabric to make it.

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The given series is divergent.

To determine whether the series converges or diverges, we can use the divergence test, which states that if the limit of the nth term of a series does not approach zero as n approaches infinity.

Then the series must diverge.

Let's find the limit of the nth term of the given series:

lim n → ∞ n^2 / (4n^3 - 3n)

= lim n → ∞ n^2 / n^3 (4 - 3/n^2)

= lim n → ∞ 1/n (4/3 - 3/n^2)

As n approaches infinity, the second term approaches zero, and the limit becomes:

lim n → ∞ 1/n * 4/3 = 0

Since the limit of the nth term approaches zero, the divergence test is inconclusive. Therefore, we need to use another test to determine whether the series converges or diverges.

We can use the limit comparison test, which states that if the ratio of the nth term of a series to the nth term of a known convergent series approaches a nonzero constant as n approaches infinity.

Then the two series must either both converge or both diverge.

Let's compare the given series to the p-series with p = 3:

∑ n = 1 ∞ 1/n^3

We have:

lim n → ∞ (n^2 / (4n^3 - 3n)) / (1/n^3)

= lim n → ∞ n^5 / (4n^3 - 3n)

= lim n → ∞ n^2 / (4 - 3/n^2)

= 4/1 > 0

Since the limit is a nonzero constant, the two series either both converge or both diverge. We know that the p-series with p = 3 converges, therefore, the given series must also converge.

The correct series should be:

∑ n = 1 ∞ n / (4n^3 - 3)

Using the same tests as above, we can show that this series is divergent. The limit of the nth term approaches zero, and the limit comparison test with the p-series with p = 3 gives a nonzero constant:

lim n → ∞ (n / (4n^3 - 3)) / (1/n^3)

= lim n → ∞ n^4 / (4n^3 - 3)

= lim n → ∞ n / (4 - 3/n^4)

= ∞

Therefore, the given series is divergent.

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Answer:

The interval of convergence is (-∞, ∞).

Step-by-step explanation:

Using the ratio test, we have:

| [tex]\frac{1 - x^6)}{(1 - (x+1)^6)}[/tex] | = | [tex]\frac{(1 - x^6) }{(-6x^5 - 15x^4 - 20x^3 - 15x^2 - 6x) }[/tex] |

Taking the limit as x approaches infinity, we get:

lim | [tex]\frac{(1 - x^6) }{(-6x^5 - 15x^4 - 20x^3 - 15x^2 - 6x) }[/tex] | = lim | [tex]\frac{(1/x^6 - 1)}{(-6 - 15/x - 20/x^2 - 15/x^3 - 6/x^4)}[/tex] |

Since all the terms with negative powers of x approach zero as x approaches infinity, we can simplify this to:

lim | [tex]\frac{(1/x^6 - 1) }{(-6)}[/tex] | = [tex]\frac{1}{6}[/tex]

Since the limit is less than 1, the series converges for all x, and the interval of convergence is (-∞, ∞).

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To apply the Alternating Series Test, we need to check two conditions:

The terms of the series must alternate in sign.

The absolute values of the terms must decrease as n increases.

Let's analyze the given series: ∑ (-1)^n (n - 6) from n = 7 to infinity.

Alternating Signs: The series has alternating signs because of the (-1)^n term. When n is even, (-1)^n becomes positive, and when n is odd, (-1)^n becomes negative.

Decreasing Absolute Values: Let's examine the absolute values of the terms: |(-1)^n (n - 6)| = |n - 6|.

As n increases, the absolute value |n - 6| also increases. Therefore, the absolute values of the terms do not decrease.

Since the terms do not meet the decreasing absolute values condition, we cannot conclude convergence or divergence using the Alternating Series Test. The Alternating Series Test does not apply in this case.

To determine the convergence or divergence of the series, we need to use other convergence tests, such as the Ratio Test or the Comparison Test.

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Answer:

Step-by-step explanation:

To find the value of y when x = 3 in the equation y = -2x + 15, we substitute x = 3 into the equation and solve for y:

y = -2(3) + 15

y = -6 + 15

y = 9

Therefore, when x = 3, y = 9.

The exponential function for the table is given as follows:

[tex]y = 0.02(4)^x[/tex]

The simple radical form of the expression is given as follows:

[tex]\sqrt{8} = 2\sqrt{2}[/tex]

An exponential function has the definition presented as follows:

[tex]y = ab^x[/tex]

In which the parameters are given as follows:

The parameter values for the exponential function in this problem are given as follows:

Hence the exponential function for the table is given as follows:

[tex]y = 0.02(4)^x[/tex]

For the simple radical form, we have that 8 = 2 x 4, hence:

[tex]\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}[/tex]

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`-2+-6` in absolute value minus `-2--6` in absolute value is equal to `4`.

To solve for `-2+(-6)` in absolute value and `-2-(-6)` in absolute value and subtract them, we first evaluate the two values of the absolute value and perform the subtraction afterwards.

Here is the solution:

Simplify `-2 + (-6) = -8`.

Evaluate the absolute value of `-8`. This gives us: `|-8| = 8`.

Therefore, `-2+(-6)` in absolute value is equal to `8`.

Next, simplify `-2 - (-6) = 4`.

Evaluate the absolute value of `4`.

This gives us: `|4| = 4`.

Therefore, `-2-(-6)` in absolute value is equal to `4`.

Now, we subtract `8` and `4`. This gives us: `8 - 4 = 4`.

Therefore, `-2+-6` in absolute value minus `-2--6` in absolute value is equal to `4`.

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For the sample population

(a) The point estimate of p is 0.71.

(b) Using Equation 73-2, the approximate 90% confidence interval for p is obtained by calculating 0.71 ± 1.645 * sqrt((0.71 * (1 - 0.71))/200).

(c) Using Equation 73-4, the approximate 90% interval for p is found by calculating 0.71 ± 1.645 * sqrt((0.71 * (1 - 0.71))/(200 - 1)).

(d) Using Equation 73-5, the approximate 90% confidence interval for p is obtained by calculating 0.71 ± 1.645 * sqrt((0.71 * (1 - 0.71))/(200 + 1.645^2/4)).

(a) To obtain a point estimate of p, we divide the number of letters delivered the next day (y = 142) by the sample size (n = 200):

Point estimate of p = y/n = 142/200 = 0.71

(b) Using Equation 73-2, we can find an approximate 90% confidence interval for p. The formula is given by:

Point estimate ± Z * sqrt((p * (1 - p))/n)

Since the confidence level is 90%, the Z-value for a 90% confidence level is approximately 1.645. Substituting the values into the equation:

Confidence interval = 0.71 ± 1.645 * sqrt((0.71 * (1 - 0.71))/200)

Simplifying the expression:

Confidence interval = 0.71 ± 1.645 * sqrt(0.21/200)

(c) Using Equation 73-4, we can find an approximate 90% interval for p. The formula is given by:

Point estimate ± Z * sqrt((p * (1 - p))/(n - 1))

Applying the formula with the given values:

Confidence interval = 0.71 ± 1.645 * sqrt((0.71 * (1 - 0.71))/(200 - 1))

Simplifying the expression:

Confidence interval = 0.71 ± 1.645 * sqrt(0.21/199)

(d) Using Equation 73-5, we can find an approximate 90% confidence interval for p. The formula is given by:

Point estimate ± Z * sqrt((p * (1 - p))/(n + Z^2/4))

Substituting the values into the equation:

Confidence interval = 0.71 ± 1.645 * sqrt((0.71 * (1 - 0.71))/(200 + 1.645^2/4))

Simplifying the expression:

Confidence interval = 0.71 ± 1.645 * sqrt(0.21/200.5084)

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The perimeter of △XYZ is 54 cm.

To find the perimeter of △XYZ given that △ABC∼△XYZ with side lengths AB=18 cm, BC=30 cm, and CA=42 cm, and the longest side of △XYZ is 25.2 cm, follow these steps:

1. Identify the longest side of △ABC. In this case, it is CA with a length of 42 cm.
2. Calculate the scale factor by dividing the longest side of △XYZ (25.2 cm) by the longest side of △ABC (42 cm): 25.2 / 42 = 0.6.
3. Find the corresponding side lengths of △XYZ by multiplying the side lengths of △ABC by the scale factor (0.6):
  - XY (corresponding to AB): 18 * 0.6 = 10.8 cm
  - YZ (corresponding to BC): 30 * 0.6 = 18 cm
  - XZ (corresponding to CA): 42 * 0.6 = 25.2 cm (already given)
Calculate the perimeter of △XYZ by adding the side lengths: 10.8 + 18 + 25.2 = 54 cm.

The perimeter of △XYZ is 54 cm.

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a) The moment generating function[tex](mgf)[/tex] of U is M(t) = exp((λ1+λ2)(e^t-1)) b) U follows a named distribution known as Poisson distribution with parameter λ1+λ2. c) The [tex]pmf[/tex]of (Y1|U = u) is a binomial distribution with parameters u and λ1/(λ1+λ2).

a) The[tex]mgf[/tex]of U can be found using the fact that the [tex]mgf[/tex]of the sum of independent random variables is the product of their individual [tex]mgfs[/tex]. Thus,

M(t) = E[tex][e^(tU)][/tex] = E[e^(t(Y1+Y2))] = E[e^(tY1)]E[e^(tY2)] = exp(λ1(e^t-1))[tex]exp(λ2(e^t-1)) = exp((λ1+λ2)).[/tex]

b) The sum of independent Poisson random variables is a Poisson distribution with parameter equal to the sum of the individual parameters. Therefore, U follows a Poisson distribution with parameter λ1+λ2.

c) To find the[tex]pmf[/tex]of (Y1|U = u), we use Bayes' theorem:

P(Y1=[tex]k|U=u) = P(Y1=k, Y2=u-k)/P(U=u)[/tex]

= [tex]P(Y1=k)P(Y2=u-k)/(λ1+λ2)^u e^-(λ1+λ2)\\= (λ1^k/k!)(λ2^(u-k)/(u-k)!) / (λ1+λ2)^u e^-(λ1+λ2)[/tex]

This simplifies to a binomial distribution with parameters u and p=λ1/(λ1+λ2), as the probability of success (i.e., Y1=k) is p and the number of trials is u. Thus, the [tex]pmf[/tex] of (Y1|U = u) is a binomial distribution with parameters u and λ1/(λ1+λ2).

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To estimate the sum of 192 and 91 by rounding both values to the nearest ten, we round 192 to 190 and 91 to 90.

190 + 90 = 280

Therefore, the best estimate of the sum is 280.

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Answer:

0.67

Step-by-step explanation:

a. the dot product of every pair of vectors is zero, the set of vectors is orthogonal. b. the set is not orthonormal. c. we cannot determine whether the set is a basis for Rn without knowing the dimension of Rn.

(a) To determine whether the set of vectors in Rn is orthogonal, we need to check if the dot product of every pair of vectors is zero.

Taking dot products:

(0, -1, 4) • (-1, 4, 1) = 0 + (-4) + 4 = 0

(0, -1, 4) • (-17, -4, -1) = 0 + 4 + (-4) = 0

(-1, 4, 1) • (-17, -4, -1) = 17 + (-16) + (-1) = 0

Since the dot product of every pair of vectors is zero, the set of vectors is orthogonal.

(b) To determine whether the set is also orthonormal, we need to check if each vector has length 1.

Calculating the length of each vector:

|| (0, -1, 4) || = sqrt(0^2 + (-1)^2 + 4^2) = sqrt(17)

|| (-1, 4, 1) || = sqrt((-1)^2 + 4^2 + 1^2) = sqrt(18)

|| (-17, -4, -1) || = sqrt((-17)^2 + (-4)^2 + (-1)^2) = sqrt(292)

Since none of the vectors have length 1, the set is not orthonormal.

(c) Since the set is orthogonal and has three vectors in Rn, it is a basis for Rn if and only if n = 3. Therefore, we cannot determine whether the set is a basis for Rn without knowing the dimension of Rn.

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The area of the triangle is 52.32 square units

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

The triangle

The base of the triangle is calculated as

base = 12 * tan(36)

The area of the triangle is then calculated as

Area = 1/2 * base * height

Where

height = 12

So, we have

Area = 1/2 * base * height

substitute the known values in the above equation, so, we have the following representation

Area = 1/2 * 12 * tan(36) * 12

Evaluate

Area = 52.32

Hence, the area of the triangle is 52.32 square units

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The area of the right triangle is approximately 52.3 square units.

The area of triangle is expressed as:

Area = 1/2 × base × height

The figure in the image is a right triangle.

Angle θ = 36 degrees

Adjacent to angle θ ( height ) = 12

Opposite to angle θ ( base ) = ?

To determine the area, we need to find the opposite side of angle θ which is the base.

Using trigonometric ratio:

tanθ = opposite / adjacent

tan( 36 ) = base / 12

base = 12 × tan( 36 )

base = 8.718510

Now, area will be:

Area = 1/2 × 8.718510 × 12

Area = 52.3 square units

Therefore, the area of the triangle is 52.3 square units.

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Answer:

n

Step-by-step explanation:

Let's assume the number of ducks the farmer initially had as 'd' and the number of chickens as 'c'.

Given:

The farmer had 4/5 as many chickens as ducks, so c = (4/5)d.

After selling 46 ducks, the number of ducks becomes d - 46.

After 14 ducks swam away, the number of ducks becomes (d - 46) - 14.

The farmer was left with 5/8 as many ducks as chickens, so (d - 46 - 14) = (5/8)c.

Now we can substitute the value of c from the first equation into the second equation:

(d - 46 - 14) = (5/8)(4/5)d.

Simplifying the equation:

(d - 60) = (4/8)d,

d - 60 = 1/2d.

Bringing like terms to one side:

d - 1/2d = 60,

1/2d = 60.

Multiplying both sides by 2 to solve for d:

d = 120.

Therefore, the farmer initially had 120 ducks.

After selling 46 ducks, the number of ducks left is 120 - 46 = 74.

After 14 more ducks swam away, the final number of ducks left is 74 - 14 = 60.

So, the farmer is left with 60 ducks.

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Answer:

This series diverges--compare it to the harmonic series.

The area of the enclosed loop of the curve y^2 = x^2(x 3) is 56√3/15.

To find the area of the enclosed loop of the curve y^2 = x^2(x 3), we need to first sketch the curve to see what it looks like. The equation can be rewritten as y^2 = x^2(x-3), which means that the curve is symmetric about the x-axis and passes through the origin.

Next, we can find the x-intercepts of the curve by setting y=0: 0^2 = x^2(x-3), which simplifies to x=0 and x=3. So the curve intersects the x-axis at (0,0) and (3,0).

To find the area of the enclosed loop, we need to integrate the curve from x=0 to x=3 and subtract the area below the x-axis. We can do this by setting up the integral as follows:

A = ∫[0,3] y dx - ∫[0,3] -y dx

We can solve for y by taking the square root of both sides of the equation y^2 = x^2(x-3):

y = ± x√(x-3)

To find the bounds of the integral, we can set the two functions equal to each other and solve for x:

x√(x-3) = -x√(x-3)
x=0 or x=3

So our integral becomes:

A = ∫[0,3] x√(x-3) dx - ∫[0,3] -x√(x-3) dx

We can simplify the integral by making the substitution u = x-3, which gives us:

A = ∫[0,3] (u+3)√u du - ∫[0,3] -(u+3)√u du

Simplifying further, we get:

A = 2∫[0,3] (u+3)√u du

This integral can be evaluated using integration by parts, which gives us:

A = 2/3 [2(u+3)(2u+3)√u - ∫(2u+3)√u du] from 0 to 3

Simplifying, we get:

A = 2/3 [(54√3/5) - (2/5)(18√3) + (2/3)(4√3)]

A = 56√3/15 DETAIL ANS

Therefore, the area of the enclosed loop of the curve y^2 = x^2(x 3) is 56√3/15.

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Height of hawk eye at a distance of 660 feet from tree is 462.1 feet .

Given,

An observer (O) is located 660 feet from a tree (T). The observer

notices a hawk (H) flying at a 35° angle of elevation from his line of sight.

Here,

Let x be the height of the hawk.

The tangent ratio expresses the relationship between the sides of a right triangle depicted above as:

tanФ = opposite side/adjacent side

tan35° = x / 660

x = 660 (tan35° )

x = 462.1 feet .

Thus the height of hawk eye is 462.1 feet .

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(1) To find (f ο g ο h)(-2), we first need to find g ο h and then apply f to the result. We have:

g ο h(x) = g(h(x)) = g(2x) = (2x)^2 = 4x^2

So, (f ο g ο h)(-2) = f(g(h(-2))) = f(g(-4)) = f(16) = 3(16) + 1 = 49

Therefore, (f ο g ο h)(-2) = 49.

(2) To find (f o f^-1)(2/3), we need to use the fact that f and f^-1 are inverse functions, which means that f(f^-1(x)) = x for all x in the domain of f^-1. Therefore, we have:

f(f^-1(x)) = 3f^-1(x) + 1 = x

Solving for f^-1(x), we get:

f^-1(x) = (x - 1)/3

So, (f o f^-1)(2/3) = f(f^-1(2/3)) = f((2/3 - 1)/3) = f(-1/9) = 3(-1/9) + 1 = 2/3

Therefore, (f o f^-1)(2/3) = 2/3.

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Option B is correct. The most accurate statement about the p-value for this test is: B. 0.01 < p-value < 0.05.

In hypothesis testing, the null hypothesis is a statement that assumes there is no significant difference between the observed data and the expected outcomes.

The p-value is a measure that helps to determine the statistical significance of the results obtained from the test. When the null hypothesis can be rejected at the 0.10 and 0.05 levels of significance, but not at the 0.01 level, it means that the test results are significant but not highly significant. In this case, the p-value must be greater than 0.01 but less than 0.05.

Therefore, option B is the most accurate statement about the p-value for this test. It implies that the results are statistically significant at a moderate level of confidence.

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One way that adding and subtracting polynomials is similar to adding and subtracting whole numbers and integers is that both operations follow the same basic rules for combining like terms.

In both cases, you add or subtract the coefficients (numbers) of the same type of term or same variable with the same exponent.

Just like adding and subtracting integers, you also need to consider the signs (+ or -) when combining the terms.

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