Identify an equation in standard form for ellipse with its center at the origin, a vertex at (3, 0), and a focus at (1, 0). Below are the options for the answer. HELP ASAP!!!

Answer:

3 and 4 ==> see work below

[tex]5. \quad\quad f^{-1}(x) = x^{1/7}[/tex]

[tex]6. \quad\quad f^{-1}(x) = -\left(\dfrac{5x}{2}\right)^{1/3}$}\\\text{We can also write this as $-\sqrt[3]{\frac{5x}{2}}$ }\\[/tex]

Step-by-step explanation:

Definition of inverse functions

If f and g are inverse functions, then f(x) = y if and only if g(y) = x

Or, in other words
If f(g(x)) = (g(f(x)) = x
then f and g are inverse functions

Q3

We have f(x) = x + 4 and g(x) = x - 4

To find f(g(x)), substitute g(x) = x - 4 wherever there is an x term in f(x)

f(g(x)) = g(x) + 4

= x - 4 + 4 = x

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

= x + 4 - 4 =x

Hence f(x) and g(x) are inverse functions

Q4

[tex]f(x) = \dfrac{1}{4}x^3\\\\g(x) = (4x)^{1/3}[/tex]

[tex]\\\begin{aligned}f(g(x)) &= \dfrac{1}{4} (g(x))^3\\\\\end{aligned}[/tex]

[tex]\begin{aligned}(g(x))^3 &= \left((4x)^{1/3} \right)^3 \\& = (4x)^{\frac{1}{3} \cdot 3}\\& = 4x\end{aligned}[/tex]

Therefore

[tex]\\\begin{aligned}f(g(x)) &= \dfrac{1}{4} (g(x))^3\\&= \dfrac{1}{4} \cdot 4x\\&= x\\\end{aligned}[/tex]

[tex]\begin{aligned}g\left(f(x)\right) & = \left(4f(x)\right)^{1/3}\\&= \left(4 \cdot \dfrac{1}{4}x^3\right)^{1/3}\\& = \left(x^3\right)^{1/3}\\& =x& \end{aligned}[/tex]

So f(x) and g(x) are inverse functions

Q5

[tex]\text{Given $f(x) = x^7 $ we are asked to find inverse $f^{-1}(x)$}[/tex]

[tex]\rm{Let \: y = f(x) = x^7}\\[/tex]

Interchange x and y:
[tex]x = y^7[/tex]

Solve for y:
[tex]y = x^{1/7}[/tex]

The right hand side is the inverse function of f(x)

[tex]f^{-1}(x) = x^{1/7}[/tex]

Q6
[tex]\rm{Given \;f(x) = -\dfrac{2}{5}x^3 \:find\:the\:inverse,\;f^{-1}(x)}[/tex]

Using the same procedure as for Q5

[tex]y=-\dfrac{2}{5}x^3\\\\x=-\dfrac{2}{5}y^3\\\\\text{Solve for y}\\[/tex]

[tex]y^3=-\dfrac{5x}{2}[/tex]

[tex]y=-\left(\dfrac{5x}{2}\right)^{1/3}\\\\\\\text{Inverse of $f(x)$ is $f^{-1}(x) = -\left(\dfrac{5x}{2}\right)^{1/3}$}\\\text{We can also write this as $-\sqrt[3]{\frac{5x}{2}}$ }\\[/tex]

Proportionately, if $170,500 profits in the second year of operation represent 576% of the company's first-year profits, the first-year profits will be $29,600.

A proportion describes two ratios equated to each other.

Proportion is a fractional value that compares one value or quantity to another.

The profits for the second year of operation = $170,500

Let the profits for the first year of operation = x

Proportionately, if $170,500 is 576% of x, x = $29,600 ($170,500 ÷ 576%)

Using equations, 5.76x = $170,500

x = $29,600 ($170,500  ÷ 5.76)

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Translating the shape S by the vector (  -1 , - 3 ) would lead to S ' with vertices of ( 1, -2 ), ( 1, -1 ), ( 3, -2 ), and (3, 1).

Given that the translation vector is negative, this means that you would have to subtract the corresponding values of the vector from the x and y coordinates of each vertex.

The vertices of the new shape S' would be :

Vertex 1: (2 , 1 ) + ( -1, -3 ) = ( 2 - 1 , 1 - 3 ) = ( 1, -2 )

Vertex 2: ( 2, 2 ) + ( -1, -3 ) = ( 2 - 1, 2 - 3 ) = ( 1, - 1 )

Vertex 3: ( 4, 1 ) + (-1 , -3 ) = ( 4 - 1 , 1 - 3) = (3 , -2)

Vertex 4: ( 4, 4 ) + (- 1, -3 ) = (4 - 1, 4 - 3 ) = (3 , 1 )

In conclusion, the translated vertices would be ( 1, -2 ), ( 1, -1 ), ( 3, -2 ), and (3, 1).

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A sample of at least 433 individuals would need to be taken to estimate the population mean with 98% confidence to within $100.

To estimate the sample size needed to estimate the population mean with 98% confidence to within $100, we can use the formula:

n = (z * σ / E)²

where:

n = sample size

z = z-score for the desired level of confidence (98% in this case), which can be found using a standard normal distribution table or calculator. For 98% confidence, the z-score is approximately 2.33.

σ = population standard deviation ($800 in this case)

E = margin of error ($100 in this case)

Plugging in the values, we get:

n = (2.33 * 800 / 100)²

n ≈ 432.64

Rounding up to the nearest whole number, we get a sample size of 433. Therefore, a sample of at least 433 individuals would need to be taken to estimate the population mean with 98% confidence to within $100.

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The quotient of functions f(x) and g(x) is given as follows:

(f/g)(x) = (x + 4)/(x - 3).

The quotient function of f(x) and g(x) is given by the division of function f(x) by function g(x), as follows:

(f/g)(x) = f(x)/g(x)

The functions for this problem are given as follows:

The functions can be factored as follows:

The term (x + 3) is common to both numerator and denominator, hence it is simplified and the quotient function is given as follows:

(f/g)(x) = (x + 4)/(x - 3).

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Hi Annie. I was interested in engaging you about your post because of my differing view. However, I think I am open to changing my views because I am not dogmatic. Do you think I am too pliable?

Throughout the Civil War, African Americans assumed crucial roles in the Union war effort. From enlisting as soldiers to acting as laborers and even spies, their contributions left an indelible impact on history.

Despite being subjected to rampant discrimination and racism from both Confederacy and Union forces, these individuals exhibited incredible heroism and bravery.

Excelling in dangerous missions that often led them to become prisoners of war, they persevered as they fought for the freedoms of themselves and others.

Hari Jones' TED talk shines a much-needed light on this overlooked aspect of history, revealing not only the extensive involvement of African American soldiers but also the pivotal role played by women in areas such as nursing, cooking, and espionage.

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If  u = - 3i + 2i,  v = 2i - 2j and w = - 4j, 5u(3v - 4w) is a scalar quantity with a value of 440.

To evaluate the expression 5u(3v - 4w), we need to first perform the scalar multiplication and then the vector subtraction. We can start by computing the scalar multiplication of 3v - 4w:

3v - 4w = 3(2i - 2j) - 4(-4j) = 6i + 14j

Next, we can perform the scalar multiplication of u and the vector 6i + 14j:

5u(3v - 4w) = 5(-3i + 2j)(6i + 14j)

Expanding the product, we get:

5(-18i² + 36ij + 42ij - 28j²)

Since i² = j² = -1, we can simplify this expression as:

5(18 + 70) = 440

Therefore, 5u(3v - 4w) is a scalar quantity with a value of 440.

The key concept used in this problem is the distributive property of scalar multiplication over vector addition and subtraction. This property allows us to simplify expressions that involve scalar multiplication and vector operations by distributing the scalar to each component of the vector.

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(a) The two dots on the graph at 53 represent two different samples of size 40 that both had a range of 53.

(b) No, the simulation does not provide evidence that the sample range is a biased estimator of the population.

Range is a statistical measure that refers to the difference between the highest and lowest values in a set of data. In other words, it is the distance between the largest and smallest values in a dataset.

(a) The two dots on the graph at 53 represent two different samples of size 40 that both had a range of 53.

(b) No, the simulation does not provide evidence that the sample range is a biased estimator of the population. A biased estimator systematically overestimates or underestimates the population parameter of interest. The fact that the distribution of sample ranges is approximately symmetric and centered around the true population range suggests that the sample range is an unbiased estimator of the population range. However, the variability of the sample range is quite large, which means that a single sample range may not be a very precise estimate of the population range.

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The matrix after the row operation is given as follows:

[tex]\left[\begin{array}{ccc}1&-\frac{4}{7}&\frac{8}{7}\\-12&7&-13\end{array}\right][/tex]

The row operation is given as follows:

Division by 7 = multiplication by 1/7, as the element at the desired position is of 7.

The matrix in the context of this problem is defined as follows:

[tex]\left[\begin{array}{ccc}7&-4&8\\-12&7&-13\end{array}\right][/tex]

We want to have element 1 into row 1 and column 1, that is, the element of 7 at row 1 and column 1 of the matrix must assume a value of -1.

An example of a valid row operation is the multiplication of the row by a constant, and as the element has a value of 7, the constant to obtain a value of 1 is:

7/k = 1

k = 7.

Dividing every element of the first row by 7, the resulting matrix is then given as follows:

[tex]\left[\begin{array}{ccc}1&-\frac{4}{7}&\frac{8}{7}\\-12&7&-13\end{array}\right][/tex]

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Answer:

16 = 6+2.5h

Step-by-step explanation:

The total cost is 16 dollars.  She spends 6 dollars for shoes and 2.50 per hour.

16 = 6+2.5h

To solve, subtract 6 from each side

16-6 = 6+2.5h-6

10 = 2.5h

Divide each side by 2.5

10/2.5 = 2.5h/2.5

4 =h

She can climb for 4 hours.

The expanded form of log y^10 is either 10 log y or 10 ln y

Assuming that the base of the logarithm is not specified, we can use the change of base formula to expand the logarithm:

log y^10 = log (y^10) / log (base)

Using base 10:

log y^10 = log (y^10) / log 10

= 10 log y / 1

= 10 log y

Using natural logarithm:

log y^10 = log (y^10) / log e

= 10 log y / 1

= 10 ln y

Therefore, the expanded form of log y^10 is either 10 log y or 10 ln y, depending on the base used.

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The type of statistical technique used here is inferential

We are given that;

Annual-sales distribution of cottage industries of the same standard=500

Number of industries=500

Now,

a) The type of statistical technique used here is descriptive statistics, which involves summarizing, organizing, and presenting data in a meaningful way. Descriptive statistics can use tables, graphs, and numerical measures such as mean, median, mode, range, standard deviation, etc. to describe the main features and patterns of a data set.

b) The type of statistical technique used here is inferential statistics, which involves drawing conclusions or making predictions about a population based on a sample. Inferential statistics can use methods such as estimation, hypothesis testing, confidence intervals, regression analysis, etc. to make inferences about the population parameters from the sample statistics.

Therefore, by statistics the answer will be inferential.

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The number of solutions for the given system of linear equations is: Infinitely many solutions.

To determine the number of solutions for the given system of linear equations:

3x - y = 4

6x - 2y = 8

We can use the algebraic method of solving the system of equations by substitution or elimination.

Let's use the algebraic method to determine the number of solutions:

3x - y = 4 (1)

6x - 2y = 8 (2)

We can simplify equation (2) by dividing both sides by 2:

3x - y = 4 (1)

3x - y = 4 (2')

Equations (1) and (2') are equivalent, which means that they represent the same line.

Therefore, the two equations represent the same line, and we have infinitely many solutions.

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Complete question

What’s the numbers that correspond to the corret number of solutions

0 solution, 1 solution, Many solutions

3x - y = 4

6x - 2y = 8

Answer:

-4x+19

Step-by-step explanation:

Otherwise known as option B :D

The value of b when the graph of function f(x) = a*bˣ passes through (2,54) and (5,16), is given by, b = 2/3.

Given the function of the graph is,

f(x) = a*bˣ

It is given that the graph of the function passes through the points with coordinates (2, 54) and (5, 16). So this points must satisfy the given function.

Substituting the point (2, 54) in the given function we get,

f(2) = 54

a*b² = 54 ..................... (i)

Substituting the point (5, 16) in the given function we get,

f(5) = 16

a*b⁵ = 16 ..................... (ii)

So dividing equation (ii) by equation (i) we get,

(a*b⁵)/(a*b²) = 16/54

b³ = 8/27

b³ = (2/3)³

b = 2/3

Hence the value of b is 2/3.

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Answer:

a) f(g(1)) = f(0) = 5

b) f(g(2)) = f(-3) = 8

c) g(f(2)) = g(3) = -8

d) g(f(3)) = g(2) = -3

e) g(g(1)) = g(0) = 2

f) f(f(3)) = f(2) = 3

The simplified expression for the perimeter of the rectangle is 10x-30

The perimeter of a shape is the addition of all the sides of the shape. If l is the length and w is the width, then the perimeter of a rectangle can be expressed as ;

p = 2(l+w)

The length is 3x-8 and the breadth is 2x-7

therefore the perimeter = 2( l+w)

= 2( 3x-8+2x-7)

= 2( 5x-15)

= 10x-30

therefore, the simplified expression for the perimeter of the rectangle is 10x-30

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The interval-notation for "all real-numbers" greater than 2 is (2, ∞).

The "Interval-Notation" is a way of expressing a set of real numbers using brackets to show which numbers are included in the set. In the case of all real numbers greater than 2, we can write this as:

(2, ∞)

The open-bracket "(" indicate that the number 2 is not included in the set, while the symbol "∞" represents all real numbers greater than 2.

This notation is used to express an open interval, meaning that the endpoints (in this case, 2 and infinity) are not included in the set.

Therefore, the required interval-notation is (2, ∞).

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The 99% confidence interval of the population mean is (352.32, 371.68).

To find the 99% confidence interval of the population mean, we need to use the following formula:

CI = x; ± zα/2 * (σ/√n)

where:

x' = sample mean (362)

zα/2 = z-score for 99% confidence level (2.576)

σ = population standard deviation (29.6)

n = sample size (40)

Substituting the values, we get:

CI = 362 ± 2.576 * (29.6/√40)

CI = 362 ± 9.68

This means that if we take many random samples of 40 hours from the population, we can be 99% confident that the true population mean of the number of customers per hour falls between 352.32 and 371.68.

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Complete question is:

A large department store found that it averages 362 customers per hour. Assume that the standard deviation is 29.6 and a random sample of 40 hours was used to determine the average. Find the 99% confidence interval of the population mean.

Answer:

x = -2/5

x = 3/4

Step-by-step explanation:

Factoring:

20x²-7x-6=0

20x²+8x - 15x - 6 = 0

4x * (5x + 2) - 3 * (5x + 2) = 0

(5x + 2) * (4x - 3) = 0

Cases:

5x + 2 = 0

4x - 3 = 0

x = -2/5

x = 3/4

Answer:

20x^2 -x(15-8)-6=0 here 6×20=120an by subtracting 15-8 we got seven where the 15×8 is also 120

Step-by-step explanation:

now,

20x^2 -15x+8x -6 =0 (here -×-is equal to +thus we write +8)

5x(4x-3) +2(4x-3)=0

(5x+2) (4x-3)=0

Either 5x=-2

X=-2/5

Or

4x=3

X=3/4

Answer: 53 ft

Step-by-step explanation:

The perimeter of a polygon is the sum of all sides. In this example, the sides are 4 ft, 20 ft, 18 ft, and 11 ft, so the perimeter is

4+20+18+11=53 ft

For the given function h(x) the points can be plotted on,

The given function h(x) is having four conditions and each condition can be clearly discussed below:

From, the above analysis we plotted the graph and attaching below.

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The vertices that produces the circuit of lowest cost is A and D. So, the correct answer is A) and D).

To apply the repeated nearest neighbor algorithm, we need to start at a vertex and repeatedly choose the nearest neighbor until all vertices have been visited. Then, we return to the starting vertex to complete the circuit.

Starting at vertex A, we can follow the path A-DE-BE-C-AD-BC-E-A. The total cost of this circuit is 3 + 1 + 13 + 7 + 6 + 5 + 3 = 38. Starting at vertex B, we can follow the path B-E-DE-BC-AD-C-A-B. The total cost of this circuit is 1 + 3 + 7 + 6 + 5 + 15 + 10 = 47.

Starting at vertex C, we can follow the path C-BC-AD-DE-E-B-CA-C. The total cost of this circuit is 5 + 6 + 1 + 13 + 3 + 15 = 43. Starting at vertex D, we can follow the path D-AD-BC-C-E-DE-BD-DA. The total cost of this circuit is 6 + 5 + 7 + 3 + 10 + 11 = 42.

Starting at vertex E, we can follow the path E-DE-BE-C-BC-AD-CA-E. The total cost of this circuit is 1 + 13 + 7 + 5 + 6 + 15 = 47. Therefore, the circuits with the lowest cost are those starting at vertex A and vertex D, both with a total cost of 38 and 42 respectively. So, the correct option is A) and D).

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If a vehicle uses 2 gallons of gasoline to travel 18¹/₂ miles, at this rate, the vehicle can travel 9¹/₄ miles per gallon of gasoline.

The rate refers to the average speed of the vehicle.

The rate also describes the unit rate or the rate expressed as a percentage or in terms of speed.

The number of gallons of gasoline used to travel 18¹/₂ miles = 2 gallons

The total distance traveled using two gallons of gasoline = 18¹/₂ miles

The average distance traveled per gallon of gasoline = 9¹/₄ miles (18¹/₂ ÷ 2)

Thus, on the average, the vehicle can travel 9¹/₄ miles miles per gallon of gasoline.

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A vehicle uses 2 gallons of gasoline to travel 18 1⁄2 miles. At this rate, how many miles can the vehicle travel per gallon of gasoline?

Joey’s monthly payment is solved to be

C. $79.70

20% down 600 to calculate the monthly payment on a 3 000 car we would pay 2 400.

the interest rate is 12% per annum which equates to 1% per month the term of the loan is three years which equates to 36 months we can use a formula to calculate the monthly payments

[tex]Monthly salary = (P * r * (1 + r)^n) / ((1 + r)^n - 1) .[/tex]

where

P is the principal (premium),

r is the monthly interest rate, and

n is the number of months.

Plug in the prices and we have:

[tex]Monthly salary = (2400 * 0.01 * (1 + 0.01)^{36} ) / ((1 + 0.01)^{36} - 1)[/tex]

= $79.65 per square foot

So Joey’s monthly payment would be $79.65.

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Answer:

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

The probability of catching a carp is 40% since 40% of the fish are carp.

We assume the fish is replaced each time so the probability of three carps is: