1. Graph the function f(x) = x² + 3 on the graph below. Then graph its inverse on the same graph.
Show the line y = x as a dashed line. (5 points each part)
a) Is the inverse a function?
b) Justify your answer.

The binomial 121y8z2 - 256x6 can be factored using the difference of two squares rule.

To factor the binomial 121y8z2 - 256x6, use the special factoring methods. Notice that the binomial has a difference of two squares, where the first term is a perfect square and the second term is the square of a binomial. This means that you can factor this binomial using the difference of two squares rule:

[tex]121y8z^2 - 256x^6 = (11y4z^2 - 16x^3) * (11y4z^2 + 16x^3)[/tex]

Therefore, the binomial 121y8z2 - 256x6 can be factored using the difference of two squares rule.

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If the dividend is 872 and the quotient is 62 and remainder is 4, the divisor is equal to 14.

In Mathematics, a quotient can be defined as a mathematical expression that is typically used for the representation of the division of a number by another number.

In Mathematics, dividend can be calculated by using this mathematical expression:

Dividend = divisor × quotient + residual

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

Dividend = divisor × quotient + residual

872 = divisor × 62 + 4

Divisor = (872 - 4)/62

Divisor = 868/62

Divisor = 14

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Complete Question;

If the dividend is 872 and the quotient is 62 and remainder is 4. What is the divisor?

Answer: it can change the amount of force that you can apply on an object.

Step-by-step explanation:

Answer:

Simple machine makes work go faster

Step-by-step explanation:

If you replace human Labour with simple machine, work is completed faster. It increases output

(a) The joint PMF of X, Y, and N can be found by multiplying the two independent PMFs of X and Y. The joint PMF is: P(X=x, Y=y, N=n) = P(X=x) * P(Y=y) = (1-p)^x * p * (1-p)^y * p = (1-p)^(x+y) * p^2.
(b) The joint PMF of X and N can be found by marginalizing over Y. The joint PMF is: P(X=x, N=n) = P(X=x, Y=n-x) = (1-p)^(x+(n-x)) * p^2 = (1-p)^n * p^2.

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3 a. The function w satisfies the equation (ex+2y)wy​−2xwx​=0.

3 b. The value of wxy​ is -2x(ex+2y)cos(ex+2xy) - 2ysin(ex+2xy)

4 . The material derivative of fatP is 0.

3. (a) To verify that the function w satisfies the equation (ex+2y)wy​−2xwx​=0, we can take the partial derivatives of w with respect to x and y, and then substitute them into the equation.

First, let's find the partial derivatives of w:

∂w/∂x = f'(ex+2xy)(ex+2y)
∂w/∂y = f'(ex+2xy)(2x)

Now, we can substitute these partial derivatives into the equation:

(ex+2y)(f'(ex+2xy)(2x)) - 2x(f'(ex+2xy)(ex+2y)) = 0

Simplifying this equation, we get:

2exf'(ex+2xy) + 4xyf'(ex+2xy) - 2exf'(ex+2xy) - 4xyf'(ex+2xy) = 0

This simplifies to 0 = 0, which is true. Therefore, the function w satisfies the equation (ex+2y)wy​−2xwx​=0.

(b) If f(u) = cosu, then we can find wxy​ by taking the partial derivative of w with respect to x and y, and then substituting f(u) = cosu:

∂w/∂x = f'(ex+2xy)(ex+2y) = -sin(ex+2xy)(ex+2y)
∂w/∂y = f'(ex+2xy)(2x) = -sin(ex+2xy)(2x)

Now, we can find wxy​ by taking the partial derivative of ∂w/∂x with respect to y:

wxy​ = ∂(∂w/∂x)/∂y = ∂(-sin(ex+2xy)(ex+2y))/∂y = -2x(ex+2y)cos(ex+2xy) - 2ysin(ex+2xy)

4. To find the material derivative of fatP, we can use the formula:

Df/Dt = ∂f/∂t + ∂f/∂x(dx/dt) + ∂f/∂y(dy/dt) + ∂f/∂z(dz/dt)

First, let's find the partial derivatives of f:

∂f/∂t = ysinx + ez
∂f/∂x = ty*cosx
∂f/∂y = tsinx
∂f/∂z = tez

Now, we can find the material derivative of fatP by substituting the point P = (π,1,0) and the derivatives of x, y, and z with respect to t:

Df/Dt = (1*sinπ + e^0) + (π*1*cosπ)(dx/dt) + (π*sinπ)(dy/dt) + (π*e^0)(dz/dt) = 0 + 0 + 0 + 0 = 0

Therefore, the material derivative of fatP is 0.

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Answer:

Let's call the weight of the book that was removed "x".

The total weight of the twelve books can be represented as 12 times the average weight of 2.75 pounds:

12(2.75) = 33

After the book is removed, there are only 11 books left, and their total weight can be represented as 11 times the new average weight of 2.70 pounds:

11(2.70) = 29.7

We can set up an equation using these two expressions and the weight of the book that was removed:

33 - x = 29.7

Solving for x, we get:

x = 33 - 29.7

x = 3.3

Therefore, the weight of the book that was removed was 3.3 pounds.

The distance from the point [tex]$(1,2)$[/tex] to the line [tex]$y=\frac{1}{2}x-3$[/tex] is approximately 3.7 units.

In mathematics, an expression is a combination of numbers, variables, and operators, which when evaluated, produces a value. An expression can contain constants, variables, functions, and mathematical operations such as addition, subtraction, multiplication, and division.

To find the distance from a point to a line, we need to find the length of the perpendicular segment from the point to the line.

The line [tex]$y=\frac{1}{2}x-3$[/tex]  can be rewritten in slope-intercept form as [tex]$y = \frac{1}{2}x - 3$[/tex], so its slope is [tex]\frac{1}{2}$.[/tex]

A line perpendicular to this line will have a slope that is the negative reciprocal of [tex]\frac{1}{2}$[/tex], which is [tex]-2$.[/tex]

We can then use the point-slope form of a line to find the equation of the perpendicular line that passes through the point [tex]$(1,2)$[/tex]:

[tex]$y - 2 = -2(x - 1)$[/tex]

Simplifying, we get:

[tex]$y = -2x + 4$[/tex]

Now we need to find the point where the two lines intersect, which will be the point on the line [tex]$y = \frac{1}{2}x-3$[/tex] that is closest to [tex]$(1,2)$[/tex]. We can do this by setting the equations of the two lines equal to each other and solving for [tex]$x$[/tex]:

[tex]$\frac{1}{2}x - 3 = -2x + 4$[/tex]

Solving for [tex]$x$[/tex], we get:

[tex]$x = \frac{14}{5}$[/tex]

To find the corresponding [tex]$y$[/tex] value, we can substitute this value of [tex]$x$[/tex] into either of the two line equations. Using [tex]$y = \frac{1}{2}x-3$[/tex], we get:

[tex]$y = \frac{1}{2} \cdot \frac{14}{5} - 3 = -\frac{7}{5}$[/tex]

Therefore, the point on the line [tex]$y = \frac{1}{2}x-3$[/tex] that is closest to [tex]$(1,2)$[/tex] is [tex]$\left(\frac{14}{5}, -\frac{7}{5}\right)$[/tex].

Finally, we can use the distance formula to find the distance between [tex]$(1,2)$[/tex] and [tex]$\left(\frac{14}{5}, -\frac{7}{5}\right)$[/tex]:

[tex]$\sqrt{\left(\frac{14}{5} - 1\right)^2 + \left(-\frac{7}{5} - 2\right)^2} \approx 3.7$[/tex]

Rounding to the nearest tenth, the distance from the point [tex]$(1,2)$[/tex] to the line [tex]$y=\frac{1}{2}x-3$[/tex] is approximately 3.7 units.

Therefore, the distance from the point [tex]$(1,2)$[/tex] to the line [tex]$y=\frac{1}{2}x-3$[/tex] is approximately 3.7 units.

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Answer:

I believe that no, it is not a rectangle because we don't know if the other 2 sides equal each other. In a rectangle 2 sets of sides are equal so no. it is not a rectangle.

thanks for coming to my ted talk

Step-by-step explanation:

The solution is, the value of x is, x=10.

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane.

here, we have,

from the given figure, we get,

the two triangles are similar

and, DE is the bisector of other  two sides,

so, we know from the property of  triangles, that,

DE = 1/2 * AB

we have,

DE = 5

and. AB =x

so, we get,

5 = x/2

or, x = 10

Hence, The solution is, the value of x is, x=10.

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Answer: None of the statements are true.

The expression has only two terms: 17 and (5)(9 + 14y).

No term in the expression equals 23.

The expression has three factors: 17, 5, and (9 + 14y).

The constant in the factor 9 + 14y is 9.

The factors in the expression are 17, 5, 9, and (9 + 14y).

Step-by-step explanation:

The area of rectangular field is   [tex]\frac{9}{20} \ \text{miles}^2[/tex].

The area a rectangle occupies is the space it takes up inside the limitations of its four sides. The dimensions of a rectangle determine its area. In essence, the area of a rectangle is equal to the sum of its length and breadth.

Here the given rectangle , length= [tex]\frac{9}{10}[/tex]miles and Breadth = [tex]\frac{1}{2}[/tex]miles

Now using area of rectangle formula,

=> A = length× breadth square unit.

=> A = [tex]\frac{9}{10}\times\frac{1}{2}[/tex]

=> A =   [tex]\frac{9}{20} \ \text{miles}^2[/tex]

Hence the area of rectangular field is  [tex]\frac{9}{20} \ \text{miles}^2[/tex].

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Answer:29 1/4

Step-by-step explanation:

Because the formula for the volume of a rectangular prism is (w)(h)(l).

The value of secant theta is √2. The solution has been obtained by using trigonometry.

Trigonometry, a subfield of mathematics, is the study of the sides, angles, and connections of the right-angle triangle.

We are given that tangent theta = -1

We know that tan θ = sin θ / cos θ

So,

⇒ -1 = sin θ / cos θ

⇒ -cos θ  = sin θ

We know that angle θ lies in the 4th quadrant i.e. between 3π/2 and 2π.

In the 4th quadrant, at 7π/4, the above is true.

So, at θ = 7π/4, we get

cos (7π/4) = √2/2

We know that

sec θ = 1 / cos θ

So,

sec θ = √2

Hence, the value of secant theta is √2.

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Quantitative, discrete, ratio

(i) Quantitative, discrete, interval
(ii) Qualitative, nominal
(iii) Qualitative, ordinal
(iv) Quantitative, discrete, ratio

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(a) The exact distance between the points is 2√10.

(b) The midpoint of the line segment is (-7, 2).

(a) To find the distance between the two given points, use the distance formula, which is:

d = √[(x₂ - x₁)² + (y₂ - y₁)²].

In this case, x₁ = -8, y₁ = 5, x₂ = -6, and y₂ = -1.

Plugging these values into the formula gives us:

d = √[(-6 - (-8))² + (-1 - 5)²]

d = √[(2)² + (-6)²]

d = √[4 + 36]

d = √40

d = 2√10

(b) The midpoint of a line segment can be found using the midpoint formula, which is:

(x₁ + x₂)/2, (y₁ + y₂)/2.

In this case, x₁ = -8, y₁ = 5, x₂ = -6, and y₂ = -1.

Plugging these values into the formula gives us:

midpoint = [(-8 + -6)/2, (5 + -1)/2]

midpoint = [(-14)/2, (4)/2]

midpoint = (-7, 2)

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FR" - Rand GR" - Rare linear functions. That is they each preserve vector additions in their own dimension and ca multiplications.  G ∘ F is a linear function.

The question is asking you to show that the composition of two linear functions, G and F, is a linear function itself.
To show that G ◦ F is a linear function, we need to show that it satisfies the two properties of linearity: preservation of vector addition and preservation of scalar multiplication.

Let's first consider preservation of vector addition:

(G ◦ F)(u + v) = G(F(u + v)) [by definition of composition]

= G(F(u) + F(v)) [since F preserves vector addition]

= G(F(u)) + G(F(v)) [since G preserves vector addition]

= (G ◦ F)(u) + (G ◦ F)(v) [by definition of composition]

Therefore, G ◦ F preserves vector addition.

Now let's consider preservation of scalar multiplication:

(G ◦ F)(ku) = G(F(ku)) [by definition of composition]

= G(kF(u)) [since F preserves scalar multiplication]

= kG(F(u)) [since G preserves scalar multiplication]

= k(G ◦ F)(u) [by definition of composition]

Therefore, G ◦ F preserves scalar multiplication.

Since G ◦ F satisfies both properties of linearity, it is a linear function that preserves vector addition and scalar multiplication.

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The solution is, that central angle Ф is, 3.49 radians.

Arc length formula is used to calculate the measure of the distance along the curved line making up the arc (a segment of a circle). In simple words, the distance that runs through the curved line of the circle making up the arc is known as the arc length.

here, we have,

The arc length formula is s = rФ,

where r is the radius and Ф is the central angle.  

We know s and r and need to calculate Ф.

now, we get,

From s = rФ

we get Ф = central angle = s/r

Here, that central angle Ф is,

(31.4 cm) / 9 cm

= 3.49 radians

Hence, The solution is, that central angle Ф is, 3.49 radians.

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Answer:

Should be the second answer.

Step-by-step explanation:

when dividing x^(4)-3x^(3)-7x^(2)-80 by x-5, the remainder will be -5.

The given polynomial equation is x^(4)-3x^(3)-7x^(2)-80 and we have to divide it by x-5 to find the remainder.

Set up the long division as follows:
```
x-5 | x^(4) - 3x^(3) - 7x^(2) + 0x - 80
```

Divide the first term of the dividend (x^(4)) by the first term of the divisor (x) to get the first term of the quotient (x^(3)):
```
x^(3)
_______
x-5 | x^(4) - 3x^(3) - 7x^(2) + 0x - 80
```

Multiply the first term of the quotient (x^(3)) by the divisor (x-5) and write the result below the dividend:
```
x^(3)
_______
x-5 | x^(4) - 3x^(3) - 7x^(2) + 0x - 80
    x^(4) - 5x^(3)
```

Subtract the result from the dividend to get the new dividend:
```
x^(3)
_______
x-5 | x^(4) - 3x^(3) - 7x^(2) + 0x - 80
    x^(4) - 5x^(3)
    --------------
         2x^(3) - 7x^(2)
```

Repeat the process with the new dividend until the degree of the remainder is less than the degree of the divisor:
```
x^(3) + 2x^(2) + 3x + 15
______________________
x-5 | x^(4) - 3x^(3) - 7x^(2) + 0x - 80
    x^(4) - 5x^(3)
    --------------
         2x^(3) - 7x^(2)
         2x^(3) - 10x^(2)
         ---------------
               3x^(2) + 0x
               3x^(2) - 15x
               ------------
                    15x - 80
                    15x - 75
                    --------
                         -5
```

The remainder is -5. Therefore, when dividing x^(4)-3x^(3)-7x^(2)-80 by x-5, the remainder will be -5.

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If the length of the flag is 12 ft more than its width, then the width of the flag is 30ft and length of the flag is 42ft.

Let "w" represent the width of the flag.

We know that the length of the flag is 12 feet more than its width, which means:

⇒ Length = w + 12

We also know that the perimeter of the flag is 144 feet,

The Perimeter of flag can be expressed as:

⇒ Perimeter = 2(Length + Width),

Substituting the expression for Length from above,

We get,

⇒ 144 = 2(w+12 + w)

Simplifying the equation:

We get,

⇒ 144 = 2(2w + 12)

⇒ 72 = 2w + 12

⇒ 60 = 2w

⇒ w = 30

So, width of flag is 30 feet. and Length = w + 12,

⇒ Length = 30 + 12

⇒ Length = 42

Therefore, the dimensions of the flag are 42feet by 30feet.

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Answer:

free

Step-by-step explanation:

28-28=0

The investor should invest $32,500 in the lowest-yielding, least-risky account, $20,000 in the medium-yielding, medium-risk account, and $17,500 in the highest-yielding, highest-risk account.

To solve this problem, we can use a system of linear equations.

We have three equations and three unknowns: x, y, and z.

The equations are: x + y + z = 50,0000.03x + 0.055y + 0.09z = 2540

We can use substitution or elimination to solve for one of the variables and then plug that value back into the other equations to find the remaining variables.

For example, we can solve for x in the first equation:

x = 50,000 - y - z

Then we can substitute this value of x into the second equation:

0.03(50,000 - y - z) + 0.055y + 0.09z = 2540

Simplifying this equation gives us:

1500 - 0.03y - 0.03z + 0.055y + 0.09z = 25400.025y + 0.06z = 1040

Now we can solve for one of the remaining variables, such as y:

y = (1040 - 0.06z) / 0.025

And we can substitute this value of y back into the first equation to find z:

50,000 - (1040 - 0.06z) / 0.025 - z = 50,000

Solving for z gives us:

z = 17,500

Finally, we can plug this value of z back into the equations for x and y to find the remaining variables:

x = 50,000 - y - 17,500 = 32,500 - y

y = (1040 - 0.06(17,500)) / 0.025 = 20,000

So the solution is x = 32,500, y = 20,000, and z = 17,500.

This means that the investor should invest $32,500 in the lowest-yielding, least-risky account, $20,000 in the medium-yielding, medium-risk account, and $17,500 in the highest-yielding, highest-risk account.

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The length of the sides x and y of the right-angled triangles using the trigonometric ratio of sine and cosine are:

11). x = 11√3 and y = 33

12). x = 4√7 and y = 8√7

The trigonometric ratios is concerned with the relationship of an angle of a right-angled triangle to ratios of two side lengths.

The basic trigonometric ratios includes;

sine, cosine and tangent.

recall that sin60° = √3/2 and cos60° = 1/2

For question 11:

sin 60° = y/22√3 {opposite/hypotenuse}

y = 22√3 × sin 60° {cross multiplication}

y = 22√3 × √3/2

y = 11 × 3

y = 33

cos 60° = x/22√3 {adjacent/hypotenuse}

x = 22√3 × cos 60° {cross multiplication}

x = 22√3 × 1/2

x = 11√3.

For question 12:

sin 60° = 4√21/y {opposite/hypotenuse}

y = 4√21/sin 60° {cross multiplication}

y = 4√21 ÷ √3/2

y = 4√21 × 2/√3

y = 4√7 × 2

y = 8√7

cos 60° = x/8√7 {adjacent/hypotenuse}

x = 8√7 × cos 60° {cross multiplication}

x = 8√7 × 1/2

x = 4√7

Therefore, the length of the sides x and y of the right-angled triangles using the trigonometric ratio of sine and cosine are:

11). x = 11√3 and y = 33

12). x = 4√7 and y = 8√7

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Answer: 0.66 cents per pen

Step-by-step explanation:

$4.65 for a pack of 7 pens. You want to divide $4.65 by 7 and you get 0.66

Both f(g(x)) and g(f(x)) equal x, we can determine that the functions f(x) and g(x) are inverses.

To determine if the functions f(x) and g(x) are inverses using composition, we need to find f(g(x)) and g(f(x)) and see if they both equal x.

First, let's find f(g(x)):
f(g(x)) = f((3)/(2)x+12)
= (2)/(3)((3)/(2)x+12)-8
= (2)/(3)(3)/(2)x + (2)/(3)(12) - 8
= x + 8 - 8
= x

Now, let's find g(f(x)):
g(f(x)) = g((2)/(3)x-8)
= (3)/(2)((2)/(3)x-8) + 12
= (3)/(2)(2)/(3)x - (3)/(2)(8) + 12
= x - 12 + 12
= x

Since both f(g(x)) and g(f(x)) equal x, we can determine that the functions f(x) and g(x) are inverses.

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The value of b that makes the area between the two functions equal to 50.5 on the interval [-6.2, 6.2] is 29.64355.

To find the value of b that makes the area between the two functions equal to 50.5 on the interval [-6.2, 6.2], we need to set up an integral and solve for b.

First, we need to find the difference between the two functions:
x2 - x1 = b - y^2

Next, we need to integrate this difference over the given interval:
∫[-6.2, 6.2] (b - y^2) dy

Using the power rule for integration, we get:
[b*y - (y^3)/3] from -6.2 to 6.2

Plugging in the values for the interval and simplifying, we get:
(6.2b - 158.488) - (-6.2b - 158.488)

Simplifying further, we get:
12.4b - 316.976

Now, we can set this equal to the given area and solve for b:
12.4b - 316.976 = 50.5

12.4b = 367.476

b = 367.476/12.4

b = 29.64354839

Rounding to five decimal places, we get:
b = 29.64355

So, the value of b that makes the area between the two functions equal to 50.5 on the interval [-6.2, 6.2] is 29.64355.

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Answer: This is a classic problem in mathematics known as the arithmetic series or the Gauss sum.

To find out how many days the person can donate with a total of Rs. 210, we need to sum the sequence of donations until we reach the total amount of Rs. 210. The sequence of donations is:

1 + 2 + 3 + 4 + 5 + ... + n

The sum of the sequence can be expressed as:

n(n+1)/2

So we need to solve the equation:

n(n+1)/2 = 210

n(n+1) = 420

n^2 + n - 420 = 0

We can solve this quadratic equation using the quadratic formula:

n = (-b ± sqrt(b^2 - 4ac)) / 2a

where a = 1, b = 1, and c = -420.

n = (-1 ± sqrt(1^2 - 4(1)(-420))) / 2(1)

n = (-1 ± sqrt(1 + 1680)) / 2

n = (-1 ± sqrt(1681)) / 2

n = (-1 ± 41) / 2

Since we are looking for a positive integer value of n, we can discard the negative solution:

n = (41 - 1) / 2

n = 40 / 2

n = 20

Therefore, the person can donate for a maximum of 20 days with a total donation of Rs. 210, starting with Rs. 1 on the first day and increasing by Rs. 1 each day.

Step-by-step explanation:

The equation of the line shown is y = 2x.

In Mathematics, a proportional relationship can be defined as a type of relationship that generates equivalent ratios and it can be modeled or represented by the following mathematical expression:

y = kx

Where:

Next, we would determine the constant of proportionality (k) as follows:

Constant of proportionality, k = y/x

Constant of proportionality, k = 2/1 = 4/2

Constant of proportionality, k = 2.

Therefore, the required equation is given by:

y = kx

y = 2x

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