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Continuity refers to the property of a function or a curve in which a small change in the input results in a small change in the output, while discontinuity is a break or interruption in a function or curve.

An example of continuity is the function f(x) = x^2, which is a continuous function because, for any value of x, there is a corresponding y-value that is obtained by squaring x. This means that if x changes slightly, the value of y will change in a corresponding way.

An example of discontinuity is the function g(x) = 1/x, which is a discontinuous function because it has a break or singularity at x=0, where the function is undefined. This means that there is a gap in the curve at x=0, and the function is not defined for any value of x that is equal to zero

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The function f(x) = x2 serves as an illustration of continuity because there is always a corresponding y-value that can be calculated by squareing x. This implies that if x marginally changes, the value of y will also change.

The function g(x) = 1/x, which has a break or singularity at x=0, where the function is undefined, is an example of a discontinuous function. This indicates that the curve has a gap at x=0 and that the function is not defined for any x value equal to zero.

The formula for a horizontal line at y=4 is as follows: There is no slope. 4.0 is the y-intercept.

What are the Slope and intercepts ?

In mathematics, a line's slope, also known as its gradient, is a numerical representation of the line's steepness and direction.

The slope expresses how quickly the dependent variable changes when the independent variable changes. The Greek capital letter D or D is frequently used by mathematicians and economics to represent change. Slope contrasts the change in the vertical axis, y, with the change in the horizontal axis, x.

On the x-axis, negative numbers are below the origin, and on the y-axis, they are to the left of the origin. If the x-value is negative, begin at the origin and go to the left; if the y-coordinate is negative, move downward.

Line graphs, also known as XY graphs, are used to visually represent data (the outcomes of an experiment) in order to demonstrate the link between various factors, such as the severity of the disease and time.

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This expression of is identical to the statement in factored form of

4 (x² - 2x) 2 (x² - 3) is 2 (x - 1) (x – 3)

Given,

4(x² – 2x) – 2(x² – 3)

We would first of all expand the expression given

This becomes

= 4x² - 8x - 2x² + 6

Rearrange to enable us simplify

4x² - 2x² - 8x + 6

= 2x² - 8x + 6

Factorize

2 (x² - 4x + 3)

factorizing further using the factors of 3 that add up to -4

2(x² - x - 3x + 3)

pick out the common factors

2 (x (x - 1) - 3 (x - 1)

2 (x - 1) (x - 3)

Hence, the correct answer is A) 2(x - 1) (x – 3)

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Answer:

B)Write down a division that has an answer of 54.8.

When the null hypothesis is rejected, the F-ratio and probability of rejection both rise.

The F-ratio is calculated using the formula F = variance between treatments/variety within treatments. An increase in the numerator will raise the F-ratio, while a reduction in the denominator will raise it even further. The probability of rejecting the null hypothesis rises with increasing F.When the null hypothesis is rejected, the F-ratio and probability of rejection both rise.

The null hypothesis for an ANOVA test is that the means of the different groups are equal. If the F-ratio is high enough (i.e., if it is significantly different from 1), then it is likely that the means of the different groups are indeed different. Therefore, if the variance between treatments increases and the variance within treatments decreases, then the F-ratio will be higher, and the likelihood of rejecting the null hypothesis will be greater.

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For a given discrete probability distribution, The random variable also contains discrete values, The mean and standard deviation are computed in terms of Expected values E(X).

The mathematical formalization of a number or object that is subject to chance events is known as a random variable. It is a function or mapping between a space of potential outcomes, commonly real numbers, to a space of measurement.

The arithmetic mean, also known as the arithmetic average or simply the mean or average, is the sum of a set of numbers divided by the total number of numbers in the set. The collection frequently consists of a series of findings from a survey, experiment, or observational study.

The standard deviation is a statistic that expresses how much variance or dispersion there is in a group of numbers. While a high standard deviation suggests that the values are dispersed throughout a wider range, a low standard deviation suggests that the values tend to be close to the established mean.

For discrete probability distribution with P(x) as the probability of the event 'x'.

Mean = E(X) = x*P(x).

Variance = [tex]E(X^{2})-(E(X))^2[/tex]

standard deviation = [tex]\sqrt {Variance}[/tex]

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The roots of the quadratic equation are 3/2 and 2.

Polynomial is an equation written as the sum of terms of the form kx^n.

where k and n are positive integers.

We have,

2x² - 7x + 6

We can find the roots by middle-term factorization.

2x² - (4 + 3)x + 6 = 0

2x² - 4x - 3x + 6 = 0

2x(x- 2) - 3(x - 2) = 0

We have,

2x - 3 = 0 and x - 2 = 0

Now,

2x = 3

x = 3/2

x - 2 = 0

x = 2

Thus,

The roots are 3/2 and 2.

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The value of the account in 2020 is given as $8622.86.

The compound interest is a form of interest where the rate of interest is applied on the amount obtained after every interval of time.

The amount of one interval of time becomes the principal to the next one.

Suppose in 2005 t = 0.

Then, in the year 2020 t = 15.

Now, the given question can be solved by finding the Amount using the expression as below,

P(1 + r/100)ⁿ

Where p, r and n are the principal, rate and time for the compounding.

Substitute all the corresponding values to obtain,

A = 5000(1 + 3.7/100)¹⁵

   = 5000 × 1.037¹⁵ = 8622.86

Hence, in 2020 the money invested in the account becomes as $8622.86.

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Julie has only 23.79 seconds left to match the record.

We know that times deals with a continued sequence of existence and events that occurs in an apparently irreversible succession from the past

From the question, the given times are:

Time to match Lisa's record = 74.6 seconds

Time to clin the wall = 10.56 seconds

Time already spent on climbing wall = 4.025 seconds

Time left is calculated thus

74.600 seconds =(40.025+10.56) seconds

=74.600-50.585

=23.79 seconds

Therefore time left is 23.79 seconds

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Answer:

d. (2,1) and (4,2)

Step-by-step explanation:

Slope is rise/run or (y2 - y1) / (x2 - x1)

We see option D, the y increase by 1, and the x increase by 2

So, the slope of this equation is 1/2, and it matches our question.

So, D is the answer!

If you can, please give me a Brainliest! Thank you, and have a good day!

Answer:

d. (2,1) and (4,2)

y = 1/2x

2y = x

x = 2y

CHECKING

x = 2y        (substitute (2,1) and (4,2)

2 = 2(1)

2 = 2          correct

x = 2y

4 = 2(2)

4 = 4         correct

Step-by-step explanation:

There is always at least one consecutive value of n for which Marvin has a successful approach.

By induction, we can demonstrate that the game must conclude after a maximum of n(n+1)/2 turns for the first half of the issue, Here Induction Means a method for demonstrating that a statement P(n) is true for every natural number n, and that the infinitely numerous examples P(0), P(1), P(2), P(3), etc. all hold, is mathematical induction. Informal analogies, such a ladder or a falling domino, are used to explain this technique .

Base case: If there is just one bowl, Marvin can get rid of it in one turn, ending the game after one.

Step one of induction: Assume that there are k bowls and that the game terminates after no more than k(k+1)/2 turns. When there are k+1 bowls, we shall demonstrate that the game terminates after at most (k+1)(k+2)/2 turns. The number of bowls lowers by one if Marvin takes a marble out of one, and the game must terminate after no more than k(k+1)/2 turns through induction hypothesis

When Marvin transfers a marble from bowl A to bowl B, where bowl A contains at least one marble and bowl B contains an equal number of marbles, bowl A loses one marble and bowl B gains one marble. The number of bowls stays the same because bowl B has at least as many marbles as bowl A. As a result, the game must still terminate after k(k+1)/2 turns at the latest.

In any scenario, the game ends after at most k(k+1)/2 turns, thus when there are k+1 bowls, the game must end after at most (k+1)(k+2)/2 turns. For the second component of the issue, we can demonstrate that for every positive integer k, there exists a n such that Marvin can devise a plan to extend the n-bowl game for at least kn turns.

Let's say Marvin wants the n-bowl match to go at least kn turns. He can accomplish this by ensuring that each bowl always contains at least k marbles. Marvin can start by putting k marbles in the first bowl, k-1 marbles in the second bowl, and so forth, down to 1 marble in the nth bowl, to accomplish this.

For the second component of the issue, we can demonstrate that for every positive integer k, there exists a n such that Marvin can devise a plan to extend the n-bowl game for at least kn turns.

Let's say Marvin wants the n-bowl match to go at least kn turns. He can accomplish this by ensuring that each bowl always contains at least k marbles. Marvin can start by putting k marbles in the first bowl, k-1 marbles in the second bowl, and so forth, down to 1 marble in the nth bowl, to accomplish this.

Marvin can select a bowl containing at least k marbles on each turn and transfer one marble to a bowl containing fewer than k marbles .Marvin can transfer one marble from the first bowl to the second bowl, for instance, if there are k+1 marbles in the first bowl and k marbles in the second bowl. This guarantees that there are at least k marbles in each bowl. The game will go for at least kn turns because Marvin can take at least one turn every bowl.

For the third component of the issue, we can demonstrate that there is always at least one value of n for which Marvin has a successful strategy.

Assume that n is a positive integer. We shall demonstrate that either Marvin or Marisa has a winning strategy for the game of n bowls or the game of (n+1) bowls.

If Marvin has a winning strategy for the N-bowl game, he can use that method to win the N-bowl game.

If Marisa has a winning strategy for the n-bowl game and Marvin does not, then Marisa does as well. In this scenario, Marvin can use Marisa's successful approach to the n-bowl game as his own successful approach to the (n+1)-bowl game.

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Answer: √3 is irrational whereas √9 is rational

Step-by-step explanation:

3x3=9. 3^2=9. but, ?x?=3 and ?^2=3. there is no number for that

The product of [tex]\sqrt{3}[/tex] and [tex]\sqrt{9}[/tex] is irrational

•

An irrational number is a type of real number which cannot be represented as a simple fraction and [tex]\sqrt{3}[/tex] falls under this category. As fractions are just ratios, being multiplied by another number will not affect the ratio, thus making the product irrational.

we can draw in finite number circle from one point. so there are n number of possibility to draw circle from on given single point.

Every point in the plane that is a certain distance away from a certain point forms a circle (center). It is, thus, a curve formed by points moving in the plane at a fixed distance from a point. A circle is a closed two-dimensional object where every pair of points in the plane are equally spaced out from the "center." A line that goes through the circle creates a specular symmetry line. At every angle, it is also rotationally symmetric about the center.

Area of circle =  

[tex]\pi r^2\\r = 10\\A = 3.14 X 20 X 20\\A = 628unit sq\\[/tex]

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Additive inverse of 16 is -16.

Multiplicative Inverse of 16 is 1/16.

To produce a result equal to 0, add the original number and the additive inverse by simply reversing the sign of the integer. On the basis of negation of the original number, the characteristics of additive inverse are listed below. For instance, if x is the starting number, then -x is x's additive inverse.

A multiplicative inverse or reciprocal for a number x is a number that, when multiplied by x, produces the multiplicative identity, or 1. It is represented by 1/x or x/1. A fraction a/b has a multiplicative inverse of b/a. Divide 1 by the real number to obtain the multiplicative inverse of the number.

A function that "undoes" another function is called an inverse. In other words, if f(x) generates y, then y entered into the inverse of f creates x. An invertible function is one that has an inverse, and the inverse is indicated by the symbol [tex]f^{-1}[/tex].

Additive inverse of 16 is -16.

Multiplicative Inverse of 16 is 1/16.

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In the inequality 3x + 1 ≤ 4 at x = {1, 0, -1, -2, -3} all the values are satisfied therefore, the graph matches with this inequality.

What is Linear Inequality?

A linear inequality is a mathematical inequality that incorporates a linear function. A linear inequality contains one of the inequality symbols. It displays data that is not equal in graph form.

Solution:

To find the answer we need to put the values of x in options

3x + 1 ≤ 4

Putting x = {1, 0, -1, -2, -3}

At x =  1

3*1 + 1 ≤ 4

At x = 0

3*0 + 1 ≤ 4

At x = -1

3*-1 + 1 ≤ 4

At x = -2

3 * -2 + 1 ≤ 4

At x = -3

3*-3 + 1 ≤ 4

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The vertical asymptote of a function can be determined by setting its denominator equal to zero (0) and solving for x.

The horizontal asymptote of a function can be found by determining the degrees of the numerator and denominator.

In Mathematics, an asymptote can be defined as a line which the graph of a given function approaches but would never touch. Additionally, the vertical asymptote of a function simply refers to the value of x which makes its denominator equal to zero (0).

In order to graph any rational function, you should determine the values for which it is undefined. This ultimately implies that, a function is considered as undefined when the value of the denominator is equal to zero, which represents vertical asymptote lines.

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If the real estate agent's average weekly sales are $850 in the city and $1340 in the suburbs, then her total average weekly sales are $850 + $1340 = $2190.

The standard deviation of her weekly sales in the city is $260, and the standard deviation of her weekly sales in the suburbs is $390. If the sales in the city and the suburbs are independent, then the standard deviation of her total weekly sales is the square root of the sum of the squares of the standard deviations of her weekly sales in the city and the suburbs, which is the square root of $260^2 + $390^2 = $67600 + $152100 = $219100. This is equal to $468.

Therefore, the mean and standard deviation of the real estate agent's total weekly sales are $2190 and $468, respectively.

The product of (2x - 5)² can be written in the form az²+bx+c as follows:

(2x - 5)² = (2x)² - 2(2x)(5) + (5)²

= 4x² - 20x + 25

In this expression, a = 4, b = -20, and c = 25.

In a case of binomial distribution of of x where n = 5 and p = 0.05, the probability that x is at least 1 is not 0.6240, it is 0.2262.

Therefore, the answer is False.

The formula for binomial distribution is

P = nCx × pˣ × (1 - p)⁽ⁿ⁻ˣ⁾

where P is the probability

n is the total number of trials

x is the number of success trial

p is the probability of success in a single trial

From given details

n = 5

p = 0.05

P( x is at least 1) = 1 - P( x is less than 1)

P( x is at least 1) = 1 - P( x is 0)

P( x is at least 1) = 1 - 5C0 × 0.05⁰ × (1 - 0.05)⁽⁵⁻⁰⁾

P( x is at least 1) = 1 - 1 × 1 × 0.95⁵

P( x is at least 1) = 1 - 0.7738

P( x is at least 1) = 0.2262

--The question is incomplete, answering to the question below--

"If x is binomial with n = 5 and p = 0.05, the chance that x is at least 1 is 0.6240. True or False."

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Answer: between 5 and 6

Step-by-step explanation: The square root of 5 is 25 and the square root of 6 is 36 . Hence, the square root of 33 lies between 5 and 6.

Answer:

Point-slope form: y + 2= 0.5(x + 3)

Slope-intercept form: y = 0.5x - 0.5

Step-by-step explanation:

The sum of the first 7 terms of the following series is 52

'A geometric sequence is a sequence where every term has a constant ratio to its preceding term. A geometric sequence with the first term a and the common ratio r and has a finite number of terms.'

According to the given problem,

Sequence = 32,40,50

Let the common ratio be r,

Let the first term of the sequence be a,

⇒ a × r = second term

⇒ 32 × r = 40

⇒ r = 40/32

⇒ r = 5/4

We know,

Sum of n number of terms in a geometric sequence can be represented by,

sn=a(1-[tex]r^{7}[/tex])/1-r

⇒ [tex]s_{7}[/tex] =32(1 - (5/4)[tex]^{7}[/tex]/1 - 5/4

⇒ [tex]s_{7}[/tex] = 32 (1 - (5/4)[tex]^{7}[/tex]/3/5

⇒ [tex]s_{7}[/tex] = 5*32(1 - (5/4)[tex]^{7}[/tex])/3

⇒ [tex]s_{7}[/tex] = 160(0.972)/3

⇒ [tex]s_{7}[/tex] = 156/3

=52

Hence, we can conclude, the sum of the first 7 terms of the geometric sequence is 52

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The surface area of the triangular prism box Makayla decorates is 454.8 square centimeters

The surface area of a solid is obtained by adding together the surface areas of the faces or boundaries that encloses the solid.

The dimensions of the triangular prism are;

Rear rectangle;

Length = 12 cm

Width = 7 cm

Dimensions of side triangles;

Base length = 9 cm

Height = 7 cm

Area of each side triangle = 9 cm × 7 cm = 63 cm²

The dimensions of the rectangle at the bottom of the rectangular prism

Length = 12 cm

Width = 9 cm

Dimensions of the slant surface rectangle at the top of the triangular prism are;

Length = 12 cm

Width = 11.4 cm

The surface area of the box, A, is the sum of the individual surface areas, which can be found as follows;

A = 84 + 126 + 108 + 136.8 = 454.8

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Answer:

Heigh of rectangle: Area divided by base

Step-by-step explanation:

Area of rectangle: Base x Height

So, to find the height, we need to take the Area divided by the Base, and we will get the height of a rectangle!

The solution of the function  2x² + 3x + 2 = 0 is the zeros obtained after factorizing them which is found to be as -2 and -1 / 2.

The given equation is ,  2x² + 3x + 2 = 0

On factorizing this we will get ,

2x² +4x - x +2 = 0

=> 2x(x +2) - ( x + 2) = 0

=> (2x-1) (x + 2) = 0

Therefore , the zeroes are -2 and -1 / 2.

Factorization is a process in which we write a given equation in terms of their prime factors. The result of factorization is known as zeros off a polynomial and are the solution for the given equation.

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The expression b/a is equivalent to the given expression b⁻²/ab⁻³ when a≠ 0 and b ≠ 0. So, option D is correct.

The two or more expressions are said to be equivalent expressions if their value is the same even if they look different.

This is the same concept as equivalent fractions.  

The given expression is b⁻²/ab⁻³.

This expression is a fraction.

Simplifying the given expression:

b⁻²/ab⁻³

we know a⁻ⁿ = 1/aⁿ

So, we can write b⁻² = 1/b² and b⁻³ = 1/b³

Then, b⁻²/ab⁻³ = (1/b²)/a(1/b³)

⇒ b⁻²/ab⁻³ = 1/b² × 1/a × b³ = b × 1/a

∴ b⁻²/ab⁻³ = b/a

Thus, the equivalent expression for the given expression is b⁻²/ab⁻³ = b/a.

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Answer:

The answer is: Acute.

Step-by-step explanation:

All sides of a equilateral triangle are 60 degrees, meaning it is acute.

Answer:

x=6/5

Step-by-step explanation:

x-4=6x-10

-5x=-6

x=6/5

y=6/5-4=-14/5

Answer:

x = 1.2

y = -2.8

Step-by-step explanation:

y = x - 4 and y = 6x - 10

so x - 4 = 6x - 10

x - 6x = 4 - 10

-5x = -6

x = -6/-5

x = 1.2

y = x - 4 = 1.2 - 4 = -2.8

Given the provided coordinates of the points, The shape that is formed is a parallelogram because the slopes of the opposite sides are same.

In mathematics, a line's slope, also known as its gradient, is a numerical representation of the line's steepness and direction. Divide the difference in elevation between two points by the distance between them to find the percent slope, then multiply the result by 100. The term "rise" refers to the difference in elevation between two sites. The run is the separation between the points. Percent slope is therefore equal to (rise / run) x 100.

A coordinate system in geometry is a method for determining the precise location of points or other geometrical objects on a manifold, such as Euclidean space, using one or more numbers, or coordinates.

slope between (1,1) and (4,4),

m1 = 4-1/4-1 =3/3 =1

slope between (4,-2) and (7,1),

m2= 1+2/7-4 = 3/3 =1

m1=m2

similarly, slope between (1,1) and (4,-2),

m3= -2-1/4-1 = -3/3 =-1

slope between (7,1) and (4,4)

m4= 1-4/7-4 = -3/3 =-1

m3=m4

So, slopes of opposite sides are same that means they are parallel, The shape is parallelogram.

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