1) Rewrite f(x)=[x] as a piecewise linear function from 6

The closure property of the whole number states that "Addition and multiplication of two whole numbers is always a whole number."

What is Property of Closure ?

A set has the closure property only under a particular operation which is if the result of the operation is always an element in the set.  When a set has the closure property under a particular operation, then we say that the set is “closed under the operation.”

a)The set of integers is closed under the operation of addition because the sum of any two integers is always another integer and is therefore in the set of integers.

b) The set of integers is not closed under the operation of division because when you divide one integer by another, you don’t always get another integer as the answer.  For example, 4 and 9 are both integers, but 4 ÷ 9 = 4/9.  4/9 is not an integer, so it is not in the set of integers!

Hence , the closure property of the whole number states that "Addition and multiplication of two whole numbers is always a whole number.

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An example of value of expression is 5 × 2a.

In the field of mathematics, a value of an expression can be described as such an expression that includes two or more two mathematical numbers. Apart from the numbers, one or more than one mathematical operation is also a part of the value of the expression.

In the example, 5×2a, 5 and 2 are the two mathematical numbers and multiplication (×) is the mathematical operation. Hence, the expression 5×2a serves as an appropriate example of the value of expression in the field of mathematics.

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The y-intercept of the equation 5x-4y = -12 is 3.

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

If a line passes through two points (x₁ ,y₁) and (x₂, y₂) ,

then the equation of line is

y - y₁ = (y₂- y₁) / (x₂ - x₁) x (x - x₁)

To find the slope;

m =  (y₂- y₁) / (x₂ - x₁)

We have the equation:

5x-4y = -12

To find the y-intercept:

First, we have to simplify the equation as subject to y.

5x-4y = -12

4y = 5x+ 12

y = (5/4)x+12/4

y =  (5/4)x + 3

Now, we know the slope-intercept form:

y = mx+b,

where m is the slope and b is the y-intercept.

Comparing, both the equation,

b = 3

3 is the y-intercept.

Therefore, y-intercept is 3.

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Answer:

[tex]a_n = \frac{1}{6}(6)^{n-1}[/tex]

Step-by-step explanation:

⭐What is the formula of a geometric sequence/progression?

We are given the geometric sequence.

Thus, we need to find the first term and the common ratio, and substitute those values into the geometric sequence formula.

The first term is [tex]\frac{1}{6}[/tex]

⭐How do we find the common ratio?

I am choosing term 1 and term 2, but you can choose any other 2 terms that are consecutive.

term 1 = [tex]\frac{1}{6}[/tex]

term 2 = [tex]1[/tex]

[tex]\frac{term2}{term1} = \frac{1}{\frac{1}{6}}[/tex]

[tex]\frac{term2}{term1} = 6[/tex]

∴ The common ratio of this sequence is 6.

Now, substitute these values (common ratio & first term) into the geometric sequence equation.

[tex]a_n = \frac{1}{6}(6)^{n-1}[/tex]

The numeric value of the expression 1/α + 1/β is given as follows:

-3/7.

The quadratic function in this problem is given as follows:

4x² + 3x + 7 = 0.

The coefficients of the quadratic function are given as follows:

a = 4, b = 3, c = 7.

The sum of the roots is given as follows:

α + β = -b/a.

Hence:

α + β = -3/4

The product of the roots is given as follows

αβ = c/a

Hence:

αβ =7/4.

These values are used to obtain the numeric value of the expression, as follows:

1/α + 1/β = (α + β)/αβ = (-3/4)/(7/4) = -3/7.

(the least common multiple of α and β is αβ, and then the least common factor is used to solve the expression).

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The common region for all the inequalities y ≥ 2, y ≤ 2x + 2and x + y ≤ 4 is represented by ΔABC (the last graph).

What is an inequality?

"It is a statement which consists of an order relationship (greater than, greater than or equal to, less than, or less than or equal to) between two mathematical expressions."

For given question,

We have been given three inequalities

y ≥ 2 ,

x + y ≤ 4 ,

y ≤ 2x + 2

If we draw the graph of these three inequalities then the common region represents the all the points that fulfill all three inequalities.

An inequality y ≥ 2 represents all the points on and above the line y = 2

An inequality x + y ≤ 4 represents all the points on and below the line       x + y = 4

An inequality y ≤ 2x + 2 represents all the points on and below the line    y = 2x + 2

Consider the following graphs.

The first graph represents the points which fulfill the inequality y ≥ 2

The graph for an inequality x + y ≤ 4 is represented by the shaded blue region.

The graph for an inequality y ≤ 2x + 2 is represented by the shaded green region.

The common region for all the inequalities is represented by ΔABC

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Using two column proof, we have been able to show that ΔABE is congruent to ΔCDE via SAS Congruency Postulate.

We are given certain statements of the given triangle and we want to get a flowchart of a proof which is a two column proof. Thus;

Statement 1; CD = AB

Reason 1; Given

Statement 2; BE = DE

Reason 2; AC is the segment bisector of BD

Statement 3; ∠ABE = ∠CDE

Reason 3; Alternate angles theorem

Statement 4; ΔABE ≅ ΔCDE

Reason 4; SAS Congruency Postulate

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The ratio that describes the number of baskets the player made to the number of baskets the player attempted is 9:15. This can be simplified to 3:5.

The ratio of the number of baskets made by the player to the number of baskets attempted is 9/15. This ratio can be simplified to 3/5, which means that for every 5 baskets attempted, the player made 3 baskets. This ratio can also be expressed as a percentage by dividing the number of baskets made by the number of baskets attempted and multiplying the result by 100%. In this case, the percentage of baskets made by the player is 60% (9 baskets made / 15 baskets attempted * 100% = 60%). This means that out of every 100 baskets attempted by the player, he made 60 baskets.

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There are two zeros of the polynomial 2x²-3x+4.

Polynomial is an expression consisting of indeterminates and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables

The given polynomial is 2x²-3x+4.

Two times of x square minus three times of x plus four.

2x²-3x+4=0

We find the roots by using quadratic equation formula.

x=-b±√b²-4ac/2a

a=2, b=-3, c=4

x=3±√9-4(8)/4

x=3±i√23/4

So the roots are x=3+i√23/4 and x=3-i√23/4.

A polynomial of degree 2 can have 2 real zeros, 1 real zero, or no real zeros

Hence, there are two zeros of the polynomial 2x²-3x+4.

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Answer:

15 people attended the party

The value of the given expression  5x²⁵ - 3x³² + 2x - 12 at x = 1 is -8.

The given expression is:

5x²⁵ - 3x³² + 2x - 12

To evaluate the value at x=1.

Substitute x = 1 in the given expression.

= 5(1)²⁵ - 3(1)³² + 2(1) - 12

Simplify:

= 5 - 3 + 2 - 12

= -8

Therefore, the value of the given expression at x = 1 is -8.

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The correct question is-

What is the value of 5x²⁵ - 3x³² + 2x - 12 at x=1?

Square Root of 120

In this mini-lesson, we will calculate the square root of 120 by the long division method along with solved examples and interactive questions. The square root of a number is the number that gets multiplied to itself to give the original number. 4 is a square root of 16 because 4 × 4 = 16 and the symbol square root is written as √ and is an integral part of mathematics.

Let us take a look at what the square root of 120 is.

Square Root of 120: √120 = 10.954

Square of 120: 1202 = 14400

Answer: 13.08

Step-by-step explanation:

i hope this helps

Answer:

h = 9

k = -8

Step-by-step explanation:

⭐ [tex]g(x) = (x-h)^2 + k[/tex] is already written in vertex form.

⭐ Vertex form: [tex]f(x) = a(x-h)^2 +k[/tex], where the vertex is (-(-h), k)

We are already given the vertex (9,-8), so we have to substitute these numbers into g(x).

[tex]g(x) = (x-9)^2 -8[/tex]

The equation of parabola is solved and the values of h and k are 9 and -8 respectively

A Parabola, open curve, a conic section produced by the intersection of a right circular cone and a plane parallel to an element of the cone. A parabola is a plane curve generated by a point moving so that its distance from a fixed point is equal to its distance from a fixed line

The equation of the parabola is given by

( x - h )² = 4p ( y - k )

y = a ( x - h )² + k

where ( h , k ) is the vertex and ( h , k + p ) is the focus

y is the directrix and y = k – p

The equation of the parabola is also given by the equation

y = ax² + bx + c

where a , b , and c are the three coefficients and the parabola is uniquely identified

Given data ,

The vertex form of a parabola function is given by:

g(x) = a(x - h)² + k

where (h, k) is the vertex.

In this case, we are given that:

g(x) = (x - h)² + k

And the vertex is located at (9, -8)

So we have:

h = 9

k = -8

Therefore, the function g(x) can be written as:

g ( x ) = ( x - 9 )² - 8

Hence , the equation of parabola is g ( x ) = ( x - 9 )² - 8

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Answer:

the answer is 20

Step-by-step explanation:

20+0=20

As the standard deviation 35 is high so the values are far from the mean.

In the statistics, standard deviation is an estimation that implies how far each available data apart from mean value. In a word, it is a measure of variances of data in relation to the mean. A low standard deviation indicates that values are closer to mean value and high standard deviation indicates the values are distributed widely.

Standard deviation of a given data set 35 means that all data set are 35 unit far from the mean value. As the value is high so all the values are dispersed widely relative to the mean value.

In the other hand, this high standard deviation implies there are enough variances in the given data in relation to the mean.

Standard deviation is one of the measures of risk assessment. Imagine an organization published a number of statistical observation data. If we find a high standard deviation, then it will be high risk of collapse of the business.

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the equation that represents the given situation is 12x + 24 = 60 and the number of batches Ruth needs to make to complete the order is 2 batches

The issue can be approached from the reverse. She already has 12 rolls, so take that out of the total. 48 will result from this. The "x" would then indicate how many more batches of 24 she needs to make in order to reach that 48 because she manufactures rolls in batches of 24. She would bake two additional batches to get 60 because 48 divided by 24 is equal to 2. If you put this concept into an equation, you get 24x + 12 = 60; 2 batches

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Answer:

No

Step-by-step explanation:

9/16 and 5/9 have a common factor is 144

So, we have

81/144  and 80/144

So, 9/16 > 5/9

Answer:

Step-by-step explanation:

no they are not equal, NO

So, an exponential function with a growth factor of 5 is of the form y = a * 5^x, where a is a constant coefficient.

An exponential function of the form y = a * b^x has a growth factor of b. Therefore, an exponential function with a growth factor of 5 is of the form y = a * 5^x, where a is a constant coefficient.

For example, if a = 2, then the function y = 2 * 5^x has a growth factor of 5.

Here is a table showing the values of y for some values of x:

x       y

0        2

1        10

2        50

3        250

4        1250

As x increases, the values of y increase exponentially at a rate determined by the growth factor of 5.

Therefore, an exponential function with a growth factor of 5 is of the form y = a * 5^x, where a is a constant coefficient.

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For example, if a = 2, then the function y = 2 * 5^x has a growth factor of 5.

Here is a table showing the values of y for some values of x:

x       y

0        2

1        10

2        50

3        250

4        1250

As x increases, the values of y increase exponentially at a rate determined by the growth factor of 5.

Therefore, an exponential function with a growth factor of 5 is of the form y = a * 5^x, where a is a constant coefficient.

Therefore , the solution to the given problem of the equation whose factors are (x+2)(x+1)(x+3).

Describe equation.

Two expressions joined by an equal sign form a mathematical statement known as an equation.

An equation is, for instance, 3x - 5 = 16. We can solve this equation and determine that the value of the variable x is 7.

According to the Given question:

x³ + 6x² + 11x +6

This pattern of coefficients 1, 6, 11, 6 is actually one that you are likely to encounter frequently.

For demonstration reasons, let's see what occurs if we begin by using the typical technique for resolving a general cubic:

We can use a linear substitution, t = x + b/3a, to simplify any cubic of the form ax3 + bx2 + cx + d, resulting in a cubic in the form a + b + et + f without a quadratic term.

This is the most straightforward type of Tschirnhaus transformation.

In our example, a = 1 and b = 6, so we want t = x + 2.

Note that:

=  + 6+ 12x +8

So we find:

+ 6+ 11x +6 =   − (x + 2)

=  - t

= t ( - 1)

= t(t − 1)(t + 1)

= (x+2)(x+1)(x+3)

Therefore , the solution to the given problem of the equation whose factors are (x+2)(x+1)(x+3).

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According to the 90° clockwise rotation, the new coordinate of the quadrilateral D' is (3, -2)

Quadrilateral

Quadrilateral refers a closed shape and a type of polygon that has four sides, four vertices and four angles.

Given,

If quadrilateral ABCD is rotated 90° clockwise about point A and then reflected across the y-axis to create quadrilateral A’B’C’D’.

Here we need to find the coordinate of D'.

In order to find the new coordinates, first we have to identify the current coordinates of the quadrilateral.

While we looking into the graph, we have identify the point of

A (0,0), B(-4, 3), C(1,5) and D(2,3)

Now, we have to use the rule of clockwise 90° rotation, states that to rotate the figure 90 degrees clockwise about a point, every point(x , y) will rotate to (y, -x).

According to this rule, the new coordinates are written as,

A(0,0) = A'(0,0)

B'(-4,3) = B'(-3,-4)

C(1,5) = C'(5, -1)

D(2,3) = D' (3, -2)

Therefore, the new coordinate of D' is (3, -2).

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Conscientiously cleaning your devices is crucial, as any residue that remains on your device may impact the accuracy of your measurements. The material your device is made of plays a significant role in the cleaning method you should use.

A tool used to measure a physical amount is called a measuring instrument. Measurement is the process of gathering and comparing physical quantities of actual things and events in the physical sciences, quality control, and engineering.

The process of measuring yields a number associating the subject under study and the referenced unit of measurement, and established standard objects and occurrences are employed as units. These numerical relations are acquired using measuring devices and rigorous test procedures that specify the instrument's use.

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To see the steps of copy and paste a formula.

Now, According to the question:

Here's how you copy and paste a formula:

1. Select the cell with the formula you want to copy.

2. Press. + C.

3. Click the cell where you want to paste the formula.

4. To quickly paste the formula with its formatting, press + V.

5. Clicking the arrow gives you a list of options.

Let's know to more:

How do I apply the same formula to multiple cells in Excel?

Fill formulas into adjacent cells

1. Select the cell with the formula and the adjacent cells you want to fill.

2. Click Home > Fill, and choose either Down, Right, Up, or Left. Keyboard shortcut: You can also press Ctrl+D to fill the formula down in a column, or Ctrl+R to fill the formula to the right in a row.

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The given data (2, 2, 3, 3, 4, 4, 5, 5, 6, 6) has no mode.

A mode is defined as the value that has the highest frequency in a given set of values. It is the value that appears the most number of times.

In statistics, the value that consistently appears in a particular collection is referred to as the mode. The mode or modal value in a data collection is sometimes referred to as the value or number that occurs most frequently in the data set. In addition, to mean and median, it is one of the three measures of central tendency.

The mode is the value that occurs the most often in a given set of data.

Given data:  2, 2, 3, 3, 4, 4, 5, 5, 6, 6.

In the given observations, data is repeated an equal number of times.  

So, the given data has no mode.

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Choose an equation, multiply a row by a constant to make it equal to 1, use row operations to make all other elements in the same column equal to 0, repeat for all elements in the matrix, when all elements are equal to 1, the 3x3 matrix is solved

To solve a 3x3 matrix for beginners, start by writing the equation in the form of Ax = B. Then, choose a set of three equations from the matrix that are independent of each other and solve the first equation for one of the unknowns, substituting the known values for the other two unknowns. Substitute the value of the unknown from the first equation into the other two equations and solve for the remaining two unknowns. Finally, check the solution by substituting the values of all three unknowns into the original equation.

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The union of two sets is to avoid counting the same elements twice.

The mathematical logic subfield of set theory investigates sets, which can be loosely defined as collections of objects. Although any object can be combined into a set, set theory, as a mathematical discipline, focuses primarily on those that are relevant to mathematics as a whole.

Given union of set

(A ∪ B),

The set of all objects that are members of either A or B, or both, is the union of the sets A ∪ B.

let us take example,

A = { 1, 2, 3, 4}

B = {3, 4, 5, 6, 7}

(A ∪ B) = { 1, 2, 3, 4} ∪  {3, 4, 5, 6, 7}

we can simply write all of A and B's elements in a single set to avoid duplicates to find A U B.

(A ∪ B) = {1, 2, 3, 4, 5, 6, 7}

Hence the  union of two sets is a set that contains all the elements of both set and avoid duplicates.

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Answer: (D) 5³ * 5⁹.

Step-by-step explanation:

The expression (5³)⁹ is the same as 5³ * 5³ * 5³ * 5³ * 5³ * 5³ * 5³ * 5³ * 5³, which can be simplified to 5^9 * 5^9 * 5^9 * 5^9 * 5^9 * 5^9 * 5^9 * 5^9 * 5^9. The only expression from the options above that is equivalent to this is 5³ * 5⁹, so the correct answer is (D) 5³ * 5⁹.

The meaure of the smallest exterior angle among the four exterior angles of a quadrilateral are x°,2x°, 4x°, and 5x° is x = 30° .

The angle formed between one side of a polygon and another line segment extending from the adjacent side is called the exterior angle of the polygon.

We have, quadrilateral with exterior angles x° , 2x° , 4x° and 5x° .

As we discuss, the sum of exterior angles of quadrilateral (polygon of four sides) = 360°

=> x° + 2x° + 4x° + 5x° = 360°

Adding all x values on left hand side,

=] 12x° = 360°

=> x = 360/12 = 30°

Now, the smallest exterior angle of quadrilateral is x° , so it's measure is 30°.

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Answer:

Solution:

Given,

Ratio of photo to to the photo frame=2:3

Length of photo frame=36cm

= 36÷3

=13

then,

Length of photo = 13 ×2

= 26cm

Step-by-step explanation:

Therefore,the length of the photo is 26cm