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Determine the domain of the following function: f(x)=x^2+3x-4/x(x^2-5)

If only the rule y = f(x) is provided, the set of all real x for which the function is defined is assumed to represent the domain.

Given that,

We have to find what is the natural domain of f(x).

We know that,

Consider the relationship between the independent variable x and the dependent variable y, y = f(x). It is considered to be in the domain of the function f if a value for x successfully enables the construction of a single value y using another value for x.

If only the rule y = f(x) is provided, the set of all real x for which the function is defined is assumed to represent the domain.

For instance, all real x has a value of zero when y = x. This is sometimes referred to as the function's natural domain.

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The quadratic equation, which is written in the form [tex]ax^{2} +bx+c=0[/tex] has two solutions because it is a second-order polynomial equation.

What is a quadratic equation?

Any equation of the form  [tex]ax^{2} + bx + c=0[/tex] where x is variable and a, b, and c are any real numbers where a ≠ 0 is called a quadratic equation.

As we know, the formula for the roots of the quadratic equation is given by :

[tex]x=[/tex][tex]\frac{-b+/-\sqrt{b^{2}-4ac } }{2a}[/tex]

where a, b, and c are the coefficients of the quadratic equation, and [tex]\sqrt{b^{2}-4ac }[/tex] represents the square root function. The plus-minus symbol (±) indicates that there are two solutions, one for each value of the symbol. These two solutions represent the roots of the equation, which are the values of x that make the equation true.

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There are commonly two answers to quadratic equations because the answers are where the parabola crosses the x -axis.

Quadratic equations are equations in two variables whose graph is a U-shaped parabola that is symmetrical about a vertical line, called the axis of symmetry. The standard form of a quadratic equation is [tex]y = ax^2 + bx + c[/tex], where [tex]a,b[/tex] and [tex]c[/tex] are real numbers, and x and y represent all (x,y) pairs that satisfy the equation. The solution(s) to a quadratic equation is the point(s) where the parabola crosses the x-axis. A quadratic expression can be written as the product of two linear factors and each factor can be equated to zero, So there exist two solution.

There are commonly two answers to quadratic equations because the answers are where the parabola crosses the x -axis.

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The value of p is 60°.

A triangle is a polygon with three sides and three vertices.

Sum of all the angles of the triangle is 180°.

Given, the triangle ABC;

∠A = p +  49° + ∠CAB = 180°  

∠CAB = 180° - p - 49°

And the sum of all the angles of the triangle is 180°.

∠CAB + ∠CBA + ∠BCA = 180°

180° - p - 49° + p - 15° + p -16° = 180°

p = 60°

Therefore, Evaluation of p is 60°

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The basic definition states that all the sides of an equilateral traingle is always same.

so equilateral traingle have same side and is justified  according to the definition of the equilateral traingle.

Now secondly we have to prove that all angles of an equilateral traingle is same  :

in the traingle PQR shown below PQ,QR,PR is the side of the traingle

we have to prove that ∠QPR = ∠PQR = ∠ PRQ.

As we know that all sides length of equilateral traingle are equal.

so PQ=QR=PR

Statement 1:

Angles opposite to equal sides QR and PR are equal so .

=>   ∠QPR = ∠PQR

Statement 2:

Angles opposite to equal sides PR and PQ are equal so .

 =>   ∠PQR = ∠ PRQ.

combining statement 1 and statement 2.

=>  ∠QPR = ∠PQR = ∠ PRQ and hence all three angles are same [Proved]

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-11/9 < c is the simplest form of the given 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:

In this question all that we have to do is to seperate the variable terms and the constant terms

It is showwn in the solution below

c - 8 < 10c + 3

-8-3 < 10c - c

-11 < 9c

-11/9 < c

as you can see after seperating the terms we can surely make an inequality

Refer to the Graph Attached

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The general case when G₁⊕G₂⊕G₃...⊕Gₙ is cyclic generated by (g₁, g₂, g₃,...,gₙ) is that the subgroups G₁, G₂, G₃,...., Gₙ are cyclic generated by  g₁, g₂, g₃, ..., gₙ respectively.

G⊕H is cyclic

That is G⊕H = <(g, h)>

Let us choose an arbitrary element g₁ ∈ G, then as G⊕H is cyclic

(g₁, h₁) ∈ G⊕H = <( g, h)>

This means that there exists an integer x such that

(g₁, h₁) = (g, h)ˣ

(g₁, h₁) = (gˣ, hˣ)

that is g₁ = gˣ

implies g₁ ∈ <g>

that is G ⊆ <g> as g₁ is an arbitrary element belonging to G

But we know <g> ⊆ G

Hence G = <g>

Thus G is a cyclic group generated by g.

Similarly we can prove that H group is cyclic generated by h, that is H = <h>

So the general case is that G₁⊕G₂⊕G₃...⊕Gₙ is cyclic generated by (g₁, g₂, g₃,...,gₙ), then the subgroups G₁, G₂, G₃,...., Gₙ are cyclic generated by  g₁, g₂, g₃, ..., gₙ respectively.

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Answer:

Out of these potential sources of bias, the most concerning the potential source of bias in the principal's poll is probably response bias.

Step-by-step explanation:

There are several potential sources of bias in the principal's poll that could have led to the discrepancy between the poll results and the principal's observations. Some possible sources of bias include:

Self-selection bias: It is possible that students who were more likely to admit to using their phone during class were more likely to participate in the poll. This could have led to an overestimation of the percentage of students who use their phone during class.

Response bias: It is possible that students who participated in the poll were not entirely truthful in their responses. For example, students who do use their phone during class might have been reluctant to admit it, leading to an underestimate of the percentage of students who use their phone during class.

Sampling bias: The principal might not have selected a truly random sample of students for the poll, leading to a sample that is not representative of the entire student body. For example, if the principal only polled students from certain classes or certain grade levels, the results might not be representative of the entire school.

Observation bias: The principal's observations might not have been completely objective. For example, the principal might have been more likely to notice students who were using their phone during class, leading to an overestimate of the percentage of students who use their phone during class.

If students are not completely truthful in their responses, it will be difficult to accurately gauge the extent to which students are using their phones during class. It is important to try to minimize response bias as much as possible in any survey or poll.

Answer:

Yes

Step-by-step explanation:

Therese infinite amount of rational numbers between 0 and 1 because of decimals.

0.9

0.99

0.999

0.9999

.....

There's an infinite amount of decimals you can add. If the sequence repeats, it is rational.

Answer:

Proportional relationships are relationships between two variables where their ratios are equivalent. Another way to think of them is that, in a proportional relationship, one variable is always a constant value for the other. That constant is known as the "constant of proportionality."

Step-by-step explanation:

The triangles can be proven to be congruent. ΔADC ≅ ΔCBA based on the HL congruence theorem.

The HL congruence theorem can be used to prove that two triangles are congruent if we can show that they are right triangles which have a pair of congruent hypotenuse and a pair of congruent legs.

Given the triangles in the image, ΔADC and ΔCBA, they have the following:

Therefore, both triangles are congruent by HL congruence theorem.

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To determine if a polynomial function is rising or falling, take the derivative of the function and evaluate it at the given point. If the derivative is positive, then the function is rising; if the derivative is negative, then the function is falling.

To determine if a polynomial function is rising or falling, take the derivative of the function. The derivative of a function at a given point is the slope of the tangent line at that point. If the slope is positive, then the function is rising; if the slope is negative, then the function is falling. To find the derivative of a polynomial function, use the power rule. This states that if f(x) = x^n, then the derivative of f(x) is nx^(n-1). Once the derivative is found, evaluate it at the given point. If the result is positive, then the function is rising; if it is negative, then the function is falling.

Let's say we have a polynomial function f(x) = x^3. To find the derivative of this function, use the power rule. The derivative of f(x) = x^3 is 3x^2. Now let's evaluate the derivative at the given point x = 2. The derivative at x = 2 is 3(2)^2 = 12. Since 12 is positive, the function is rising at that point.

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To identify corresponding parts of congruent triangles, you can use the transformations of translation, rotation, and reflection.

Translation involves moving a figure a certain distance in a certain direction, and all parts of the figure move the same distance in the same direction. If you translate two congruent triangles, their corresponding parts will still be corresponding.

The term "trigon" is occasionally (though not very frequently) used to refer to a triangle, a 3-sided polygon. Three sides and three angles, some of which may be the same, characterize every triangle.

The term "acute triangle" refers to a triangle with all three angles being smaller than. The term "obtuse triangle" refers to a triangle with any one angle greater than. The terms "right triangle" and "right-angled triangle" both refer to triangles with at least one angle of.

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The integer set that makes the inequality 3p - 10 > 7p + 6 true is given as follows:

{- infinity, ..., -6, -5}.

The inequality in this problem is defined as follows:

3p - 10 > 7p + 6

To solve the inequality, we treat it similarly to an equality, isolating the variable of interest. The difference is that the solution is a set with infinity values, and not a single value.

Hence the solution of the inequality is obtained as follows:

3p - 10 > 7p + 6

-4p > 16

4p < -16 (multiplying by -1 due to the negative coefficient of x, the sign also changes)

p < -16/4

p < -4.

Hence the integer set of the solution of the inequality is given as follows:

{- infinity, ..., -6, -5}.

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Answer: 4 cats

Step-by-step explanation:

10% = 0.1

Take 40 times 0.1 = 4 cats

z-score in normal distribution is a standardized value of raw score,i.e., random variable X when mean is 0 and standard deviation is 1.

What is standard deviation?

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The first ten even natural numbers are 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20. The mean of these numbers is the sum of the numbers divided by the number of numbers in the set. The sum of these numbers is 110 and there are 10 numbers in the set, so the mean is 11.

The mean of the numbers 9, 10, and 11 is 10. The mean of the numbers 12, 13, and 14 is 13. So the mean of the numbers 9, 10, 11, 12, 13, 14, 15, 16, 17, and 18 is (10+13)/2 = 11.5.

Standard Numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11 are the numbers we use to count or arrange things in numerical order. Detailed numbers the set of integers that includes zero and the natural numbers. neither a fraction nor a decimal.

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The area of the shaded region of the figure composed of a trapezoid and triangle is 26.25 m².

The figure shown is composed of a triangle within a trapezoid. The area of the shaded portion can be found as follows:

The area of the shaded portion is the subtraction of area of the triangle from the area of the trapezoid.

Hence,

area of the shaded portion = area of the trapezoid - area of the triangle

area of the trapezoid  = 1 / 2 (5 + 9)6

area of the trapezoid  = 1 / 2 (14)6

area of the trapezoid  = 42 m²

area of the triangle = 1 / 2 × 7 × 4.5

area of the triangle = 1 / 2 × 31.5

area of the triangle = 15.75 m²

Therefore,

area of the shaded portion = 42 - 15.75

area of the shaded portion = 26.25 m²

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The conditions for the discontinuity is illustrated as follows.

The term discontinuity in math is defined as the functions f(x) that are not continuous at a point of its domain.

If the function fall under the discontinuity if it satisfies any one of the following condition,

If the function will satisfies any one of these condition, then the function must discontinuous one.

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We're given the equation,

[tex]\longrightarrow\sin\theta+\csc\theta+2=0[/tex]

Take [tex]\sin\theta=x,[/tex] then [tex]\csc\theta=\frac{1}{x}.[/tex] Then the equation becomes,

[tex]\longrightarrow x+\dfrac{1}{x}+2=0[/tex]

Multiply both sides by [tex]x,[/tex] assuming [tex]x\neq 0.[/tex] Then we get,

[tex]\longrightarrow x^2+2x+1=0[/tex]

[tex]\longrightarrow (x+1)^2=0[/tex]

[tex]\Longrightarrow x=-1[/tex]

We're asked to find the value of,

[tex]\longrightarrow y=\sin^{2019}\theta+\csc^{2020}\theta[/tex]

As we've taken [tex]\sin\theta=x,[/tex] this expression becomes,

[tex]\longrightarrow y=x^{2019}+\dfrac{1}{x^{2020}}[/tex]

Now putting [tex]x=-1,[/tex]

[tex]\longrightarrow y=(-1)^{2019}+\dfrac{1}{(-1)^{2020}}[/tex]

[tex]\longrightarrow y=-1+\dfrac{1}{1}[/tex]

[tex]\longrightarrow y=-1+1[/tex]

[tex]\longrightarrow\underline{\underline{y=0}}[/tex]

Hence 0 is the answer.

In mathematics the expression of i² refers to or has a value equal to -1

We are going to define complex numbers, since "i" is a letter that denotes that it is an imaginary number.

Complex numbers are known as imaginary numbers and can be defined as an extension of the set of real numbers, since in its structure it has a real number followed by an imaginary number denoted with the letter "i".

The value of "i" starts from the square root of (-1)

√(-1) = i

Then if we want to know the value of i squared we do the following:

√(-1) = i

[√(-1)]² = i²

(-1) = i²

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The vertical asymptotes for the function f(x) = (x^2)(x-6)(x+3)^-1 are x=6 and x=-3.

The vertical asymptotes for the function f(x) = (x^2)(x-6)(x+3)^-1 are x=6 and x=-3. This is because the denominator can be factored into two linear expressions: x-6 and x+3. When either of these expressions equals 0, the fraction becomes undefined and a vertical asymptote is created. To find the asymptotes, set each linear expression in the denominator equal to 0 and solve for x. For x-6, x=6; for x+3, x=-3. Therefore, the vertical asymptotes of the function are x=6 and x=-3. When x is close to either of these values, the fraction will become very large, making the graph approach the vertical asymptotes without ever touching them.

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The solution of the given system of equations 3x + 6y = 2 ,  15x -30y = 10 is equal to x = 2/3 and y = 0.

As given in the question,

Given system of equations are :

3x + 6y = 2      ____(1)

15x -30y = 10   _____(2)

Multiply equation (1) by 5 and add it to equation (2) to eliminate y and get the value of x we have,

15x + 30y = 10

15x - 30y = 10

30x      = 20

⇒x = 20 / 30

⇒ x = 2 / 3

Substitute value of x = 2/3 in equation (1) to get the value of y,

3(2/3) + 6y = 2

⇒ 2 + 6y = 2

⇒ 6y = 0

⇒ y = 0

Therefore, the solution of the given system of equations is equal to x = 2/3 and y =0.

The above question is incomplete, the complete question is:

What is the solution of the system of equations 3x + 6y = 2 ,  15x -30y = 10?

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The term "difference" relates to subtraction, thus since they are looking for an unknown number's "difference," we can utilize the variable "x." X-7

Subtraction is an operation to find the difference of two numbers; yet, the outcome of subtraction is a difference.

The greater-than sign is represented by the letter >. Therefore, 9>7 is understood to mean "9 is greater than 7." The sign for less than is. (greater than or equal to) and are two more comparison symbols (less than or equal to).

The larger than, less than, and equal to signs are referred to as our "at least" signs.

7 divided by a number equals x-7.

Four times is four or four.

At the very least = -52 = Response negative 52

Connect them all,

x-7(4)<-52

4(x - 7) ≥ -52

4x - 28 ≥ -52

4x ≥ -24

x ≥ -6.

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The value of the given expression ( 402 ) ^ (1/2) using binomial theorem is equal to option a. 20.05.

As given in the question,

Given expression is equal to :

( 402 ) ^(1/2)

Standard form of binomial theorem is:

( a + b )ⁿ = ⁿC₀aⁿb⁰ +  ⁿC₁aⁿ⁻¹b¹ + ⁿC₂aⁿ⁻²b² +..............+  ⁿCₙa⁰bⁿ

Rewrite the 402 = 400 + 2 and represent it in binomial theorem form:

( 400 + 2 ) ^(1/2)

Here take out 400 as common factor :

= (400)^(1/2) ( 1 + 2/400)^(1/2)

= 20 ( 1 + 1/200) ^(1/2)

Using one of the result of binomial theorem to get the approximate value if other values are extremely small.

( 1 + x )ⁿ = 1 + nx

Here n = 1/2 and x = 1/ 200

= 20 [ 1 + (1/2) (1/200)]

= 20 ( 1 + 1 /400)

= 20 ( 401/400)

= 401 / 20

= 20.05

Therefore, the value of (402)^(1/2) using binomial theorem is equal to option a. 20.05.

The above question is incomplete , the compete question is:

The value of (402)^(1/2) according to the binomial theorem is

a.  20.05        b. 20.01         c. 20.00          d.  20.2​

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this is just base (b) times width (w) times height (h)

you can put them into an equation that looks like this:

b × w × h = area (a)

now lets put those numbers for the letters.

8 × 2 × 4 = a

a = 64

you're welcome :)

64 should be added to the equation to make it perfect square. The correct option is A.

A perfect square is a number that can be expressed as the product of an integer by itself or as the second exponent of an integer. Perfect square trinomials are three-term polynomials with a quadratic equation of ax^2 + bc + c

In order to find the c, we must divide 16 in the equation 16x , and raise it to the power of 2.

This will be:

(16/2)²

= 8²

= 64

Therefore the answer is 64.

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Complete question

Which term should be inserted in X² − +16 to make it a perfect square?

A. 64

B. 6

C. 4

D. 2

Answer:

80π feet

Step-by-step explanation:

The question is asking us for the circumference of a circle with diameter 80 ft.

The formula for the circumference of a circle is πd. Therefore, the circumference is π * 80 = 80π ft!

if this helps you, it would help me a lot if you could mark this as brainliest :)

Answer:

The circumference of a circle is equal to the diameter times pi, which is approximately 3.14. Therefore, the distance of one complete loop on a circular path with a diameter of 80 feet is approximately 80 * 3.14 = 251.2 feet.

Step-by-step explanation:

The system of linear equations has no solution. Then the number of the solution of the equations will be zero.

A connection between a number of variables results in a linear model when a graph is displayed. The variable will have a degree of one.

The linear equation is given as,

y = mx + c

Where m is the slope of the line and c is the y-intercept of the line.

The equations are given below.

y = 0 and y = - 7

The lines are parallel to each other. Then they will never intersect with each other.

The system of linear equations has no solution. Then the number of the solution of the equations will be zero.

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The horizontal distance from the point where it was launched is; Option C: 7.14 feet

Horizontal distance is defined as the distance between two points measured at a zero percent slope.

We are told that the path of the marshmallow can be modeled by the quadratic equation;

y = −0.07x² + x + 2.2

The maximum point will occur when dy/dx = 0

Thus;

dy/dx = -0.14x + 1

At dy/dx = 0,

-0.14x + 1 = 0

1 = 0.14x

x = 7.14

This value of x is the x-coordinate of the vertex and it represents the horizontal distance from the point where it was launched.

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If the volume of a rectangular pyramid is known, we can determine the length of the pyramid if we get the base area or the base dimensions. Here the length of the pyramid is the height of the rectangular pyramid.

The volume of a right rectangular pyramid of base l×b and height h can be determined by the formula

V = l×b×h/3

Therefore we can obtain the length or height of the rectangular pyramid given the volume if we know other two dimensions or their product, that is the base area of the pyramid. In which case we can obtain the height be substituting.

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