how many permutations of abcde are there in which the first character is abc and the last character is cde

The expression f (0) in a function is used to represents the y-intercept on the graph of the function f (x).

The point where the graph cuts the y - axis is known as the y - intercept whereas the point where it cuts the x - axis is known as x - axis. The graph cuts y- axis when the value of x coordinate is 0.

The values which satisfies the equation is known as the zeros of the function. The graph of a function is the set of all points in the plane of the form (x f(x)).

The graph of f could similarly be defined as the graph of the equation y = f(x). As a result, the graph of a function is a subset of the graph of an equation.

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Suppose that f and g are inverse functions, with f(g(x)) = g(f(x)) = x. A function's inverse retrieves the initial value, and g(y) = (y-5)/2 = x is the inverse of f. (x).

The inverse function theorem in mathematics provides a necessary requirement for a function to be invertible in the vicinity of a point in its domain, meaning that its derivative is continuous and non-zero at the point. This condition is relevant to differential calculus.

The derivative of the inverse function is also given a formula by the theorem. This theorem in multivariable calculus can be extended to any continuously differentiable, vector-valued function whose Jacobian determinant is nonzero at a point in its domain, providing a formula for the Jacobian matrix of the inverse.

The inverse function theorem also applies to complex holomorphic functions, differentiable functions between Banach spaces, differentiable functions between manifolds, and so forth.

Picard and Goursat used an algebraic approach to prove the theorem in the beginning.

Hence, Suppose that f and g are inverse functions, with f(g(x)) = g(f(x)) = x. A function's inverse retrieves the initial value, and g(y) = (y-5)/2 = x is the inverse of f. (x).

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The triangle has sides of 12, 36, and 37 and is not a right triangle.

What are Pythagoras's triples of the right triangle?

Pythagorean triples are defined as a² + b² = c², where a, b, and c are positive integers. These triples are denoted as (a,b,c). The perpendicular is a the base is b, and the hypotenuse is c of a right-angled triangle.

We have,

a triangle has sides of 12, 36, 37

A Pythagorean triple consists of three positive integers a, b, and c, such that a² + b² = c². Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k.

(12)² + (36)² = (37)²

1440 = 1369

S0, LHS ≠ RHS

Hence, the triangle has sides of 12, 36, and 37 is not a right triangle.

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The fraction Lee give the dog Gilligan is 7/10

From the question, we have the following parameters that can be used in our computation:

Initial amount of water = 5/7 gallons

Amount of water given to the dog = 1/2 gallons

The fraction given to the dog is then calculated as

Fraction given to the dog = Amount of water given to the dog/Initial amount of water

Substitute the known values in the above equation, so, we have the following representation

Fraction given to the dog = (1/2)/(5/7)

Express as products

Fraction given to the dog = 1/2* 7/5

Evaluate the products

Fraction given to the dog = 7/10

Hence, the fraction is 7/10

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Consider a cube of side a units, with sides along x, y and z axis and one vertex at origin, as in the figure.

Here AB and CD are two diagonals of the cube, because A and B are diagonally opposite corners, also C and D.

In the figure, coordinates of A is (0, 0, 0) and that of B is (a, a, a). Then vector AB is given by,

[tex]\vec{\rm{AB}}=\rm{\left < a,\ a,\ a\right > }[/tex]

Coordinates of C is (a, 0, 0) and that of D is (0, a, a). Then vector CD is given by,

[tex]\vec{\rm{CD}}=\rm{\left < -a,\ a,\ a\right > }[/tex]

The diagonals each have a magnitude of a√3.

[tex]|\vec{\rm{AB}}|=|\vec{\rm{CD}}|=\rm{a\sqrt3}[/tex]

In the figure θ is the angle between the diagonals.

We take the dot product of the vectors AB and CD to get angle between them.

[tex]\longrightarrow\vec{\rm{AB}}\cdot\vec{\rm{CD}}=|\vec{\rm{AB}}|\cdot|\vec{\rm{CD}}|\cdot\cos\theta[/tex]

[tex]\longrightarrow\rm{\left < a,\ a,\ a\right > \cdot\left < -a,\ a,\ a\right > =a\sqrt3\cdot a\sqrt3\cdot \cos\theta}[/tex]

[tex]\longrightarrow\rm{-a^2+a^2+a^2=3a^2\cos\theta}[/tex]

[tex]\longrightarrow\rm{a^2=3a^2\cos\theta}[/tex]

[tex]\longrightarrow\rm{\cos\theta=\dfrac{a^2}{3a^2}}[/tex]

[tex]\longrightarrow\rm{\underline{\underline{\cos\theta=\dfrac{1}{3}}}}[/tex]

The angle between the diagonals satisfy this equation.

Hence Proved!

The altitude of the triangle is determined by, a = √(c² - b²)

Given,

Altitude of triangle;

A triangle's altitude is determined by drawing a perpendicular line from its vertex to its opposite side. The altitude, sometimes referred to as the triangle's height, forms a right-angle triangle with the base.

Here,

The simplest way is to use the Pythagorean theorem if you are aware of two additional right triangle sides:

a² + b² = c²

Where,

a is the altitude, b is the base and c is the hypotenuse

Then,

If leg a is the missing side, change the equation such that a is on one side and compute the square root of that number:

a = √(c² - b²)

That is,

We can find the altitude of the triangle by, a = √(c² - b²)

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Red to tiny will have a probability of 12/20.

Describe probability.

In other terms, probability is a number that indicates the likelihood of an event occurring. Probability is defined as the ratio of favorable outcomes to all outcomes.

Probability is calculated as follows: N = N samples / N favorable outcomes

There are 20 blue numbers, of which the large ones are 17 and the small ones 3. Likewise, there are 20 red digits; the large ones are 8 and the small ones are 12.

The likelihood of receiving a little red number will be determined as follows:

Probability is calculated as follows: N = N samples / N favorable outcomes

12 good outcomes total.

There are 20 samples total.

Fill out the formula above with the value.

12/20 is the probability.

As a result, 12/20 will be the likelihood of red to little.

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Answer: 24

Step-by-step explanation:

1*4=4

2*4=8

3*4=12

6*4=24

9*4=36

V = 1/3πr²h

•Mutiply both sides by 3 to cancel out the fraction

3V = πr²h

•Divide both sides by πh 3V/πh = r²

•Find the square root of both sides

√3V/πh = r

•Switch the sides to make it easier to read

r = √3V/πh

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1/3 is the slope of the line represented by the equation y = 1/3x + 4.

In other words, the slope is the ratio of the change in height to the horizontal distance between any two locations on the line. It is calculated as the ratio of the "rise" divided by the "run" between two points on a line. The angle that a straight line forms with the positive x-axis direction is known as its inclination. The slope or gradient of a line is the tangent of the angle it creates with the positive x-axis direction. A line's slope is typically represented by the symbol m.

Given

Equation

y = 1/3x + 4

Using slope intercept form,

y = mx + b

m = slope

m = 1/3

1/3 is the slope of the line represented by the equation y = 1/3x + 4.

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The rational zero theorem states that each rational zero(s) of a polynomial with integer coefficients f(x) = [tex]a_{n} x^{n} +a_{n-1} x^{n-1} +...........+a_{1} x^{} +a_{0}[/tex]

is of the form p/q, where:

->p is a factor of the constant

->q is a factor of the leading coefficient

->p and q are relatively prime (i.e., the fraction p/q is in its simplest form)

The rational root theorem, also known as the rational root test, is an algebraic theorem that states that for a polynomial equation in one variable with integer coefficients to have a solution (root) that is a rational number, both the constant term (the term without a variable) and the leading coefficient (the coefficient of the highest power) must be divisible by the fraction's denominator.

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 The y-intercept is the point on the graph where it crosses the y-axis, and has coordinates of (0, b). The y-coordinate is the value of b in the slope-intercept form. The Slope is - ⅖.

Y-intercept = (0, 4), where b = 4

Given the linear equation in standard form, 2x + 5y = 20, where A = 2, B = 5, and C = 20:

Start by transforming the standard equation into its slope-intercept form, y = mx + b, where m = slope, and b = y-intercept.

Subtract 2x from both sides:

2x -2x + 5y = - 2x + 20

5y = -2x + 20

Divide both sides by 5 to isolate y:

y = - ⅖x + 4 ⇒ This is the slope-intercept form where the slope, m = -⅖, and the y-intercept, b = 4.  The y-intercept is the point on the graph where it crosses the y-axis, and has coordinates of (0, b). The y-coordinate is the value of b in the slope-intercept form. Therefore, the y-intercept is (0, 4).

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Answer:

k = - 0.4 , y = - 0.4x

Step-by-step explanation:

given y varies directly as x then the equation relating them is

y = kx ← k is the constant of variation

to find k use the condition y = 0.4 when x = - 1 , then

0.4 = - k ( multiply both sides by - 1 )

- 0.4 = k

then constant of variation is k = - 0.4

y = - 0.4x ← equation of variation

The unit vector in the same direction will be equal to  (6 i/5.1 - 3 j/5.1).

A vector is a quantity with both magnitude and direction in physics. It is often depicted by an arrow with the same direction as the amount and a length proportionate to the size of the quantity.

A vector lacks position, but has magnitude and direction. A vector is therefore unaffected by displacement parallel to itself as far as its length is unaltered.

As per the given data in the question,

The unit vector is calculated as:

[tex]\vec u_a=\frac{\vec a}{||\vec a||}[/tex]

[tex]\vec a= x \vec i+y \vec j[/tex]

Now, calculate the magnitude of the vector,

[tex]\vec a[/tex] = √(x² + y²)

[tex]|\vec a|[/tex] = √6² - 3²

= √36 - 9

= 5.1

So, the unit vector will be,

[tex]\vec u_a[/tex] = (6 i - 3 j)/5.1

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Answer:

c. 49

Step-by-step explanation:

(g o f) (3)

f(x) = 3x - 2

g(x) = x²

(g o f) (x) = (3x - 2)²

          = (3 × 3 - 2)²

          = (9 - 2)²

          =7²

          =49

several sixteens. Numerous numbers that are multiples of 16 include 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, and 176.

In mathematics, a multiple is any number that results from multiplying a given number by an integer. As an illustration, multiples of 5 include: 10, 15, 20, 25, 30, etc. 14, 21, 28, 35, 42, 49, and so on are all multiples of 7.

A number is a multiple of a number if another natural number and the given number are multiplied together.

Divide the digits into pairs starting at the end, then multiply the final group by 1, the group before it by 2, the group before that by 4, then 8, 16, and so on. Include every component. If the outcome is a multiple of 7, then so was the original number. Since 91 (137) is a multiple of 7, so is 21553.

16-digit multiples. There are 16 multiples, including 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, and 176.

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Angles can be measured using one of two standard units. Degrees are the most widely used unit of measurement.

∠ABD = 39°

Therefore, the correct option is c) 39°

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We need to use the concept of Finding zeros of the polynomial function. The zeros of the function are 1/2,1+2i,1-2i.

The values of x that fulfill the formula f(x) = 0 are the zeros of a polynomial. The polynomial's zeros are the x values for which the function's value, f(x), equals zero in this case. The degree of equation f(x) = 0 determines how many zeros a polynomial has. The zeros of the polynomial are all such domain values of the function for which the range is zero.

The locations where the graph of y = f(x) crosses the x-axis are known as the polynomial's zeros graphically. In the information that follows, we will learn more about how to express a polynomial's zeros on a graph.

Take f(x)=2x^3 -5x^2+12x-5

So in the question he mentioned that (2x-1) is factor of the polynomial.

So take out (2x-1) from the polynomial function

f(x)=(2x-1)(x^2-2x+5)

We know that zeros of a polynomial can be found when f(x)=0

⇒(2x-1)=0

∴x=1/2

⇒(x^2-2x+5)=0

By using the concept of finding roots of a quadratic equation

(-b±√b2-4ac)/2a

⇒(2±4i)/2

∴x=(1+2i) and (1-2i)

∴1/2,(1+2i),(1-2i) are the zeros.

We need to use the concept of Finding zeros of the polynomial function. The zeros of the function are 1/2,1+2i,1-2i.

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The relationship of the volume of the pyramid to the volume of prism: The volume of a pyramid is 1/3 the volume of a prism with the same base area and height.

In this question we need to find the relationship of the volume of the pyramid to the volume of prism.

The volume of a pyramid is ________ the volume of a prism with the same base area and height.

We know that the formula for the volume of pyramid.

V1 = 1/3 * base area * height

And the formula of the volume of prism is:

V2 = base area * height

We have been given a pyramid and prism have the same base area and height.

So,  volume of pyramid = 1/3 * volume of prism

Consider a ratio the volume of the pyramid to the volume of prism.

(the volume of the pyramid) / (the volume of prism) = 1/3

Therefore, the volume of a pyramid is 1/3 the volume of a prism with the same base area and height.

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There is a 4/5 probability of catching a T-shirt that is not a size small.

Simply put, the probability is the likelihood that something will occur.

When we don't know how an event will turn out, we can discuss the likelihood or likelihood of several outcomes.

Statistics is the study of events that follow a probability distribution.

So, there were 50 shirts thrown into the throng in total.

There were 10 little shirts thrown.

15 medium-sized shirts were hurled.

The quantity of extra-large and big shirts.

= Total shirts - ( 10 + 15)

= 50 - 25 = 25

So, a total of 25 shirts are scattered throughout the large/extra large shirts.

A number of shirts that are NOT SMALL size are now available.

= Total shirts - Shirts with small size = 50 - 10 = 40

Probability of getting a not small t-shirt = The total not small t-shirt/The total number of t-shirts

= 40/50

= 4/5

Therefore, there is a 4/5 probability of catching a T-shirt that is not a size small.

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According to the SAS rule, the two triangles are congruent.

In geometry  SAS rules states that if two triangle are congruent if two sides and the included angle of one triangle are equal to the two sides and the included angle of the other triangle.

Here we have to prove that the two triangle given here under are congruent using SAS rule.

While we looking into the following diagram, we have identified that the following are given,

In the given triangle, we have know that that the triangle △GHJ and △RTS, has the side values HJ =TS

Here we also know that the angles ∠GHJ=∠STR are equal.

Then the sides of the triangle HG = TR

Therefore, the △GHJ Congruent to △RTS

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To determine which graph is greater, you need to compare the y-values of the two graphs at the same x-values or look at the overall shape of the graphs.

You can compare the y-values of the two graphs at the same x-values to determine which graph is greater if one graph has a higher y-value than the other graph at a particular x-value, then the first graph is greater at that x-value.

For example, suppose you have two graphs: one is a straight line with a slope of 2 and the other is a straight line with a slope of -3. If you compare the two graphs at the same x-values, you will see that the graph with the slope of 2 has a higher y-value than the graph with the slope of -3 for all x-values. This means that the graph with the slope of 2 is always greater than the graph with the slope of -3.

You can also compare the graphs by looking at the overall shape of the graphs. For example, if one graph is always above the other graph, then it is clear that the first graph is greater. On the other hand, if one graph crosses over the other graph, then there may be some x-values for which one graph is greater and some x-values for which the other graph is greater.

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The statement that correctly describes related variables is given as follows:

Two variables are related when a discernible pattern exists between them.

Two variables are said to be related if one variable can be written as a function of another, that is, the output variable can be written as a function of the input variable.

There are multiple relations between variables, such as linear, quadratic and exponential relations. What is common for all these relations is that a pattern exists between these variables, meaning that the third option gives the correct option.

The pattern is identified from a sample of observations of these two variables, in which it is verified how the dependent variable varies according to the independent variable.

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The best represents causation is D, As the sale of winter coats increase the amount of hot chocolate sold decreases.

Causation can be defined as the capacity of one variable to influence another. The first variable may bring the second into existence.

There are two types of causation:

Therefore the correct option is D because the best way to represent causation is when a sale of winter coat increases the amount of hot chocolate sold  decreases.

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The Square root function is written in the form of f(x) = √x .

The Square Root function is defined as the function in which the parent function is written in the form of f(x) = √x .

and the inverse of the Square root function is square function , f(x) = x² .

The Domain and range of the root function is [0 , ∞) .

which means the range is the set of all positive real numbers and

hence in the square root function, result is always positive.

Some Examples of Root Functions are , √2 , √3 , √8 etc.   .

Therefore , Square Root function is explained .

The given question is incomplete , the complete question is

What is the Square Root Function ?

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The zeros of the function x^2 + 5x + 5  = 0 are (-5 ± sqrt (5))/2.

We know that the general form of a quadratic equation is:

ax^2 + bx + c =0

The equation we have is:

x^2 + 5x + 5  = 0

Comparing the given equation with the general form of a quadratic equation, we get that:

a = 1, b= 5, and c = 5

To find the zeros of the any quadratic function, we use the formula:

x = (-b ± sqrt (b^2 - 4ac))/(2a)

Therefore, the zeros of the equation x^2 + 5x + 5  = 0 are:

x = (-5 ± sqrt (5^2 - 4x(1)x(5))/(2x1) = (-5 ± sqrt (5))/2.

The question is incomplete, the complete question is:

What are the zeros of the function x^2 + 5x + 5  = 0?

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The number of pages Robert reads each day form an arithmetic sequence 27, 54, 81, 108,... After 8 days  he has read 216 pages.

The n-th term of an arithmetic sequence is given by:

a(n) = a0 + (n - 1) d

Where:

a0 = first term

d = common difference

In the given problem, the number of pages form an arithmetic sequence:

27, 54, 81, 108, ....

with:

first term = a0 = 27

common difference = d = 54 - 27 = 27

Hence, for n = 8

a(8) = 27 + (8 - 1) x 27

      = 27 + 7 x 27 = 216

Hence, after 8 days he has read 216 pages.

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Answer:

x=5

Step-by-step explanation:

supplementary angles adds to 180

25x+5+10x=180

35x+5=180

35x=180-5

35x=175

divide

x=5

The antiderivative of f(x) = tan⁻¹(x) is x tan⁻¹(x) − ln(x²+1)/2 + C

The term antiderivative in math is referred as a function that reverses what the derivative does

Here we have given the expression f(x) = tan⁻¹(x).

Now, the antiderivative of the expression is calculated by doing the integral

=> ∫tan⁻¹(x)dx

By using the integration by parts ∫udv = uv−∫vdu

Here let us consider that u=tan⁻¹(x) and dv=dx

And the value of du is calculated as,

=> du =  ∫tan⁻¹(x)dx =  dx/(x² + 1)

And the value of v = x.

Then the expression is written as,

=>  ∫tan⁻¹(x)dx = x tan⁻¹(x) - ∫x/(x² + 1)dx

As per the concept of the constant of integration, it can be simplified as,

=> x tan⁻¹(x) − ln(x²+1)/2 + C

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3x + 1 is called the Collatz problem.

"The 3x+1 problem, often referred to as the Syracuse problem, Collatz problem, Kakutani's problem, Hasse's algorithm, and Ulam's problem, concerns the behaviour of the iterates of the function that takes odd numbers n to 3n+1 and even integers n to n/2. One of the most well-known unsolved mathematical puzzles is the Collatz conjecture. The conjecture asks if every positive integer will eventually become 1 if two basic arithmetic processes are repeated.

Given

3x + 1 is called the Collatz problem.

"The 3x+1 problem, often referred to as the Syracuse problem, Collatz problem, Kakutani's problem, Hasse's algorithm, and Ulam's problem, concerns the behaviour of the iterates of the function that takes odd numbers n to 3n+1 and even integers n to n/2.

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