1
A number x is multiplied by 4, and 2
is added to it; the result is less than 82.
Find the range of values of x.

Answer:

60 inches

Step-by-step explanation:

5 ft x 12 in/ft = 60 in

Answer:

To write 9/11 in the form √a, we need to simplify the fraction so that it has a square root in the numerator or the denominator. We can do this by multiplying the numerator and denominator by √11:

9/11 × (√11/√11) = (9√11)/(11√11) = (√(9 × 11))/(√(11 × 11)) = √99

Therefore, 9/11 can be written in the form √99, where a = 99.

Answer:

Step-by-step explanation:

1) To solve the equation √(6x) = x, we can first square both sides of the equation to eliminate the square root:

(√(6x))^2 = x^2

6x = x^2

Next, we can rearrange the equation into standard quadratic form by subtracting 6x from both sides:

x^2 - 6x = 0

Now, we can factor out an x from the left-hand side of the equation:

x(x - 6) = 0

Setting each factor equal to zero, we find two solutions:

x = 0 or x - 6 = 0

Therefore, the solutions to the equation √(6x) = x are x = 0 and x = 6.

2)To solve √(5x-6) = x, we can square both sides of the equation:

(√(5x-6))^2 = x^2

5x-6 = x^2

Rearranging this quadratic equation to the standard form ax^2 + bx + c = 0, we get:

x^2 - 5x + 6 = 0

This can be factored into:

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

Therefore, the solutions to the equation √(5x-6) = x are x = 2 and x = 3.

We should check if these solutions satisfy the original equation:

When x=2: √(5x-6) = √(5(2)-6) = √4 = 2, which satisfies the equation.

When x=3: √(5x-6) = √(5(3)-6) = √9 = 3, which also satisfies the equation.

Therefore, the solutions are x = 2 and x = 3.

3) To solve the equation √(2x-1) = x-3, we can square both sides of the equation to eliminate the square root:

(√(2x-1))^2 = (x-3)^2

Simplifying the left-hand side gives:

2x-1 = (x-3)^2

Expanding the right-hand side gives:

2x-1 = x^2 - 6x + 9

Rearranging the equation gives:

x^2 - 8x + 10 = 0

We can solve this quadratic equation :

x^2 - 8x + 10 = 0

Now we can solve this quadratic equation by using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 1, b = -8, and c = 10. Substituting these values into the formula:

x = (-(-8) ± √((-8)^2 - 4(1)(10))) / 2(1)

Simplifying:

x = (8 ± √(64 - 40)) / 2

x = (8 ± √24) / 2

x = 4 ± √6

Therefore, the solutions to the equation √(2x - 1) = x - 3 are x = 4 + √6 and x = 4 - √6.

4) has no real solutions.

when you have to remove the square root you have to power the both sides of the equation.

1)6x= x^2

 0 = x^2 - 6x

0= x(x-6)

x=0 or x-6=0

            x=6

2) [tex]\sqrt{5x-6}[/tex] = x

   5x-6=x^2

   x^2 -5x+6=0

   (x-3)(x-2)=0

    x-3=0 or x-2= 0

    x=3 or x=2

3)

[tex]\sqrt{2x-1}[/tex] = x-3

2x-1 = (x-3)^2

2x-1= x^2 - 6x +9

x^2 -8x +10=0

this equation can't solve by factoring method easily. so we have to use the completing square method.

x^2-8x+10=0

x^2-8x= -10

x^2 - 8x+(-4)^2 = -10+ ( -4)^2

(x-4)^2 = 6

[tex]x= 4 +\sqrt{6}[/tex] or  [tex]x= 4 -\sqrt{6}[/tex]

4) [tex]x= 2 + \sqrt{2x-11}[/tex]

x^2= 4+4 [tex]\sqrt{2x-11}[/tex] + 2x-11

x^2= 7+ 2x +4[tex]\sqrt{2x-11}[/tex]

x^2-2x-7= 4[tex]\sqrt{2x-11}[/tex]

(x^2-2x-7)^2= 16 (2x-11)

x^4 +4x^2 +49-4x^3 + 28x - 14x^2 = 32x-176

x^4 - 4x^3 - 10x^2 - 4x + 225= 0

x(x^3 -4x^2 - 10x-4)+225=0

this is a power 4 th equation this equation can solve by normal steps.

The prοbability οf getting either 2 οr οdd number is B)4/49.

Prοbability is a way οf calculating hοw likely sοmething is tο happen. It is difficult tο prοvide a cοmplete predictiοn fοr many events. Using it, we can οnly fοrecast the prοbability, οr likelihοοd, οf an event οccurring. The prοbability might be between 0 and 1, where 0 denοtes an impοssibility and 1 denοtes a certainty.

Here the Tοtal number οf οutcοmes = 7

When spinner is spun first we can get οn favοrable οutcοme , which is 2 οr οdd number.

If prοbability οf getting 2

=> Number οf favοrable οutcοme/Tοtal number οf οutcοme

=> 1/7

Nοw number οf οdd number frοm 1 tο 7 = 4

Prοbability οf getting οdd number in secοnd spun = 4/7

Then prοbability οf getting either 2 οr οdd number = [tex]\frac{1}{7}\times\frac{4}{7}[/tex] = [tex]\frac{4}{49}[/tex].

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Answer:

We can use the double angle identity for sine to find sin2θ:

sin2θ = 2sinθcosθ

Since cosθ = √10/8 and 3π/2 < θ < 2π, we know that θ is in the fourth quadrant, where sine is negative. We can use the Pythagorean identity to find the value of sinθ:

sin²θ + cos²θ = 1

sin²θ = 1 - cos²θ

sinθ = -√(1 - cos²θ)

Substituting the value of cosθ, we get:

sinθ = -√(1 - (√10/8)²) = -√(1 - 5/16) = -√(11/16) = -√11/4

Now we can plug in the values of sinθ and cosθ into the double angle identity for sine:

sin2θ = 2sinθcosθ = 2(-√11/4)(√10/8) = -√110/16 = -√(11/4)(10/16) = -√(55/8)

Therefore, sin2θ = -√55/8 when cosθ = √10/8 and 3π/2 < θ < 2π.

Answer:

$200.00

Step-by-step explanation:

We are asking what is 75% of what number is 150.  If we take off 1/4 or 25% we are leaving on 3/4 or 75%

.75 x = 150  Divide both sides by .75

[tex]\frac{.75x}{.75}[/tex] = [tex]\frac{150}{.75}[/tex]

x = 200

Check:

200 x .25 = 50  To find the discount.

200 - 50 = 150   Subtract the discount from the original price to find the sale's price.

Helping in the name of Jesus.

Answer: Let's assume that the cost to make one item of clothing is directly proportional to the number of hours it takes to make it. We can use the equation:

Cost = k * Hours

where "Cost" is the cost to make one item of clothing, "Hours" is the number of hours it takes to make one item of clothing, and "k" is the proportionality constant that relates the cost to the number of hours.

In other words, if it takes x hours to make one item of clothing, then the cost to make one item is:

Cost = k * x

The value of k will depend on factors such as the cost of materials, the wages of the workers, and other expenses involved in the production process. The value of k can be determined based on the actual cost and number of hours for a particular item of clothing.

Step-by-step explanation:

Answer:

9π

Step-by-step explanation:

if the circumference=6π, then you can solve for the radius. C=2πr

6π=2πr

Divide 2π from both sides to get r=3

Area=πr^2

A=π*9

Area= 9π

The solutions to the absolute value inequalities 1 to 20 are shown below

|x + 8| < 16

So, we have

-16 < x + 8 < 16

-24 < x < 8

|r + 1| ≤ 2

So, we have

-2 ≤ r + 1 ≤ 2

-3 ≤ r ≤ 1

|2c - 1| ≤ 7

So, we have

-7 ≤ 2c - 1 ≤ 7

-6 ≤ 2c ≤ 8

-3 ≤ r ≤ 4

Using the above as a guide, the solution to the other inequalities are

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The wοrk dοne in pushing the car up the incline is 39,672 fοοt-pοunds.

The dοt prοduct is a mathematical οperatiοn that takes twο vectοrs and prοduces a scalar. It is the prοduct οf the magnitudes οf the twο vectοrs and the cοsine οf the angle between them.

Tο calculate the wοrk dοne, we need tο find the fοrce applied tο the car and the distance it was mοved.

First, let's calculate the fοrce required tο push the car up the incline. We can use the weight οf the car and the angle οf the incline tο find the fοrce required.

The weight οf the car is given as 1850 pοunds. The fοrce required tο push the car up the incline is equal tο the cοmpοnent οf the weight that acts parallel tο the incline. This is given by:

fοrce = weight * sin(angle)

where the angle is 7 degrees.

Plugging in the values, we get:

fοrce = 1850 * sin(7) = 220.4 pοunds

Nοw, we need tο find the distance that the car was pushed up the incline. This is given as 180 feet.

Using the dοt prοduct, the wοrk dοne can be calculated as:

wοrk = fοrce * distance * cοs(angle)

where the angle is the angle between the fοrce and the displacement vectοrs. In this case, the fοrce and displacement are in the same directiοn, sο the angle is 0 degrees and cοs(0) = 1.

Plugging in the values, we get:

wοrk = 220.4 * 180 * cοs(0) = 39,672 fοοt-pοunds

Therefοre, the wοrk dοne in pushing the car up the incline is 39,672 fοοt-pοunds.

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The measure of the angle A as required to be determined in the task content is; 16.7°.

It follows from the task content that the measure of the angle A is to be determined.

By using the trigonometric ratio;

tan (A) = BC / AC

Hence, by substitution; it follows that;

tan (A) = 6 / 20

Hence, by simplification; we have that;

tan A = 3/10

tan A = 0.3

Therefore, to determine A;

A = tan-¹ (0.3)

A = 16.7°.

Ultimately the measure of angle A as required to be determined is; 16.7°.

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Answer:

Rounding to the nearest whole degree, the rescue boat should travel at an angle of approximately 41 degrees off of due east to take the shortest route to the ship.

Step-by-step explanation:

We can use trigonometry to solve this problem. Let's draw a diagram to represent the situation:

            A (rescue boat)

              | \

              |   \

              |     \ C (disabled ship)

   24 mi |       \

              |        \

              |          \

              |θ          \

              B-----------D (Port Lincoln)

               48 mi

In the diagram, point B represents Port Lincoln, point C represents the disabled ship, and point A represents the rescue boat. Point D is the intersection of the eastward path and the northward path taken by the ship.

We are given that BD = 24 miles and CD = 48 miles. We want to find the angle θ, which is the angle between the line segments AB and AD.

To find θ, we can use the law of cosines:

cos(θ) = (BD² + CD² - AD²) / (2 x BD x CD)

Substituting the given values, we get:

cos(θ) = (24² + 48² - AD²) / (2 x 24 x 48)

Simplifying, we get:

cos(θ) = 0.75

To solve for θ, we can take the inverse cosine of both sides:

θ = cos⁻¹(0.75)

Using a calculator, we get:

θ ≈ 41.41°

Rounding to the nearest whole degree, the rescue boat should travel at an angle of approximately 41 degrees off of due east to take the shortest route to the ship.

The answers to each question are:

(a) a1 = 3, a2 = 5, a4 = 17 are prime numbers.

(b)  9 is not a prime number,

What are the prime numbers?

A prime number is a positive integer greater than 1 that has no positive integer divisors other than 1 and itself. In other words, it can only be divided evenly by 1 and itself. Examples of prime numbers include 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and so on.

(a) [tex]a_{k} = 2 ^ k + 1[/tex]

[tex]a_{1} = 2^1 + 1 = 3[/tex]

[tex]a_{2} = 2 ^ 2 + 1 = 5[/tex]

[tex]a_{4} = 2 ^ 4 + 1 = 17[/tex]

a1, a2, a4 are prime numbers.

(b) We know when k = 3 an [tex]a_{3} = 2 ^ 3 + 1 = 9[/tex].

9 is not a prime number, it can be divide by 3 * 3 so the Sequence does not always produce a prime number.

Hence, the answers to each question are:

(a) a1 = 3, a2 = 5, a4 = 17 are prime numbers.

(b)  9 is not a prime number,

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Complete question:

It is suggested that the sequence [tex]a_{k} = 2 ^ k + 1[/tex], k...I produces only prime numbers.

(a) Show that a, a, and a, produce prime numbers.

(b) Prove by counter-example that the sequence does not always produce a prime number.

The width of the arch is 20 meters and the height of the arch is 0 meters.

The height of an arc is the sagitta. It is the line perpendicular to the chord's midpoint and to the arc itself.

The equation for the arch is provided as y = -0.04x2. The junction points (-10, 4) and (10, -4) can be used to calculate the arch's width and height.

The breadth of the arch, which is equal to the separation between the two x-intercepts, is first determined:

Width is equal to 10 minus 10 metres.

Then, we determine the arch's height, which is its highest point. We know that the largest value of y occurs at the parabola's vertex since the equation for the arch has the form y = ax2, where an is a negative constant. The vertex's x-coordinate is provided by the formula x = -b/2a, where a = -0.04 and b = 0. (since there is no linear term in the equation). The vertex's x-coordinate is as a result:

x = -b/2a = -0/2(-0.04) = 0

We change x = 0 in the equation of the arch to obtain the equivalent y-coordinate of the vertex.

y = -0.04x² = -0.04(0)² = 0

As a result, the parabola's vertex is (0, 0), and the arch's greatest height is 0 metres.

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√49 is a perfect square and therefore rational.

√50 is also a perfect square and therefore rational.

∛127 is a non perfect cube and therefore irrational.

∛125 is rational because it is equal to a whole number.

It can be expressed as the product of two equal integers. Perfect squares are also known as 'whole squares' or 'complete squares'.

√49 is a perfect square and therefore rational.

This is because the square root of 49 can be expressed as a common fraction 7/7. The square root of 49 = 7, which is a whole number.

√50 is also a perfect square and therefore rational.

The square root of 50 can be expressed as a common fraction 5/5. The square root of 50= 5, which is a whole number.

∛127 is a non perfect cube and therefore irrational.

This is because the cube root of 127 cannot be expressed as a common fraction. The cube root of 127 = approximately 4.9, which is not a whole number.

∛125 is rational because it is equal to a whole number.

The cube root of 125 can be expressed as a common fraction 5/5. The cube root of 125 = 5, which is a whole number.

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Answer:

b) The graph will shift down 5 units from its parent graph.

Step-by-step explanation:

Your answer is in the image.

Let's use variables to represent the width and length of the rectangle:

Let w be the width of the rectangle (in cm)

Then, the length of the rectangle can be expressed as 5w - 18 (since it is 18 cm less than five times its width).

The area of the rectangle can be calculated as:

A = w * (5w - 18)

We are given that the area of the rectangle is 35 square cm. Substituting this value into the equation above, we get:

35 = w * (5w - 18)

Expanding the right-hand side, we get:

35 = 5w^2 - 18w

Rearranging the terms and setting the equation equal to zero, we get:

5w^2 - 18w - 35 = 0

To solve for w, we can use the quadratic formula:

w = (-b ± sqrt(b^2 - 4ac)) / 2a

where a = 5, b = -18, and c = -35.

Plugging in the values, we get:

w = (-(-18) ± sqrt((-18)^2 - 4(5)(-35))) / 2(5)

w = (18 ± sqrt(784)) / 10

w = (18 ± 28) / 10

So w = 2.6 or w = -1.4

Since the width cannot be negative, we reject the solution w = -1.4.

Therefore, the width of the rectangle is approximately 2.6 cm.

To find the length, we can use the expression we derived earlier:

length = 5w - 18

length = 5(2.6) - 18

length = 13 - 18

length = -5

Since the length cannot be negative, we know that there is an error in our calculation. We made an assumption that the width is less than the length, which is not necessarily true. We should check our work and try again using the positive root instead of the negative one.

Using the positive root for the width, we get:

w = (18 + 28) / 10

So w = 4.6

Therefore, the width of the rectangle is 4.6 cm.

To find the length, we can use the expression we derived earlier:

length = 5w - 18

length = 5(4.6) - 18

length = 23 - 18

length = 5

So the length of the rectangle is 5 cm.

Therefore, the dimensions of the rectangle are width = 4.6 cm and length = 5 cm.

Using the pattern given, we found that the colour of the lightbulb in the 543rd position of a string of lightbulbs is green.

What is meant by the pattern?

A recurring arrangement of numbers, shapes, colours and other elements is known as a pattern. The Pattern can be connected to any kind of occasion or thing. When a group of numbers are arranged in a particular way, the arrangement is referred to as a pattern. Patterns can also occasionally be referred to as a series. The number of patterns can be limitless or finite. There are many distinct kinds of number patterns, including geometric, Fibonacci, and algebraic or arithmetic patterns. In mathematics, number patterns are quite prevalent.

Given,

The number of lights on a string of lights = 1000

The order in which the lights repeat is red, yellow, green and blue.

After every four lights, the order repeats.

We can get the pattern as follows.

The blue lights are in positions 4,8,12,........,4n.

We have to find the multiple of 4 close to 543.

Now we can check if 542 is a multiple of 4.

542/4 = 135.5

So it is not a multiple of 4.

Now check if 544 is a multiple of 4.

544/4 = 136

So the bulb in the 544th position is blue.

Then the bulb in the 543rd position should be green.

Therefore using the pattern given, we found that the colour of the lightbulb in the 543rd position of a string of lightbulbs is green.

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The simplified expression is: cos(13π/35)

Trigonometry is a branch of mathematics that studies the relationships between angles and the sides of triangles. It deals with functions such as sine, cosine, and tangent and their inverses, and is used extensively in fields such as engineering, physics, and astronomy.

We can simplify the given expression using the identity:

cos(A - B) = cos(A) cos(B) + sin(A) sin(B)

We can rewrite the given expression as:

cos(4π/7) cos(π/5) - sin(4π/7) sin(π/5)

= cos(4π/7 - π/5) [using the identity]

To simplify further, we can find a common denominator for 4/7 and 1/5:

4/7 = 20/35 and 1/5 = 7/35

Then, we can write:

4π/7 - π/5 = (20π/35) - (7π/35) = 13π/35

This expression cannot be further simplified using trigonometric identities, and is written only in terms of the cosine function.

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[tex]~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ A(\stackrel{x_1}{-4}~,~\stackrel{y_1}{-3})\qquad B(\stackrel{x_2}{x}~,~\stackrel{y_2}{y}) \qquad \left(\cfrac{ x_2 + x_1}{2}~~~ ,~~~ \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left(\cfrac{ x -4}{2}~~~ ,~~~ \cfrac{ y -3}{2} \right) ~~ = ~~\stackrel{\textit{\LARGE M} }{(-6~~,~~-4)} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{ x -4 }{2}=-6\implies x-4=-12\implies \boxed{x=-8} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{ y -3 }{2}=-4\implies y-3=-8\implies \boxed{y=-5}[/tex]

Therefore, the solutions for θ in the interval 0° ≤ θ ≤ 360° are approximately 131.81° and 228.19°.

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles, particularly right triangles. It is used to study and analyze various phenomena that involve periodic functions, such as waves, oscillations, and sound. Trigonometry also has practical applications in fields such as physics, engineering, navigation, and surveying. It involves the use of trigonometric functions such as sine, cosine, tangent, cosecant, secant, and cotangent to calculate the sides and angles of triangles and other geometric figures.

Here,

Let's solve for θ:

First, let's substitute u = cos(θ), so we have:

3u² - 5u - 4 = 0

Now we can use the quadratic formula:

u = [ -(-5) ± √((-5)² - 4(3)(-4))] / (2*3)

u = [ 5 ± √(49) ] / 6

u1 = (5 + 7) / 6 = 2

u2 = (5 - 7) / 6 = -2/3

Since the cosine function has a range of -1 ≤ cos(θ) ≤ 1, we can discard the solution u1 = 2.

Now we can solve for θ:

cos(θ) = u2 = -2/3

θ = cos⁻¹(-2/3) ≈ 131.81°  

θ = 360° - cos⁻¹(-2/3) ≈ 228.19°

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Answer: 7 marbles

Step-by-step explanation:

Lin starts with 27 and creates 3 equal groups.

When separating, you divide.

27 divided by 3 is 9.

Each group has 9 marbles.

This means that by giving one group to Jill, she receives 9 marbles.

Jill gives away 2 of her 9 marbles, thus leaving her with 7 marbles.

9-2=7.

Answer:

Step-by-step explanation:

A bandana is in the shape of a triangle. The base of the bandana is 30 in. wide and the height is 12 in.

Triangle Area = 1/2 b x h

1/2 30 x 12 =

15 x 12 = 180 in^2

Answer:

Step-by-step explanation:

Let [tex]x,y[/tex] be the two numbers. Then we get

    [tex]2x+3y=-1[/tex]     [tex](a)[/tex]

       [tex]x+4y=2[/tex]       [tex](b)[/tex]

We solve equations [tex](a)[/tex] and [tex](b)[/tex] simultaneously:

    equation [tex](b)\times2[/tex] :

       [tex]2x+8y=4[/tex]        [tex](c)[/tex]  

   [tex](c)-(a)[/tex] gives:

       [tex]5y=5\rightarrow y=1[/tex]

    Sub [tex]y=1[/tex] into [tex](a)[/tex] :  

       [tex]2x+3\times1=-1\rightarrow2x=-4\rightarrow x=-2[/tex]

The solution is -2,1

The sum of the first 10 terms in the sequence is 195.The sequence given above is an arithmetic sequence.

What is an arithmetic sequence?

An arithmetic sequence is a sequence of numbers in which each successive number is found by adding a fixed number, called the common difference, to the previous number in the sequence. The common difference is the same for each pair of consecutive numbers.

An arithmetic sequence is a sequence of numbers where each successive number is obtained by adding a fixed amount, the common difference, to the preceding number. In this particular sequence, the common difference is 5.

The first term in the sequence is a₁ = -3, and the general term is an = 3an-1 + 5, where n > 2. To find any term in the sequence, we can use the formula an = a₁ + (n - 1)d, where d is the common difference. In this case, a₁ = -3 and d = 5, so the formula becomes an = -3 + (n - 1)5.

To find the 9th term in the sequence, we can substitute n = 9 into the above formula and solve for an. This gives us an = -3 + (9 - 1)5 = -3 + 40 = 37. Therefore, the 9th term in the sequence is 37.

To find the sum of the first 10 terms in the sequence, we can use the formula Sₙ = (n/2)(2a₁ + (n - 1)d). In this case, a₁ = -3 and d = 5, so the formula becomes Sₙ = (n/2)(2(-3) + (n - 1)5). Substituting n = 10 into the formula gives us Sₙ = (10/2)(2(-3) + (10 - 1)5) = (10/2)(-6 + 45) = (10/2)(39) = 195. Therefore, the sum of the first 10 terms in the sequence is 195.

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Right Question:-

a₁ = -3,

an=3an-1+5

for

n>2.

For given sequence, Find the sum of first 10 terms?

By answering the presented question, we may conclude that Thus function  f(x) = y = (5, 7, 1, 4, 9, 5, 3, 2) and x = (3, 8, 9, 13, 10, 6, 5, 4).

Mathematicians examine numbers and their variations, equations and associated structures, forms and their locations, and prospective positions for these things. The term "function" refers to the relationship between a collection of inputs, each of which has a corresponding output. A function is a connection of inputs and outputs in which each input leads to a single, identifiable outcome. Each function is assigned a domain, a codomain, or a scope. The letter f is widely used to denote functions (x). The symbol for entry is an x. The four primary types of accessible functions are on functions, one-to-one capabilities, so multiple capabilities, in capabilities, and on functions.

A. Assume f(x) = (4x - 7). To obtain the inverse of f(x), we must swap x and y in the function and solve for y.

As a result, g(x) = (x + 7)/4 is the inverse of f(x).

B. We must demonstrate that f(g(x)) = x for any x in g's domain.

f(g(x)) = f((x + 7)/4) = 4((x + 7)/4) - 7 = x + 7 - 7 = x + 7 - 7 = x.

As a result, f(g(x)) = x, confirming that g(x) is really the inverse of f. (x).

C. Assuming the function:

| x | 5 8 9 13 10 6 5 4 |\s|—-|—-|—-|—-|—-|—-|—-|—-|\s| f(x) | 3 7 1 4 9 5 3 2 |

Thus f(x) = y = (5, 7, 1, 4, 9, 5, 3, 2) and x = (3, 8, 9, 13, 10, 6, 5, 4).

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Answer:

The probability of winning the lottery, denoted by P(A), can be calculated as follows:

P(A) = number of winning outcomes / total number of possible outcomes

Since there is only one winning four-digit number and there are 2,352 possible four-digit numbers, we have:

P(A) = 1/2352

Using a calculator, we get:

P(A) ≈ 0.00042

Therefore, the value of P(A) is approximately 0.00042 or 4.2 × 10^(-4) (rounded to five decimal places).

Step-by-step explanation:

The graph of inequalities are attached accordingly.

Graphs of inequalities are visual representations of solutions to inequality equations.

These graphs consist of shaded regions on a coordinate plane that indicate all the points that satisfy a given inequality. In real life, graphs of inequalities are used in many areas, including business, finance, and science.

For example, a company may use a graph of inequalities to determine the feasible production and pricing strategies that will maximize profits. Similarly, a biologist may use a graph of inequalities to analyze the optimal conditions for the growth of a particular species. The graphs of inequalities provide a quick and easy way to understand and visualize complex mathematical relationships.

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The after-tax rate of return on a taxable bond is calculated by multiplying the pre-tax yield by one minus the tax rate. In this case, it would be 9.1% x (1 - 0.21) = 7.19%. The municipal bond is tax-free, so its after-tax rate of return is equal to its pre-tax yield of 7.35%. Therefore, you should buy the municipal bond as it has a higher after-tax rate of return than the taxable bond.