Can someone please provide a step-by-step explanation for the answer? I would really appreciate it.

Let f(x)= 2(x-1) / x²-2x-3 - 1/ x-3, x ЄR, x > 3.


(a) Show that f(x) = 1/ x+1

(b) Find the inverse function of ƒ (x).

(c) Find the domain of ƒ−¹(x).

(d) Given that g(x) = 2x² – 3, where x Є R. Solve (ƒ o g)(x) = 1/8.

Answer:

Step-by-step explanation:

The areas of the triangles are listed below:

Triangle A: A = 30 cm²

Triangle B: A = 14 cm²

Triangle C: A = 15 cm²

Triangle D: A = 40.997 cm²

In this problem we have a composite figure in the form of a square and formed by three right triangles (A, B, C) and a non-right triangle (D). The area of each triangle must be found, the areas of the each right triangle can be determined by this formula:

A = 0.5 · b · h

Where:

And the area of the non-right triangle can be found by Heron's formula:

A = √[s · (s - a) · (s - b) · (s - c)]

s = 0.5 · (a + b + c)

Where:

First, determine the areas of the right triangles:

Triangle A

A = 0.5 · (10 cm) · (6 cm)

A = 30 cm²

Triangle B

A = 0.5 · (7 cm) · (4 cm)

A = 14 cm²

Triangle C

A = 0.5 · (3 cm) · (10 cm)

A = 15 cm²

Second, find the side lengths of the non-right triangle by Pythagorean theorem:

a = √[(10 cm)² + (6 cm)²]

a = 2√34 cm

b = √[(7 cm)² + (4 cm)²]

b = √65 cm

c = √[(3 cm)² + (10 cm)²]

c = √109 cm

Third, calculate the semiparameter of the non-right triangle:

s = 0.5 · (2√34 cm + √65 cm + √109 cm)

s = 15.082 cm

Fourth, calculate the area of the non-right triangle:

A = √[(15.082 cm) · (15.082 cm - 2√34 cm) · (15.082 cm - √65 cm) · (15.082 cm - √109 cm)]

A = 40.997 cm²

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Step-by-step explanation: there are 3 y's so the answer is 3y

and there are 2 x's so therefore the answer is 2x

and then you just add them together to get 3y + 2x

Based on the information, we can infer that the area of the floor is 130 square inches. Besides, we can infer that 2 tiles are needed to cover the entire floor area.

To calculate the area of the floor we must divide the figure into fragments to find the area more easily. First of all we can find the area of the left zone that measures 10 base by 9 height.

10 * 9 = 90

For the right part the area would be 10 * 4. So:

10 * 4 = 4040 + 90 = 130

According to the above, the area of the place would be 130. On the other hand, we can infer that 2 tiles are needed to cover the entire floor area.

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The volume of the bottle is 25 fluid ounce of iced tea.

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

Volume of bottle = 750 milliliters of iced tea.

Also, we have

1 fluid ounce = 30 milliliters

This means that

1/30 fluid ounce = 1 milliliters

So, we have

Volume of bottle = 750 * 1/30 fluid ounce of iced tea.

Evaluate

Volume of bottle = 25 fluid ounce of iced tea.

Hence, the volume of bottle is 25 fluid ounce of iced tea.

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1. The weight of the patient in kilograms is 27.216 kg. (60 lb * 0.4536 kg/lb = 27.216 kg)

2. The total dosage of medication required for the patient is 13.608 mg. [tex](0.5 mg/kg \times 27.216 kg = 13.608 mg)[/tex]

3. The patient should be administered 1.3608 mL of the stock medication. (13.608 mg / 10 mg/mL = 1.3608 mL)

To calculate the necessary values based on the given information, let's follow the steps below:

Determine the weight of the patient in kilograms:

Given that the patient weighs 60 lb, we can convert this to kilograms using the conversion factor of 1 lb = 0.4536 kg.

Weight (in kg)[tex]= 60 lb \times 0.4536 kg/lb = 27.216 kg.[/tex]

Calculate the total dosage of medication required for the patient:

The medication order is 0.5 mg/kg, and the patient weighs 27.216 kg.

Total dosage [tex]= 0.5 mg/kg \times 27.216 kg = 13.608 mg.[/tex]

Determine the amount of stock medication required in milliliters (mL):

The stock medication is available in a concentration of 10 mg/mL.

To find the volume required, we need to divide the total dosage by the concentration of the stock medication.

Volume (in mL) = Total dosage (in mg) / Concentration (in mg/mL) = 13.608 mg / 10 mg/mL = 1.3608 mL.

Therefore, based on the given information, the weight of the patient is 27.216 kilograms, the total dosage of medication required is 13.608 milligrams, and 1.3608 milliliters of the stock medication should be administered to the patient.

Please note that when administering medication, it is crucial to follow the guidance of a healthcare professional and consider other factors such as the specific medication instructions, patient's condition, and any allergies or contraindications.

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Question: A patient weighs 60 lb, and the medication order is 0.5 mg/kg. The stock medication is available in a concentration of 10 mg/mL.

Based on this information, calculate the following

What is the weight of the patient in kilograms? (1 lb = 0.4536 kg)

What is the total dosage of medication required for the patient?

How many milliliters (mL) of the stock medication should be administered to the patient?

Please provide the necessary calculations and steps to find the answers based on the given information.

If the value of the truck is at $23,000 which is 23% off the original price. Then the original price of the truck will be $ 29,870.13.

What is the percentage?

The amount of something is expressed as if it is a part of the total which is a hundred. The ratio can be expressed as a fraction of 100. The word percent means per 100. It is represented by the symbol ‘%’.

A 4th of July sale at your local truck dealership is advertising big discounts on pickup trucks.

The truck you've had your eye on is now listed at $23,000 which is 23% off the original price.

Then the original price of the truck will be given as

Let the original price of the truck will be x. Then we have

0.77 × x = $ 23,000

         x = $ 29,87

The contractor should lower the ramp by approximately 4.62° to meet safety regulations.

We have,

To determine the angle created by the bottom of the ramp and the ground, we can use the trigonometric function of sine.

In the given triangle, the height is 6 ft and the base is 16 ft.

The angle we want to find is opposite the height (let's call it θ).

The sine of an angle is defined as the ratio of the opposite side to the hypotenuse.

sin(θ) = opposite/hypotenuse

sin(θ) = 6/16

sin(θ) = 0.375

To find the angle θ, we can take the inverse sine (also known as arcsine) of 0.375:

θ = arcsin(0.375)

θ ≈ 22.62°

To meet safety regulations, the angle should be no more than 18°. Therefore, the contractor needs to lower the ramp by:

22.62° - 18° ≈ 4.62°

Thus,

The contractor should lower the ramp by approximately 4.62° to meet safety regulations.

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The data-set in this problem is classified as follows:

D) Negatively skewed.

Negative skew refers to a longer or fatter tail on the left side of the distribution, that is, the majority of the values are on the lower end of the distribution.

Positive skew refers to a longer or fatter tail on the right side of the distribution, that is, the majority of the values are on the right end of the distribution.

For a symmetric distribution, we have that the majority of the values are at the center of the distribution, with an equal proportion at both tails.

For this problem, we have that the larger percentage is less than $4.00, which is at the left tail of the distribution, hence the distribution is negatively skewed.

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Using the z-score, mean and sample mean, the standard deviation of the sample is 7.57

A z-score describes the position of a raw score in terms of its distance from the mean when measured in standard deviation units. The z-score is positive if the value lies above the mean and negative if it lies below the mean.

The formula is given as;

z = x - μ / σ

Substituting the values into the formula;

-2.13 = 19.3 - 29 / σ

Cross multiply both sides;

-2.13σ = 19.3 - 29

-2.13σ = -9.7

Divide both sides by the coefficient;

-2.13σ / -2.13 = -9.7 / -2.13

σ = 7.57

The standard deviation is 7.57

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Answer:

To find out how many jars Jazmine can fill with her 3 1/3 cups of jam, we need to divide the total amount of jam by the amount of jam that each jar can hold.

First, we need to convert 3 1/3 cups into an improper fraction:

3 + 1/3 = (3 x 3 + 1) / 3 = 10/3

So, Jazmine has 10/3 cups of jam.

Next, we need to convert the jar size of 2/5 cup into a fraction with the same denominator as 10/3:

2/5 = 6/15

Now we can divide the total amount of jam by the amount of jam that each jar can hold:

(10/3) ÷ (6/15)

To divide fractions, we need to multiply the first fraction by the reciprocal of the second fraction:

(10/3) x (15/6)

Simplifying:

(10/3) x (5/2) = (50/6)

So Jazmine can fill (50/6) jars with her 3 1/3 cups of jam.

This is an improper fraction, so we can convert it to a mixed number:

(50/6) = 8 2/6 = 8 1/3

Therefore, Jazmine can fill 8 and 1/3 jars with her 3 1/3 cups of jam.

Step-by-step explanation:

To produce 120 cupcakes, Lihle needs to make 5 batches if one batch produces 24 cupcakes. However, it is important to plan for a little more than needed in case of any failures or discrepancies, so making an extra batch might be helpful. It is also important to consider the amount of frosting needed for 120 cupcakes, which can vary depending on the recipe. As a basic estimate, 1 cup of frosting is suggested for 12 cupcakes.

Answer:

Step-by-step explanation:

To express 6√3 + 6i in polar form, we first need to find the magnitude and argument of the complex number.

The magnitude (or modulus) of the complex number is given by the formula:

|z| = √(a^2 + b^2)

where a and b are the real and imaginary parts of the complex number, respectively.

In this case, a = 6√3 and b = 6, so:

|6√3 + 6i| = √((6√3)^2 + 6^2) = √(108 + 36) = √144 = 12

The argument (or angle) of the complex number is given by the formula:

arg(z) = tan^-1(b/a)

In this case, arg(z) = tan^-1(6/6√3) = tan^-1(1/√3) = π/6

Therefore, the polar form of 6√3 + 6i is:

12(cos(π/6) + i sin(π/6))

or

12e^(iπ/6)

Answer:

error

Step-by-step explanation:

I'm sorry, but I need more information to calculate the average budget across the four quarters. Can you please provide me with the values of the budget for each quarter? Once I have this information, I can calculate the average and then determine the percentage difference between the average and the estimated budget for next year.

Triangle ABC is congruent to triangle BAD because all their corresponding angles are equal.

Congruent triangles are triangles with equal corresponding angles and equal corresponding sides. This means that the angles will be equal and their corresponding sides will be equal.

Since AC is equal to BD , then the corresponding angles facing the two lines will be equal and angle A is equal to angle B

Therefore angle C will be equal to angle D, therefore we can say triangle ABC is congruent to triangle BAD.

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The given expressions can be identified as:

1. Trinomial

2. Monomial

3. Polynomial

4. Binomial

The given expressions can be identified as:

1. x^3 + y^2 - 3: This is a trinomial since it consists of three terms.

2. 7y^2: This is a monomial since it consists of only one term.

3. x^3 + 2x^2 - x + 10: This is a polynomial since it consists of four terms.

4. x + 7: This is a binomial since it consists of two terms.

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Answer:

$224.42

Step-by-step explanation:

P is the principal balance

r is the interest rate (as a decimal)

n is the number of times interest is compounded per year

t is the number of years

Input all of this into the equation you have.

[tex]A=P(1+\frac{r}{n} )^{nt}[/tex]

[tex]A=160(1+\frac{.07}{1})^{1(5)}[/tex]

Do the math.

[tex]A=160(1+.07)^{5}[/tex]

[tex]A=160(1.07)^{5}[/tex]

[tex]A=160(1.403)[/tex]

[tex]A=224.418[/tex]

The correct proof of the equation is cos3θ =cos(2θ + θ) =  cos²θ - 3cosθsin²θ (Proved)

Given the trigonometry identity below:

cos3∅=cos²∅-3cos∅sin²∅

We are to prove that both sides of the equation are equal.

cos 3θ = cos(2θ + θ)

Using the double angle formula for cosine, we can expand the first term:

cos(2θ + θ) = cos2θ cos θ - sin2θ sinθ

Since cos2θ = cos²θ - sin²θ

cos(2θ + θ) =  cos²θ - sin²θ - 2cosθsinθsinθ

cos(2θ + θ) = cos²θ - sin²θ - 2cosθsin²θ

On simplifying, we can see that;

cos3θ =cos(2θ + θ) =  cos²θ - 3cosθsin²θ (Proved)

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The correct answer is:

B. cosθ = √5/3 and tanθ = 2/√5

C. cosθ = -√5/3 and tanθ = 2/√5

To solve this question, we can use the given information that sinθ = 2/3 and determine the values of the other trigonometric functions based on that.

Recall the trigonometric identity: sin^2θ + cos^2θ = 1

Since sinθ = 2/3, we can substitute this value into the identity and solve for cosθ:

(2/3)^2 + cos^2θ = 1

4/9 + cos^2θ = 1

cos^2θ = 1 - 4/9

cos^2θ = 5/9

Taking the square root of both sides, we get:

cosθ = ±√(5/9) = ±(√5/3)

Now, let's analyze the given options:

A. secθ = 3/√5 and tanθ = 2/√5

B. cosθ = √5/3 and tanθ = 2/√5

C. cosθ = -√5/3 and tanθ = 2/√5

D. secθ = -3/2 and tanθ = 2/√5

From our calculations, we found that cosθ can be either √5/3 or -√5/3. So, option B and C are valid.

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The exact value of cos theta if the terminal side of theta in standard position contains the point (6,-8) is, 3 / 5

We have to given that;

the terminal side of theta in standard position contains the point (6,-8)

Hence, For given condition we get;

The exact value of cos theta if the terminal side of theta in standard position contains the point (6,-8) is,

Here, x = 6, y = - 8

Hence, We get;

r = √x² + y²

r  = √6² + 8²

r = √36 + 64

r = √100

r = 10

So, The exact value of cos theta if the terminal side of theta in standard position contains the point (6,-8) is,

cos θ = x / r

cos θ = 6 / 10

cos θ =  3 / 5

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Answer:

220 adults and 250 children

Step-by-step explanation:

we require to set up 2 equations and solve simultaneously.

let a be the number of adults and c the number of children , then

a + c = 470 → (1) based on number of people

3a + c = 910 → (2) based on ticket sales

subtract (1) from (2) term by term to eliminate c

(3a - a) + (c - c) = 910 - 470

2a + 0 = 440

2a = 440 ( divide both sides by 2 )

a = 220

substitute a = 220 into (1) and solve for c

220 + c = 470 ( subtract 220 from both sides )

c = 250

that is 220 adults and 250 children attended

The cost of each pound of grass seed is $6.

A graph of the solution for the cost of each pound of grass seed is shown below.

In order to determine the cost of each pound of grass seed, we would assign a variable to the cost of each pound of grass seed and then translate the word problem into an algebraic linear equation.

Let the variable x represent the cost of each pound of grass seed.

Since Lori bought 3.6 pounds of grass seed and she was charged $22.90, including a tax of $1.30, an algebraic linear equation that best represent this situation is given by;

3.6x + 1.30 = 22.90

3.6x = 22.90 - 1.30

3.6x = 21.60

x = 21.60/3.6

x = $6.

In conclusion, we would sue an online graphing calculator to plot the solution for the cost of each pound of grass seed as shown in the image attached below.

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The  probability of rolling two dice that have a sum of 7 is: 1/6.

When rolling two dice each die will have 6 possible outcomes

Total number of possible outcomes= 6 * 6

Total number of possible outcomes = 36

Now let's roll a sum of 7 dice to determine the amount of favourable results. There are 6 various ways that the number 7 may be combined, and they are as follows:
(1, 6), (2, 5), (3, 4), (4, 3), (5, 2) and (6, 1)

The number of favorable outcomes is 6.

Let find the  probability of rolling two dice that have a sum of 7 :

Probability = (Number of favorable outcomes) / (Total number of possible outcomes)

Probability = 6 / 36

Probability = 1 / 6

Therefore the probability  is 1/6.

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The area of the sector is 30.375 ft squared.

The  sector of a circle has a diameter of 18 feet and an angle of 3/4π radians.

Therefore, the area of the sector can be found as follows:

area of the sector = ∅ / 2 r²

where

∅ = central angle in radian

r = radius of the circle

Therefore,

∅ = 3 / 4 π

r = 18 / 2 = 9 ft

area of the sector = 3 / 4 π ÷ 2 × 9²

area of the sector = 3 / 8π × 81

area of the sector = 243 / 8π

area of the sector = 30.375 ft²

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The area of the sector is 95.43 square feet.

The area of sector when the angle is given in radians is given by the formula:

Area of sector = (1/2) × r²θ

where θ is the angle subtended at the center and r is the radius of the circle.

We have:

θ = 3π/4

r = 18/2 = 9 feet

Substituting into the formula:

Area of sector = (1/2) × 9² × 3π/4

Area of sector = 95.43 square feet

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The solution of expression is,

⇒ y = - 39

We have to given that,

An expression is,

⇒ - 5 = (y - 7) / 9

Now, We can simplify as,

An expression is,

⇒ - 5 = (y - 7) / 9

Multiply by 9,

⇒ - 5 × 9 = (y - 7) × 9 / 9

⇒ - 45 =y - 7

Add 7 both side,

⇒ - 45 + 7 = y

⇒ y = - 39

Therefore, The solution of expression is,

⇒ y = - 39

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Answer:

x = 4

Step-by-step explanation:

a line with an undefined slope is a vertical line with equation

x = c ( c is the value of the x- coordinates the line passes through )

the line passes through the point (4, - 3 ) with x- coordinate 4 , then

x = 4 is the equation of the line

note this equation cannot be expressed in slope- intercept form

The solution is : 22 meters would be saved if it were possible to walk through the pond.

Here, we have,

There is a pond between two points A and B.

To avoid the pond, one must walk to the south from point A to point C (say) by 34 meters and then to the east from point C to point B by 41 meters.

Now, it is clear that Δ ABC is a right triangle with AB as the hypotenuse that is the minimum distance from A to B through the pond.

The two legs of the right triangle are AC and CB.

Applying Pythagoras Theorem,

AB² = AC² + CB²

      = 34² + 41²

      = 2837

⇒ AB = 53 meters (Rounded to the nearest meter)

Therefore, (34 + 41) - 53 = 22 meters will be saved if it were possible to walk through the pond.

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complete question:

To get from point A to point B you must avoid walking through a pond. To

avoid the pond, you must walk 34 meters south and 41 meters east. To the

nearest meter, how many meters would be saved if it were possible to walk

through the pond?

Answer:

Set your calculator to Degree mode.

[tex]x = {sin}^{ - 1} \frac{73}{83} = 61.6 \: degrees[/tex]

Angle x measures about 61.6°.

Answer: The acceleration of the car is -6 m/s²

Step-by-step explanation:

The acceleration of the car can be found by using the given below equation:

Acceleration = (final velocity - initial velocity) / time

Given:

Initial velocity (u) = 30 m/s

Final velocity (v) = 0 m/s

Time (t) = 5 seconds

Putting the values in the equation:

acceleration = (0 - 30) m/s / 5 s

acceleration = -30 m/s / 5 s

acceleration = -6 m/s²

Therefore, the acceleration of the car is -6 m/s² (negative sign indicates deceleration).

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People in this country own approximately 111 million dogs, 123 million cats, and 258 million fish as pets.

Let's solve the problem step by step to determine how many of each type of pet people in this country own.

Let's assume the number of dogs as D.

According to the information given, the number of cats would be D + 12 million since there are 12 million more cats than dogs.

The number of fish would be the sum of the number of cats and dogs plus 12 million, as there are 12 million more fish than the total number of cats and dogs combined.

So, the equation representing the total number of pets is:

D + (D + 12 million) + (D + 12 million) = 356 million

Simplifying the equation, we have:

3D + 24 million = 356 million

Subtracting 24 million from both sides, we get:

3D = 332 million

Dividing both sides by 3, we find:

D = 110.67 million

Since the number of pets cannot be in fractions, we can round D to the nearest whole number. In this case, D ≈ 111 million.

Now, we can calculate the number of cats:

Cats = D + 12 million

= 111 million + 12 million

= 123 million

And the number of fish:

Fish = (D + Cats) + 12 million

= (111 million + 123 million) + 12 million

= 246 million + 12 million

= 258 million

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