Find the trigonometric ratios as simplified fractions and as decimals to the nearest thousandths (as necessary).



Sin X: Fraction ______ Decimal _______

Cos X: Fraction ______ Decimal ______

Tan X: Fraction _______ Decimal ______

(30 points)

The exact value of the inverse-trigonometric expression arccos( [tex]cos\frac{\pi }{2}[/tex] ) is found to be  [tex]\frac{\pi }{2}[/tex].

These are also called as arc functions. These trigonometric functions are inverse operations of given trigonometric functions. The cosine of the angle indicates the ratio of base of right triangle and hypotenuse of the same triangle and the inverse of the cosine is the ratio of base and hypotenuse will give the respective angle. Inverse functions can be used to unknown angles, angle of elevation or depression in any right triangle.

The given expression is arccos( [tex]cos\frac{\pi }{2}[/tex] )

we know that arc cosine or inverse cosine= [tex]cos^{-1}(-x)[/tex] = [tex]\pi - cos^{-1} - (x)[/tex]

also we know that cos(-x)=cosx

arccos( [tex]cos\frac{\pi }{2}[/tex] ) = arccos( [tex]-cos\frac{\pi }{2}[/tex] )

                    = [tex]\pi - cos^{-1} (cos \frac{\pi }{2})[/tex]

                    = [tex]\pi - \frac{\pi }{2}[/tex]

                   = [tex]\frac{\pi }{2}[/tex]

∴ The value of arccos( [tex]cos\frac{\pi }{2}[/tex] ) is  [tex]\frac{\pi }{2}[/tex].

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Answer: 5/8 as a fraction 0.625 as a decimal

Step-by-step explanation:

The probability of a person being 20 years or younger and preferring strawberry ice cream is: 10 / 50 = 0.2 or 20%, So the answer is 20%.

8 + 10 + 6 + 8 + 6 + 12 = 50

The number of people who are 20 years or younger and prefer strawberry ice cream is 10.

Therefore, the probability of a person being 20 years or younger and preferring strawberry ice cream is:

10 / 50 = 0.2 or 20%

So the answer is 20%.

Among people over 20 years old, chocolate ice cream is the most preferred flavor with a total of 12 votes.

The total number of people who prefer vanilla ice cream is 16 (8 in the 20 years and younger group + 8 in the over 20 years group).

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Answer:

x = [tex]-\frac{2}{7}[/tex]

Step-by-step explanation:

Answer:

Step-by-step explanation:

there is an equation written in the question and one in the figure, to be sure I'll solve them both

Solve the equation.

7x +(4x - 5) = 3 + 2(x − 3)

7x + 4x - 5 = 3 + 2x - 6

7x + 4x - 2x = 3 - 6 + 5

9x = 2

x = 2/9  ( or 0.2repeating)

----------------------------------------------------------------------

7x + 1/2(4x - 5) = 3/2 + 2(x - 3)

7x + 2x - 5/2 = 3/2 + 2x - 6

7x - 5/2 = 3/2 - 6

14x - 5 = -9/2

14x = -9 + 5

14x = -4

x = -2/7 ( or 0.285714 repeating)

Answer: $505,000

Step-by-step explanation:

The salary difference between a teacher and a cashier is:

$42,630 - $17,380 = $25,250

This means that a teacher earns $25,250 more per year than a cashier.

To find out how much more a teacher would earn over a 20-year career, we need to multiply the salary difference by the number of years:

$25,250 x 20 = $505,000

Therefore, a teacher would earn $505,000 more than a cashier over a 20-year career.

Answer:

y = -50x + 500

Step-by-step explanation:

The equation is y = mx + b

m = the slope

b = y-intercept

Slope = rise/run or (y2 - y1) / (x2 - x1)

Pick 2 points (5,250) (7,150)

We see the y decrease by 100, and the x increase by 2, so the slope is

m = -100/2 = -50

Y-intercept is located at (0,500)

So, the equation is y = -50x + 500

If I eat too much, then I will get sick." The reverse of this statement would be "If I get sick, then I ate too much." However, this is not always true, as getting sick can have multiple causes and not just be Attributed to eating too much

An if-then statement is a type of logical statement that relates two propositions, where the second proposition is the consequence of the first. An example of an if-then statement is "If it rains, then the ground will be wet." The reverse of this statement would be "If the ground is wet, then it rained."

Using everyday knowledge, an example of an if-then statement whose reverse is not correct is "If I eat too much, then I will get sick." The reverse of this statement would be "If I get sick, then I ate too much." However, this is not always true, as getting sick can have multiple causes and not just be attributed to eating too much.

Another example could be "If I study hard, then I will pass the test." The reverse of this statement would be "If I pass the test, then I studied hard." This may not always be true, as there could be other factors that contributed to passing the test. Therefore, it is important to be mindful of the validity of if-then statements and their reverses.

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(A)  the annual rate of change between 1992 and 2006 was 0.0804

(B) r = 0.0804 * 100% = 8.04%

(C) value in 2009 = $11,650

What is the rate of change?

The rate of change is a mathematical concept that measures how much one quantity changes with respect to a change in another quantity. It is the ratio of the change in the output value of a function to the change in the input value of the function. It describes how fast or slow a variable is changing over time or distance.

A) The initial value is $44,000 and the final value is $15,000. The time elapsed is 2006 - 1992 = 14 years.

Using the formula for an annual rate of change (r):

final value = initial value * [tex](1 - r)^t[/tex]

where t is the number of years and r is the annual rate of change expressed as a decimal.

Substituting the given values, we get:

$15,000 = $44,000 * (1 - r)¹⁴

Solving for r, we get:

r = 0.0804

So, the annual rate of change between 1992 and 2006 was 0.0804 or approximately 0.0804.

B) To express the rate of change in percentage form, we need to multiply by 100 and add a percent sign:

r = 0.0804 * 100% = 8.04%

C) Assuming the car value continues to drop by the same percentage, we can use the same formula as before to find the value in the year 2009. The time elapsed from 2006 to 2009 is 3 years.

Substituting the known values, we get:

value in 2009 = $15,000 * (1 - 0.0804)³

value in 2009 = $11,628.40

Rounding to the nearest $50, we get:

value in 2009 = $11,650

Hence, (A)  the annual rate of change between 1992 and 2006 was 0.0804

(B) r = 0.0804 * 100% = 8.04%

(C) value in 2009 = $11,650

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Therefore , the solution of the given problem of function comes out to be 3x – 6 as a result.

The questions on the midterm exam will cover every topic, including created and actual places and also algebraic variable design. a diagram showing the relationships between different elements that cooperate to create the same result. A service is composed of numerous distinctive components that cooperate to create distinctive results for each input.

Here,

We can easily add the two functions to obtain f(x) + g(x):

=> F(x) = (x² + 6x - 5) + (-x² - 3x - 1) + G(x)

By condensing similar words, we can say:

=>  x² + 6x - 5 - x² - 3x - 1 = f(x) plus g(x).

=>  (x² - x²) + (6x - 3x) + f(x) + g(x) (-5 - 1)

=>  f(x) + g(x) = 3x - 6

=>  F(x) + G(x) = 3x – 6 as a result.

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Answer:

b

Step-by-step explanation:

a. the probability that a randomly selected green turtle migrates a distance of less than 1900 kilometers is approximately 0.3156 or 31.56%.

b. the probability that a randomly selected green turtle migrates a distance between 2000 kilometers and 2500 kilometers is approximately 0.3556 or 35.56%.

c. the probability that a randomly selected green turtle migrates a distance greater than 2450 kilometers is approximately 0.3446 or 34.46%.

As the decimal system uses kilometres as the unit of measurement, distance is expressed in kilometres instead of miles. Using a unit with decimals improves accuracy and convenience. Miles are not decimal system units.

Just three nations—the United States, Liberia, and Myanmar—continue to use the imperial measurements, which relies on measurements of length, weight, height, or area that ultimately refer to human parts or commonplace objects.

We can use the standard normal distribution to solve this problem, by transforming the distances into z-scores using the formula:

z = (x - μ) / σ

Where:

(a) To find the probability that a randomly selected green turtle migrates a distance of less than 1900 kilometers, we need to find the z-score for 1900 using the formula:

z = (1900 - 2200) / 625 = -0.48

We can then use a standard normal table or calculator to find the probability corresponding to this z-score, which is approximately 0.3156.

Therefore, the probability that a randomly selected green turtle migrates a distance of less than 1900 kilometers is approximately 0.3156 or 31.56%.

(b) To find the probability that a randomly selected green turtle migrates a distance between 2000 kilometers and 2500 kilometers, we need to find the z-scores for both distances:

z₁ = (2000 - 2200) / 625 = -0.32

z₂ = (2500 - 2200) / 625 = 0.48

We can then use a standard normal table or calculator to find the probability corresponding to each z-score and subtract them to find the probability in between, which is approximately 0.3556.

Therefore, the probability that a randomly selected green turtle migrates a distance between 2000 kilometers and 2500 kilometers is approximately 0.3556 or 35.56%.

(c) To find the probability that a randomly selected green turtle migrates a distance greater than 2450 kilometers, we need to find the z-score for 2450 using the formula:

z = (2450 - 2200) / 625 = 0.4

We can then use a standard normal table or calculator to find the probability corresponding to this z-score and subtract it from 1 to find the probability of a distance greater than 2450, which is approximately 0.3446.

Therefore, the probability that a randomly selected green turtle migrates a distance greater than 2450 kilometers is approximately 0.3446 or 34.46%.

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Ahmed placed a chocolate slice is $1.75.

Arithmetic is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots.

Here, we have

Given: Ahmed and Tiana buy a cake for $14 that is half chocolate and half vanilla. They cut the cake into 8 slices.

Denoting the value of the slice of chocolate as C and V as the vanilla slice then

4 slices × C + 4 slices *V = $14

if Ahmed  likes 4 times more the chocolate than the vanilla slice, then he finds C four times more valuable than V, thus

C = 4*V

4 slices × 4V + 4 slices ×V = $24

20 slices × P = $14

P = $0.7/slice

V = 4 × P = 4 × $0.7/slice  = $2.8/slice

for a slice that is half chocolate and half vanilla

value= 1/2 slice × C + 1/2 slice × V = 1/2 slice ($0.7 /slice + $2.8/slice) =  $1.75

Hence,  Ahmed placed a chocolate slice is $1.75.

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Answer:

what trapezoid? please provide the trapezoid so I can help you

According to the question there are 54 different breakfast specials that can be ordered from this menu.

Multiplication in mathematics is the addition of equal groups. As we proliferate, the multitude of issues in the group increases. The product as well as the two elements are part of a multiplication issue. 6 and 9 are factors in the multiplication equation 6 9 = 54, and 54 is the result.

Given,

It seems like the breakfast special menu consists of four categories, with different options for each category.

Juice: Apple, Orange, Grapefruit

Toast: White, Brown

Eggs: Scrambled, Fried, Poached

Beverage: Coffee, Tea, Milk

To order the breakfast special, a customer must choose one option from each category. For example, a customer might choose Orange juice, Brown toast, Scrambled eggs, and Coffee for their breakfast special.

The total number of different breakfast specials that can be ordered is the product of the number of options in each category. So, in this case, the total number of breakfast specials is:

3 (options for juice) x 2 (options for toast) x 3 (options for eggs) x 3 (options for beverage) = 54

Therefore, there are 54 different breakfast specials that can be ordered from this menu.

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Using binomial expansion, the sum of the fourth roots of 17 and 15 is approximately 3.9985.

We are given that x is so small that its fourth and higher powers may be neglected. Therefore, we can use the binomial expansion to approximate the fourth roots of (1+x) and (1-x) as follows:

∜((1+x)) ≈ 1 + (1/4)x - (1/32)x² + O(x³)

∜((1-x)) ≈ 1 - (1/4)x - (1/32)x² + O(x³)

where O(x³) represents the neglected terms of x³ and higher powers.

Adding these approximations, we get:

∜((1+x)) + ∜((1-x)) ≈ 2 + (1/16)(-2x²) + O(x³)

Simplifying:

∜((1+x)) + ∜((1-x)) ≈ 2 - (1/8)x²

Comparing this with the given equation a - bx², we see that a = 2 and b = 1/8.

Now, we can substitute x=1/16 into the approximate expression we obtained for the sum of the fourth roots:

∜((1+(1/16))) + ∜((1-(1/16))) ≈ 2 - (1/8)(1/16)²

Simplifying:

∜(17/16) + ∜(15/16) ≈ 3.9985

Therefore, the sum of the fourth roots of 17 and 15 is approximately 3.9985.

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The sector has a central angle of 270° and the radius is 9m. So, the area of the sector is:

A_sector = (270/360) * π * (9m)^2 = 57.15m²

To find the area of the triangle, we need to find the height of the triangle. We can use the Pythagorean theorem to find the height:

h = √[(9m)^2 - (4.5m)^2] = √(81m^2 - 20.25m^2) = √60.75m^2 = 7.8m

The base of the triangle is 4.5m (half of the diameter), so the area of the triangle is:

A_triangle = (1/2) * 4.5m * 7.8m = 17.55m²

Therefore, the area of the shaded segment is:

A_shaded segment = A_sector - A_triangle = 57.15m² - 17.55m² = 39.6m²

Rounded to the nearest tenth, the area of the shaded segment is 39.6m².

well, since all interior angles in a triangle add up to 180°, by definition A = 44°, since B and A are complementary anyway.

[tex]\cos(46^o )=\cfrac{\stackrel{adjacent}{9}}{\underset{hypotenuse}{c}}\implies c=\cfrac{9}{\cos(46^o)}\implies c\approx 13.0 \\\\[-0.35em] ~\dotfill\\\\ \tan(46^o )=\cfrac{\stackrel{opposite}{b}}{\underset{adjacent}{9}}\implies 9\tan(46^o )=b\implies 9.3\approx b[/tex]

Make sure your calculator is in Degree mode.

The missing values in the figure is solved using Inscribed Angle Theorem to get

x = 7

arc KL = 92 degrees

arc KLJ = 225 degrees

The arc length KL is solved with the knowledge that sum of arc length in a circle is 360 degrees

135 + 133 + KL = 360

KL = 360 - 135 - 133

KL = 92

Solving for x

inscribed angle = 0.5 * intercepted arc

5x + 11 = 0.5 * 92

5x + 11 = 46

5x = 46 - 11

5x = 35

x = 7

Arc KLJ

= 92 + 133

= 225

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The numerical sign of the limiting value is significant because it indicates the general direction of the function's behavior (increasing or decreasing) within the given interval,

Hi there! In Exploration 5.3.2, Question 3 required finding multiple average rates of change for different intervals, and in Question 6, you discussed the limiting value that those average rates are approaching. The significance of the numerical sign of the limiting value found in Question 6, addressed in Question 7, is as follows:

The sign of the limiting value (positive or negative) indicates the general direction of the function's behavior during the given interval. A positive limiting value means that the function is generally increasing, while a negative limiting value suggests that the function is generally decreasing.

Understanding the sign of the limiting value is essential as it provides insight into the overall trend of the function in the specified range. This information can be useful in various applications, including predicting future behavior and identifying potential maximum or minimum points of the function.

In summary, providing valuable information about the function's characteristics and trends.

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Answer:

50

Step-by-step explanation:

P= 2l + 2w Formula for a perimeter

P= 184 given

l= 4x+2 given

w= 5x given

184= 2(4x+2) + 2(5x) substitution

184= 8x+4+ 10x distributive property

184= 18x+4 combine like terms

184 -4 =18x +4 -4 subtract 4 from both sides to get 18x alone

180=18x

180/18= 18x/18 divide both sides by 18 to get x alone

10= x

x=10

Side PS= 5x since it is parallel to RQ which is 5x

5(x)=5(10)=50

Answer:

The answer is the first option

Step-by-step explanation:

The reason why it is the first option is because if we add up the total amount of things sold (candy, snacks, automotive supplies, and beverages) we would see that together the store sold 500 items. Using that we can diving the amount of candy, snacks, automotive supplies, and beverages by the total amount of things sold to get the percentages.

Ex: 75/500 = 0.15*100 = 15%
     100/500 = 0.20*100 = 20%
     etc.

Answer:

the correct answer is the last one i think

Step-by-step explanation:

Will put in chat if correct or not if i remember  to do it.

The experimental probability of landing on a number greater than 5 is 8/60, which simplifies to 2/15.

Experimental probability is the probability of an event based on the results of an experiment or observation.

It is calculated by dividing the number of times the event occurs by the total number of trials or observations.

It is also called empirical probability or observed probability.

Experimental probability can be used to estimate the theoretical probability of an event, which is the probability of the event based on mathematical reasoning or modeling.

As the number of trials or observations increases, experimental probabilities tend to approach theoretical probabilities. 

The experimental probability of landing on a number greater than 5 is the number of times a number greater than 5 (i.e., 6) is obtained divided by the total number of tosses.

From the table, we see that 6 is obtained 8 times out of 60 tosses.

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For A (15) (0.4) (0.6) (15-i) For B. Consequently, there is a 0.196 percent chance For C. As a result, there is a roughly 0.382 percent chance

The number of successes in a certain number of independent trials with the same chance of success are described by the binomial distribution, which is a probability distribution. The two parameters n and p define the binomial distribution. The parameters n and p represent the number of trials and the probability of success in each trial, respectively.

Let's say that 40 percent of recent college grads want to earn a graduate degree. We choose fifteen recent college grads at random.

a. Using the binomial distribution formula, we can determine the likelihood that no more than four of the college grads intend to pursue a graduate degree:

P(X ≤ 4) = Σ(i=0 to 4) Choose (15) (0.4) (0.6) (15-i)

where X represents the proportion of recent college graduates who intend to pursue a graduate degree. We can determine that using a calculator or software that:

P(X ≤ 4) ≈ 0.0001

As a result, there is a roughly 0.0001 chance that no more than four of the college graduates will go on to get a graduate degree.

b. We can once more use the data to determine the likelihood that precisely seven of the college grads intend to pursue a graduate degree.

use the formula for the binomial distribution:

P(X = 7) = (15 choose 7) (15 choose 7) (0.4)⁷ (0.6⁸

We can determine that using a calculator or software that:

P(X = 7) ≈ 0.196

C . The cumulative binomial distribution function can be used to calculate the likelihood that at least six but no more than nine of the college graduates intend to pursue a graduate degree:

P(6 X 9) = (i=6 to 9) (15 choose I (0.4)i (0.6) (15-i)

We can determine that using a calculator or software that:

P(6 ≤ X ≤ 9) ≈ 0.382

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Answer:

---------------------------

Given equation:

Solve it in below steps:

This equation has one solution.

Answer: First, we need to find the length of the line segment MN. Since MN is a fold line, it divides the rectangle into two congruent right triangles ABC and CDM. We can use the Pythagorean theorem to find the length of MN:

AC² + CM² = AM²

Since AC = 16 cm and CM = 2 cm (half the width of the rectangle), we have:

16² + 2² = AM²

256 + 4 = AM²

260 = AM²

AM = sqrt(260) = 2sqrt(65) cm

Since MN is the hypotenuse of right triangle ACM, we have:

MN = 2AM = 4sqrt(65) cm

Now, let's draw a line segment from B to MN, perpendicular to MN, and let the intersection point be E. Since triangle ABN is similar to triangle CDM, we have:

BN/DM = AB/CD

BN/2 = 4/16

BN = 1 cm

Since triangle BEN is a right triangle, we can use the Pythagorean theorem to find the length of BE:

BE² + EN² = BN²

BE² + (MN - DM)² = 1²

BE² + (4sqrt(65) - 2)² = 1

BE² + 64*5 - 16sqrt(65) + 4 = 1

BE² = -64*5 + 16sqrt(65) - 3

BE = sqrt(-64*5 + 16sqrt(65) - 3)

Note that BE is an imaginary number, which means that point E is actually below line segment MN. Therefore, the area of pentagon ABNMD' is zero.

Step-by-step explanation:

================================================

Explanation:

Grab some graph paper or use GeoGebra.

Place point A at the origin (0,0). Move 16 units to the right to plot B at (16,0).

Then move 4 units up to get to C(16,4). Then move 16 units left to arrive at D(0,4)

Here are the four points so far:

Next draw a line through A and C.

The equation of line AC is y = 0.25x; I'll skip the steps showing how I got that equation. But let me know if you need to see those steps.

The perpendicular bisector of segment AC is the equation y = -4x+34. Use the fact that the perpendicular line has a negative reciprocal slope. Meaning the slope 0.25 has the negative reciprocal -4. Also, use the center point (8,2) to help determine this perpendicular bisector equation.

Why is the perpendicular bisector so important? It's the mirror line. We'll reflect C over this line to land on A.

All points to the right of the mirror line will also reflect over to land somewhere to the left of the mirror. This will form the pentagon ABNMD where segment NM is the mirror line.

-----------

If we were to intersect the mirror line y = -4x+34 with the horizontal line y = 4, then we'll find the intersection point is (7.5,0) which is the location of point M in the diagram below.

Intersect y = 0 (aka the x axis) with y = -4x+34 to find the location of point N(8.5, 0)

So we should have

M = (7.5, 0)

N = (8.5, 0)

-----------

Pentagon ABNMD is composed of the following triangles

But notice carefully that triangle NPQ has folded over mirror line MN to land exactly on top of triangle ABN. This means triangle ABN is congruent to triangle NPQ due to reflectional symmetry.

Also due to the symmetry of the fold, triangle ADM = triangle NPQ

Because of symmetry we have:

Apply the transitive property to find triangle ADM = triangle ABN

-----------

area of triangle ADM = 0.5*base*height

area of triangle ADM = 0.5*MD*AD

area of triangle ADM = 0.5*7.5*4

area of triangle ADM = 15

Therefore, triangle ABN is also 15 square cm as well.

area of triangle MNA = 0.5*base*height

area of triangle MNA = 0.5*AN*4

area of triangle MNA = 0.5*8.5*4

area of triangle MNA = 17

-----------

Summary of the triangle areas:

Therefore,

pentagon ABNMD = (triangle ADM)+(triangle MNA)+(triangle ABN)

pentagon ABNMD = (15)+(17)+(15)

pentagon ABNMD = 47 square cm is the final answer.

The diagram is shown below. I used GeoGebra to make it. The diagram is to scale.

Notes:

Answer:

8

Step-by-step explanation:

Answer:

angle 3 is 127 and angle 4 is 31

Step-by-step explanation:

If you see the base of the triangle looks like an angle measurer of 180 degrees. so if 2 is 53 then subtract 180-53 to get 127. Once you have 3 just subtract 180 - 22 and 127 to get 31  

:D

The range for the skull size of Welsh Corgis is given by the compound inequality 54.8 ≤ D ≤ 59.7.

We can solve the given inequality 25 ≤ 1.05D – 32.7 ≤ 30 for D by adding 32.7 to all three parts of the inequality:

25 + 32.7 ≤ 1.05D – 32.7 + 32.7 ≤ 30 + 32.7

57.7 ≤ 1.05D ≤ 62.7

Then, we can divide all three parts of the inequality by 1.05:

57.7/1.05 ≤ D ≤ 62.7/1.05

54.76 ≤ D ≤ 59.71

Mathematical statements that compare two quantities are known as inequality. They represent the relationship between the quantities with symbols like, >,, and. An inequality may have one or more variables, and based on the values of those variables, the inequality may be true or false. The collection of all values for the variables necessary to make an inequality true is the answer to the inequality. In real-world applications, inequalities are frequently used to indicate limitations, limits, or ranges of values.

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the function is symmetric about the vertical line [tex]x = 3[/tex]  . This means that the inverse function should also be symmetric about this line. We choose the positive solution for y:This gives us the inverse of H(x). We can write it as[tex]H^(-1)(x) = 3 ± sqrt((x - 4) / 2)[/tex]

To find the inverse of a function, we need to switch the positions of x and y and solve for y.

[tex]Let y = H(x) = 2(x-3)^2+4.[/tex]

We can begin by subtracting 4 from both sides to get:

[tex]y - 4 = 2(x-3)^2[/tex]

Next, we can divide both sides by 2 to get:

[tex](y - 4) / 2 = (x-3)^2[/tex]

Taking the square root of both sides, we obtain:

[tex]±sqrt((y - 4) / 2) = x - 3[/tex]

Adding 3 to both sides gives:

[tex]x = 3 ± sqrt((y - 4) / 2)[/tex]

Therefore, This gives us the inverse of H(x). We can write it as:

[tex]H^(-1)(x) = 3 ± sqrt((x - 4) / 2)[/tex]

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