what is the equation of a line that passes through the points (2,5) and (4,3)

Part A:

A(t) is a linear function, and 12∙J(t) is an exponential function. The straight line nature of A(t) indicates linear growth, while the curve of 12∙J(t) suggests exponential growth.

Part B:

The intersection of the 12∙J(t) curve and the A(t) curve represents the point where the accumulated profit from sales is sufficient to pay off the entire loan in one lump-sum payment.

Part C:

As P increases while keeping r and J(t) constant, the relationship between the two curves shifts, indicating a higher profit requirement to cover the increased loan amount. This is dangerous for business financial health and banks have safeguards to prevent excessive debt and defaults.

Part A:

From the given information, we have two functions: A(t) and 12∙J(t). Let's analyze their features to determine their types.

A(t) is a linear function:

A linear function is characterized by a constant rate of change and forms a straight line on a graph.

In the given graph, A(t) is represented by a straight line, indicating a linear relationship between the variables.

This suggests that A(t) is a linear function.

12∙J(t) is an exponential function:

An exponential function is characterized by a constant ratio or base and shows exponential growth or decay.

In the given graph, 12∙J(t) is represented by a curve that exhibits exponential growth.

The increasing rate of change as time progresses indicates an exponential relationship.

Therefore, 12∙J(t) is an exponential function.

Part B:

The intersection of the 12∙J(t) curve and the A(t) curve represents the point at which the profit generated from sales is sufficient to pay off the entire loan in one lump-sum payment. In other words, it represents the point in time when the accumulated profit matches the amount of the loan.

At this intersection point, the profit generated from sales has reached a level where it can fully cover the principal and interest owed on the loan. This indicates that the business has generated enough funds to repay the loan in its entirety.

Part C:

As the principal (P) is increased while keeping the interest rate (r) and 12∙J(t) constant, the relationship between the two curves changes. Specifically, the intersection point between the curves shifts to the right on the graph.

This change in the relationship between the curves signifies that the business needs a higher level of profit to cover the increased loan amount. It suggests that the business has taken on a larger loan, which requires a higher profit to repay.

This situation is considered dangerous for the financial health of the business because it increases the risk of not generating sufficient profit to repay the loan. If the business fails to generate enough profit to cover the loan payments, it may lead to financial instability and potential default on the loan.

To mitigate such risks, banks put safeguards in place to prevent businesses from taking on excessive debt. These safeguards include conducting thorough evaluations of the business's financial health, setting limits on loan amounts based on the business's income and creditworthiness, and assessing the repayment capacity of the borrower. These measures aim to minimize the risk of defaults and protect both the borrower and the lender.

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The length of segment OP, which represents the distance from the origin to point P on the unit circle, is always equal to 1.

To determine the length of segment OP on the unit circle, we need to use trigonometry. Let's break down the problem step by step:

Definition: The unit circle is a circle with a radius of 1 centered at the origin (0, 0) in the Cartesian coordinate system.

Initial Ray: The initial ray is a line segment that starts from the origin (0, 0) and extends to a point on the unit circle. It forms an angle with the positive x-axis.

Rotation: We are rotating the initial ray counterclockwise by θ degrees. This means we are essentially finding a new point on the unit circle based on the angle θ.

Trigonometric Functions: The trigonometric functions sine (sin) and cosine (cos) are particularly useful for calculating the coordinates of points on the unit circle.

sin(θ) gives the y-coordinate of a point on the unit circle.

cos(θ) gives the x-coordinate of a point on the unit circle.

Coordinates of Point P: Since we are rotating the initial ray counterclockwise by θ degrees, the coordinates of point P on the unit circle can be obtained as follows:

x-coordinate of P: cos(θ)

y-coordinate of P: sin(θ)

Distance from the Origin (Length of Segment OP):

Using the coordinates of point P, we can calculate the distance between the origin (0, 0) and point P using the distance formula.

The distance formula states that for two points (x1, y1) and (x2, y2), the distance between them is given by:

d = √((x2 - x1)² + (y2 - y1)²)

In this case, point P has coordinates (cos(θ), sin(θ)), and the origin is (0, 0). Thus, the distance (length of segment OP) is:

d = √((cos(θ) - 0)² + (sin(θ) - 0)²)

= √(cos²(θ) + sin²(θ))

= √(1) [Using the trigonometric identity: sin²(θ) + cos²(θ) = 1]

= 1

Therefore, the length of segment OP, which represents the distance from the origin to point P on the unit circle, is always equal to 1.

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Answer:  A    -6x² + 2x -4

Step-by-step explanation:

[tex]\frac{-12x^{3}+ 4x^{2} -8x}{2x}[/tex]                             >Divide each of the top terms by 2x[tex]=\frac{-12x^{3}}{2x} +\frac{4x^{2} }{2x} -\frac{8x}{2x}[/tex]

= -6x² + 2x -4

Answer:

At the time the ball was thrown, it was 8 feet above the ground.

h'(x) = -32x + 64 = 0, so x = 2

The ball reaches its maximum height after 2 seconds.

As you can see below, both expressions result in the same value of 105. This demonstrates the application of the associative law in regrouping terms and maintaining the equivalence of the expression.

The associative law in mathematics states that the grouping of numbers in an addition or multiplication operation does not affect the result. In other words, you can regroup terms within parentheses without changing the value of the expression.

Using the associative law, we can regroup the terms in the expression s + (r + 75) by removing the parentheses and rearranging the terms:

s + (r + 75) = (s + r) + 75

The expression (s + r) + 75 is equivalent to s + (r + 75) because the addition operation is associative.

Let's take an example to illustrate this:

Suppose s = 10 and r = 20.

Using the original expression:

s + (r + 75) = 10 + (20 + 75) = 10 + 95 = 105

Using the expression with regrouped terms:

(s + r) + 75 = (10 + 20) + 75 = 30 + 75 = 105

As you can see, both expressions result in the same value of 105. This demonstrates the application of the associative law in regrouping terms and maintaining the equivalence of the expression.

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Answer: [tex]3x^4yz^3[/tex]

I'm hoping this is your equation: [tex](9x^{8}y^{2}z^{6})^{1/2}[/tex]

and not: [tex](9x^{8y^{2z^{z^6}}})[/tex]

Step-by-step explanation:

The square root of a number can be shown as a 1/2 power

We'll use the exponent rule:

= [tex]9^{1/2}(x^8 )^{1/2}(y^2)^{1/2}(z^6){1/2}[/tex]

Then we'll do each term individually

the square root of 9 is 3

for [tex](x^8)^{1/2}[/tex] the exponents multiply which give us [tex]x^4[/tex]

[tex](y^2)^{1/2}[/tex] gives us y

[tex](z^6)^{1/2}[/tex] gives us z^3

After doing all this we get [tex]3x^4yz^3[/tex]

Answer: Unfactorable

Step-by-step explanation:

5v^2-30v+70

5(v^2-6v+14)

You can't factor v^2-6v+14 with rational numbers

Answer:

Step-by-step explanation:

The length of PQ is 3√5 and its slope is -2

The length of SR is 3√5 and its slope is -2

The length of SP is 5√2 and its slope is -7

The length of RQ is 5√2 and its slope is -1

So PQ ≅ SR and SP ≅ RQ. By the Perpendicular Bisector theorem, adjacent sides are perpendicular. By the selection of  side, ∠PSR, ∠SRQ, ∠RQP and ∠QPS are right angles. So, the quadrilateral is a rectangle.

To find the lengths and slopes of the sides of the quadrilateral PQRS, we apply the distance formula:

D = [tex]\sqrt{(x_{2} - x_{1})^2 + (y_{2} - y_{1})^2}[/tex]  

and the slope formula:

m = [tex]\frac{y_{2} - y_{1}}{x_{2} - x_{1}}[/tex]

1. Length PQ:

Using the distance formula, the length PQ can be calculated as follows:

PQ = [tex]\sqrt{(x_{2} - x_{1})^2 + (y_{2} - y_{1})^2}[/tex]

  = √((3 - 0)² + (-4 - 2)²)

  = √(3² + (-6)²)

  = √(9 + 36)

  = √45

  = 3√5

2. Length SR:

Using the distance formula, the length SR can be calculated as follows:

SR = [tex]\sqrt{(x_{2} - x_{1})^2 + (y_{2} - y_{1})^2}[/tex]

  = √((1 - (-2))² + (-5 - 1)²)

  = √((1 + 2)² + (-6)²)

  = √(3² + 36)

  = √(9 + 36)

  = √45

  = 3√5

3. Length SP:

Using the distance formula, the length SP can be calculated as follows:

SP = [tex]\sqrt{(x_{2} - x_{1})^2 + (y_{2} - y_{1})^2}[/tex]

  = √((1 - 0)² + (-5 - 2)²)

  = √(1² + (-7)²)

  = √(1 + 49)

  = √50

  = 5√2

4. Length RQ:

Using the distance formula, the length RQ can be calculated as follows:

RQ = [tex]\sqrt{(x_{2} - x_{1})^2 + (y_{2} - y_{1})^2}[/tex]

  = √((-2 - 3)² + (1 - (-4))²)

  = √((-2 - 3)² + (1 + 4)²)

  = √((-5)² + 5²)

  = √(25 + 25)

  = √50

  = 5√2

Now, let's calculate the slopes of the sides:

1. Slope PQ:

The slope of PQ can be calculated using the slope formula:

m = [tex]\frac{y_{2} - y_{1}}{x_{2} - x_{1}}[/tex]

  = (-4 - 2) / (3 - 0)

  = -6 / 3

  = -2

2. Slope SR:

The slope of SR can be calculated using the slope formula:

m = [tex]\frac{y_{2} - y_{1}}{x_{2} - x_{1}}[/tex]

  = (-5 - 1) / (1 - (-2))

  = -6 / 3

  = -2

3. Slope SP:

The slope of SP can be calculated using the slope formula:

m =[tex]\frac{y_{2} - y_{1}}{x_{2} - x_{1}}[/tex]

  = (-5 - 2) / (1 - 0)

  = -7 / 1

  = -7

4. Slope RQ:

The slope of RQ can be calculated using the slope formula:

m = [tex]\frac{y_{2} - y_{1}}{x_{2} - x_{1}}[/tex]

  = (1 - (-4)) / (-2 - 3)

  = 5 / (-5)

  = -1

Therefore, the lengths and slopes of the sides of the quadrilateral PQRS are:

Length PQ: 3√5

Length SR: 3√5

Length SP: 5√2

Length RQ: 5√2

Slope PQ: -2

Slope SR: -2

Slope SP: -7

Slope RQ: -1

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Subtraction and Division

Inverse operations help find the solution to equations.

Defining Inverse Operations

Firstly, let's define an operation. An operation in math is a function that can manipulate a value. Inverse operations are operations that are opposite operations that undo each other. For example, addition and subtraction are inverse operations because subtraction undoes addition. Multiplication and division are also inverse operations.

Solving the Equation

The equation 16 = 3x + 1 involves both addition and multiplication. So, to solve this, we can use the inverse operations of subtraction and division. First, subtract 1 from both sides.

Then, divide both sides by 3.

This shows that by using subtraction and division, we can undo the addition and multiplication used in the equation. This allows us to find the value of x.

The minimum value of [tex]f(x) = -2^3 \sqrt\(5-4x+3)[/tex] on the interval [1, 8] is -3 and the maximum value is 5. To find the minimum value, we can start by finding the critical points of the function.

The critical points are the points where the derivative of the function is equal to zero. In this case, the derivative of the function is

[tex]f'(x) = -2^3 \times (5-4x+3) ^(-3/2) \times (-4)[/tex]

The critical points of the function are x = 1 and x = 5.

We can now evaluate the function at each critical point and at the endpoints of the interval to find the minimum and maximum values. The values of the function at the critical points and at the endpoints are

x | f(x)

-- | --

1 | -3

5 | 5

8 | 9

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ΔFSH ≅ ΔFSI by the rule of angle-angle-side theorem, or AAS.

The angle-angle-side theorem, or AAS, tells us that if two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the triangles are congruent.

If we consider triangle FSH and triangle FSI, we will observe the following;

So based on the  angle-angle-side theorem, or AAS, we can see that triangle FSH is congruent to triangle FSI.

Thus, our answer will be; ΔFSH ≅ ΔFSI by the rule of angle-angle-side theorem, or AAS.

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The function A(x) = 100 + 4x describes the scenario of an ant colony that starts with an initial population of 100 and experiences a constant rate of increase of 4 ants per year.

We have,

The function A(x) = 100 + 4x represents a linear relationship between the variable x (representing time in this case) and the variable A(x) (representing the population of the ant colony).

The term 100 in the function represents the initial population of the ant colony.

It indicates that at the starting point (x = 0), the population is 100.

The term 4x in the function represents the rate at which the population increases over time. Since the coefficient of x is positive (4), it indicates that the population is increasing.

For every unit increase in x (in this case, for every year that passes), the population increases by 4.

Therefore,

The function A(x) = 100 + 4x describes the scenario of an ant colony that starts with an initial population of 100 and experiences a constant rate of increase of 4 ants per year.

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a. It will take 7 months to pay off the credit card.

b. it will take 4 months to pay off the credit card.

Since, APR stands for Annual Percentage Rate. It is the interest rate charged on a loan or credit card, expressed as a yearly percentage rate. The APR takes into account not only the interest rate, but also any fees or charges associated with the loan or credit card.

a. If you put half of the available money each month toward the credit card, then you are paying $150.00 per month towards the credit card balance.

We can use the formula for the present value of an annuity to find how many months it will take to pay off the credit card:

PV = PMT × ((1 - (1 + r)⁻ⁿ) / r)

where:

PV is the present value of the debt

PMT is the payment amount per period

r is the monthly interest rate

n is the number of periods

Substituting the values, we get:

754.43 = 150 × ((1 - (1 + 0.011333)⁻ⁿ) / 0.011333)

Simplifying and solving for n, we get:

n = log(1 + (PV ×r / PMT)) / log(1 + r)

n = log(1 + (754.43×0.011333 / 150)) / log(1 + 0.011333)

n = 6.18

Therefore, it will take 7 months to pay off the credit card if you put half of the available money each month toward the credit card.

b. If you put all of the available money each month toward the credit card, then you are paying $300.00 per month towards the credit card balance.

754.43 = 300 ×((1 - (1 + 0.011333)⁻ⁿ) / 0.011333)

Simplifying and solving for n, we get:

n = log(1 + (PV × r / PMT)) / log(1 + r)

n = log(1 + (754.43× 0.011333 / 300)) / log(1 + 0.011333)

n = 3.43

Therefore, it will take 4 months to pay off the credit card if you put all of the available money each month toward the credit card.

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The solution to the equation 3x = 8 - 2x is x = 1.6.

To solve the equation 3x = 8 - 2x by graphing, we can plot the two sides of the equation as functions of x and find the point(s) where they intersect. Here's a step-by-step explanation:

Express the equation in the form of y = f(x). Rearrange the equation:

3x + 2x = 8

5x = 8

x = 8/5 or 1.6

Graph the functions y = 3x and y = 8 - 2x on the same coordinate plane. The line represented by y = 3x is upward sloping, and the line represented by y = 8 - 2x is downward sloping.

Plot the points (1.6, 3(1.6)) and (1.6, 8 - 2(1.6)) on the graph.

The point of intersection represents the solution to the equation. In this case, the lines intersect at (1.6, 4.8).

Therefore, the solution to the equation 3x = 8 - 2x is x = 1.6.

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Answer:

Step-by-step explanation:

May Kawasaki's required minimum distribution (RMD) is $3,808. This is calculated by dividing her IRA balance of $98,000 by the distribution period of 26.2, which is found in the Uniform Life Table on page 45 for a 72-year-old.

b) If May Kawasaki fails to take her RMD, she will incur a 50% penalty on the amount she should have withdrawn. In this case, the penalty would be $1,904.

c) If May Kawasaki had made an early withdrawal of $10,000 to take a vacation, she would have paid a 10% penalty on the amount withdrawn. In this case, the penalty would have been $1,000.

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1. May Kawasaki is 72 and has an IRA with a fair market value of $98,000. a) Use the Uniform Life Table on p. 45 to determine her required minimum distribution. b) What penalty would she incur if she failed to take the distribution? c) What penalty would she have paid if she had made an early withdrawal of $10,000 to take a vacation?

Options 2, 3, and 5 are correct representations of the line passing through the given points.

The equation y = -5/3(x - 2) represents a line passing through the points (-4, 10) and (-1, 5).

Let's verify each option:

y = -5/3x - 2: This equation does not represent the same line. The constant term is different (-2 instead of +10/3).

y = -5/3x + 10/3: This equation represents the same line. It has the same slope (-5/3) and the same y-intercept (10/3).

3y = -5x + 10: This equation represents the same line. It can be simplified by dividing both sides by 3, resulting in the same slope (-5/3) and the same y-intercept (10/3).

3x + 15y = 30: This equation does not represent the same line. The coefficients of x and y are different, resulting in a different slope.

5x + 3y = 10: This equation represents the same line. It has the same slope (-5/3) and the same y-intercept (10/3).

Therefore, options 2, 3, and 5 are correct representations of the line passing through the given points.

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a) The  two equations in terms of x and y: one for the perimeter and one for the area of the rectangle are:

y = 0.5x + 3.5

6y + 3 = x² + 14x + 48

b) The area and perimeter of the rectangle are:

Perimeter = 30 m

Area = 15 m²

The formula to find the area of a rectangle is:

Area = Length * Width

The formula to find the perimeter of a rectangle is:

Perimeter = 2(Length + Width)

We are given that:

Perimeter = 8y meters

Area = (6y + 3) square meters

From the image, we see that:

Length = x + 6

Width = x + 8

Thus:

Perimeter equation is:

8y = 2(x + 6 + x + 8)

8y = 4x + 28

y = 0.5x + 3.5

Area equation is:

6y + 3 = (x + 6)(x + 8)

6y + 3 = x² + 14x + 48

Thus:

6(0.5x + 3.5) + 3 = x² + 14x + 48

3x + 24 = x² + 14x + 48

x² + 11x + 24 = 0

Using quadratic equation calculator gives:

x = -8 or -3

Thus, we will use x = -3 and we have:

Length = -3 + 6 = 3 m

Width = -3 + 8 = 5 m

Perimeter = 2(3 + 5) = 30 m

Area = 3 * 5 = 15 m²

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The function representing the number of bacteria after t hours is N(t) = 5300 * (1 + 0.0592)^t, and the growth rate per hour is approximately 5.92%.

To represent the number of bacteria after t hours, we can use the exponential growth formula:

N(t) = N₀ * (1 + r)^t,

where N(t) is the number of bacteria after t hours, N₀ is the initial number of bacteria, r is the hourly growth rate, and t is the time in hours.

In this case, the initial number of bacteria is 5300, and the hourly growth rate can be determined from the doubling time of 12 hours. The growth rate can be calculated using the formula:

r = 2^(1/t_double) - 1,

where t_double is the doubling time in hours.

Substituting the given values, we have:

r = 2^(1/12) - 1 ≈ 0.0592.

Now we can write the function for the number of bacteria after t hours:

N(t) = 5300 * (1 + 0.0592)^t.

To determine the percentage of growth per hour, we can calculate the relative growth rate as a percentage:

percentage_growth = r * 100 ≈ 5.92%.

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Answer:

Step-by-step explanation:

3x3x3= 27

Answer:

An improper fraction for 1 3/4 is 7/4.

Step-by-step explanation:

[tex]\frac{7}{4}[/tex]

To find the improper fraction of a mixed number fraction. You first have to remove the whole number from the fraction (the big one bending the fraction)

Do this by multiplying the denominator (4) by the whole number (1)

4 x 1 = 4

Then add this number with the numerator (top number) which is 3.

4+3 = 7

Seven is our new numerator, our denominator stays the same (4)

So our new improper fraction is:

[tex]\frac{7}{4}[/tex]

The volume of the tarp shelter is 65.37 ft³ .

Given,

Conic tarp shelter with radius 5.1 ft and height 7.2 ft .

Now,

Volume of cone = 1/3 × π × r² × h

Substitute the values in the formula,

Volume of cone = 1/3 ×3.14 × (5.1)² × 7.2

Volume of cone = 65.37 ft³ .

Hence volume of tarp shelter will be 65.37 ft³ .

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Answer:

208 units

Step-by-step explanation:

The first term is given as 16, which means a = 16.

The second term can be obtained by adding the common difference to the first term: 16 + d = 48.

The third term is obtained by adding the common difference to the second term: 48 + d = 80.

The fourth term is obtained by adding the common difference to the third term: 80 + d = 112.

We can solve these equations to find the value of 'd':

16 + d = 48

d = 48 - 16

d = 32

48 + d = 80

32 + 48 = 80 (valid)

80 + d = 112

32 + 80 = 112 (valid)

Therefore, the common difference is 32.

Now that we have the common difference, we can find the distance the body falls in the seventh second.

The formula for finding the nth term of an arithmetic progression is:

a_n = a + (n - 1) * d

where a_n is the nth term, a is the first term, n is the position of the term, and d is the common difference.

Plugging in the values, we can find the seventh term:

a_7 = 16 + (7 - 1) * 32

a_7 = 16 + 6 * 32

a_7 = 16 + 192

a_7 = 208

Therefore, the distance the body falls in the seventh second is 208 units.

Answer:

Jim needs to bike 64 miles on the last day to maintain an average of 59 miles per day.

Step-by-step explanation:

To find out how many miles Jim needs to bike on the last day to maintain an average of 59 miles per day, we can use the concept of averages.

The total distance Jim needs to bike over the 6 days to maintain an average of 59 miles per day can be calculated as follows:

Total distance = Average distance per day × Number of days

Total distance = 59 miles/day × 6 days = 354 miles

Jim has already biked a total of 59 + 52 + 66 + 45 + 68 = 290 miles over the first five days.

To find out how many miles Jim needs to bike on the last day, we subtract the distance he has already biked from the total distance needed:

Distance needed on the last day = Total distance - Distance already biked

Distance needed on the last day = 354 miles - 290 miles = 64 miles

Therefore, Jim needs to bike 64 miles on the last day to maintain an average of 59 miles per day over the 6-day cross-country biking challenge.

Answer:

0.30

Step-by-step explanation:

To find the probability that a randomly selected individual would spend between 30 and 40 minutes on the treadmill, we need to calculate the z-scores corresponding to these values and then use the z-table or a statistical calculator to find the probability.

First, we calculate the z-scores using the formula:

z = (x - μ) / σ

where x is the value (in this case, 30 and 40), μ is the mean (42.5), and σ is the standard deviation (4.8).

For x = 30:

z = (30 - 42.5) / 4.8 ≈ -2.604

For x = 40:

z = (40 - 42.5) / 4.8 ≈ -0.521

Next, we look up the probabilities associated with these z-scores in the z-table or use a statistical calculator.

From the z-table or calculator, the probability corresponding to z = -2.604 is approximately 0.0047, and the probability corresponding to z = -0.521 is approximately 0.3015.

To find the probability between 30 and 40 minutes, we subtract the probability associated with z = -2.604 from the probability associated with z = -0.521:

P(30 ≤ x ≤ 40) = P(z = -0.521) - P(z = -2.604)

≈ 0.3015 - 0.0047

≈ 0.2968

Therefore, the probability that a randomly selected individual would spend between 30 and 40 minutes on the treadmill is approximately 0.2968, which is equivalent to 29.68%. Rounding up we will get 0.30.

Hope this helps!

The domain and range of the graph above in interval notation include the following:

Domain = [-6, 3]

Range = [-3, 3]

In Mathematics and Geometry, a domain refers to the set of all real numbers (x-values) for which a particular function (equation) is defined.

In Mathematics and Geometry, the horizontal portion of any graph is used to represent all domain values and they are both read and written from smaller to larger numerical values, which simply means from the left of any graph to the right.

By critically observing the graph shown in the image attached above, we can reasonably and logically deduce the following domain and range:

Domain = [-6, 3] or -6 ≤ x < 3.

Range = [-3, 3] or -3 < y < 3

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The solution of the given equation is x = -10.

We are given that;

The equation 1/27^{4-x}=9^{2x-1}

Now,

This is an exponential equation that can be solved by using the properties of exponents and logarithms. Here are the steps to solve it:

Rewrite both sides of the equation using the same base. Since 27 and 9 are both powers of 3, we can use 3 as the base. We have:

(3(-3))(4-x) = (32)(2x-1)

Apply the power rule of exponents to simplify the expressions. The power rule states that (ab)c = a^(bc). We have:

3^(-12+3x) = 3^(4x-2)

Since the bases are equal, we can set the exponents equal to each other and solve for x. We have:

-12 + 3x = 4x - 2 -10 = x

Therefore, by the given equation the answer will be x = -10.

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