(a) Find a counterexample which shows that WAT is not true if we replace the closed interval [a,b] with the open interval (a,b).(b) What happens if we replace [a,b] with the closed set [a,\infty). Does the theorem hold?

equivalent integral with the order of integration reversed ST 2-3 F(x,y) dydc = o g(y) F(x,y) dedy+ So k(y) F(x,y) dardy Jh(v) a- he C- f(y) = g(y) = h(g) = k(y) = By reversing the order of integration, you've found an equivalent integral to the original one provided.

step-by-step explanation to achieve this, using the terms "integral," "reversed," and "equivalent" in the answer.

Step 1: Identify the original integral
The original integral is given as ∫∫ F(x, y) dy dx, where the integration limits are not explicitly provided. In this case, let's assume the limits of integration for y are from a(x) to b(x), and for x, they are from c to d.

Step 2: Sketch the region of integration
To reverse the order of integration, it's helpful to sketch the region of integration, which is the area in the xy-plane where the function F(x, y) is being integrated.

Step 3: Determine the new limits of integration
After sketching the region, determine the new limits of integration by considering the range of x for a given y value, and the range of y values. Let's assume the new limits for x are from g(y) to h(y), and for y, they are from e to f.

Step 4: Write the equivalent reversed integral
Now, you can write the equivalent integral with the order of integration reversed. In this case, it will be ∫∫ F(x, y) dx dy, with the new limits of integration. The complete reversed integral will look like:

∫(from e to f) [ ∫(from g(y) to h(y)) F(x, y) dx ] dy

By reversing the order of integration, you've found an equivalent integral to the original one provided.

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The volume of the campfire pyramid is approximately 8,175 cubic centimeters. This is found by using the formula for the volume of a pyramid and plugging in the given values for the height and base area of the pentagonal pyramid.

The formula for the volume of a pyramid is

V = (1/3) * base_area * height

where V is the volume, base_area is the area of the base, and height is the height of the pyramid.

In this case, the height of the pyramid is given as 27 cm, and the base area is given as 965 square cm. To find the volume, we can plug these values into the formula

V = (1/3) * 965 cm² * 27 cm

V = 8,175 cm³

Therefore, the volume of the campfire pyramid is 8,175 cubic centimeters.

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Hunter has 32 different possible outcomes if he rolls a pyramid-shaped die with four sides, spins a spinner with four equal-sized sections, and flips a coin.

There are different methods to approach this problem, but one possible way is to use the multiplication principle of counting, which states that if there are m ways to perform one task and n ways to perform another task, then there are m x n ways to perform both tasks in sequence.

In this case, Hunter has three tasks: rolling the die, spinning the spinner, and flipping the coin.

For the first task, rolling the die, there are four possible outcomes.

For the second task, spinning the spinner, there are four possible outcomes as well.

For the third task, flipping the coin, there are two possible outcomes.

Using the multiplication principle, the total number of possible outcomes is:

4 x 4 x 2 = 32

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The angular velocity of the elbow during the elbow flexion exercise was approximately 523.81 degrees/s

To calculate the angular velocity of the elbow during an elbow flexion exercise, we'll use the information given about the relative angle at different times. Here's the step-by-step explanation:
First, find the change in relative angle: Δθ = Final angle - Initial angle = 120 degrees - 10 degrees = 110 degrees.
Next, find the change in time: Δt = Final time - Initial time = 0.71s - 0.5s = 0.21s.
Now, calculate the angular velocity: ω = Δθ / Δt = 110 degrees / 0.21s ≈ 523.81 degrees/s.
So, the angular velocity of the elbow during the elbow flexion exercise was approximately 523.81 degrees/s.

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The expression that describes the length of one side of the rooftop is therefore: x - 9.

In mathematics, an expression is a combination of one or more variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. An expression can be as simple as a single variable or constant, or it can be a more complex combination of variables and operations.

Here,

The expression for the area of the rooftop is (x + 9)², where x is a variable representing the length of one side of the rectangle. To find the factors of this expression, we can expand it using the identity (a+b)² = a² + 2ab + b².

Expanding (x + 9)², we get:

(x + 9)² = x² + 18x + 81

Now, we need to find the factors of this expression that are also factors of the length of the sides of the rectangle. Since the sides of the rectangle must have a common factor of x, we can factor out x from the expression:

x² + 18x + 81 = x(x + 18) + 81

The factors of (x + 9)² are x(x + 18) + 81, (x + 9)(x + 9), (x - 9)(x - 9), and -(x + 9)(x + 9).

Since we are looking for factors that represent the length of one side of the rooftop, we can eliminate the negative factor and the factor (x + 9)(x + 9), since the sides of a rectangle must be positive.

That leaves us with x(x + 18) + 81 and (x - 9)(x - 9).

The expression describes the length of one side of the rooftop: x - 9

This is because the sides of a rectangle must be positive, and (x - 9) is a factor of (x + 9)² that represents a positive length.

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It would take approximately 95 minutes for the wick to have 74 mm remaining.

1) Finding the slope (mm per minute):
The wick was initially 150 mm in length and reduced to 110 mm after 50 minutes. To find the slope, we use the formula:

Slope = (change in length) / (change in time)

Slope = (110 mm - 150 mm) / (50 minutes - 0 minutes)
Slope = (-40 mm) / (50 minutes)
Slope = -0.8 mm/minute

2) Constructing the linear equation:
We now have the slope (-0.8) and the initial length (150 mm) to create a linear equation:

Length (L) = initial length + slope × time (t)
L = 150 - 0.8t

3) Predicting the time to have 74 mm remaining:
To find the time, plug in 74 mm for the length (L) in the equation and solve for t:

74 = 150 - 0.8t
76 = 0.8t
t = 95 minutes

So, it would take approximately 95 minutes for the wick to have 74 mm remaining.

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To find out how much Peter eats in one week (which is 7 days), we need to multiply this expression by 7.

Peter eats 3 carrot sticks and 1 cup of peanut butter before lacrosse practice every day, so in one day he eats:

3 + p

To find out how much he eats in one week (which is 7 days), we need to multiply this expression by 7:

7(3 + p)

Distributing the 7, we get:

21 + 7p

So the equivalent expressions that represent how much Peter eats before practice in one week are:

3 + 4p + 3p

4(3 + p)

21 + 7p

7(3p + 1)

So the correct answers are:

4(3 + p)

21 + 7p

7(3p + 1)

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The radius of the circle is 5.

To complete the proof, we need to find the radius of the circle centered at the origin and containing the point (5, 0). We can use the distance formula to find the distance between the origin (0, 0) and the point (5, 0):

distance = √((5 - 0)^2 + (0 - 0)^2) = √25 = 5

Therefore, the radius of the circle is 5.

Now, to determine whether the point (, −3) lies on the circle, we need to find the distance between the origin and the point (, −3):

distance = √((-3 - 0)^2 + (0 - 0)^2) = √9 = 3

Since the distance between the origin and the point (, −3) is not equal to the radius of the circle, which is 5, we can conclude that the point (, −3) does not lie on the circle centered at the origin and containing the point (5, 0).

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The area of the shaded region of the circle is 89.75 mi².

The area of a sector of a circle, you can use the formula:

A = (θ/360) × π × r²

Where A is the area of the sector, θ is the central angle of the sector, r is the radius of the circle, and π is a constant approximately equal to 3.14.

From the diagram, angle of the unshaded sector equals 150 degree.

Angle of the shaded region = 360 - 150 = 210 degree

Radius r = 7 miles.

We can substitute these values into the formula and solve for the area A.

A = (θ/360) × π × r²

A = ( 210/360 ) × 3.14 × 7²

A = ( 210/360 ) × 3.14 × 49

A = 89.75 mi²

Therefore, the area of the sector is approximately 89.75 mi².

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To approximate √125.04 using linear approximation, first find the linearization of f(x) = ³√x at x = 125. Then use the point-slope form of the equation to find the equation of the tangent line and plug in x = 125.04 to get the approximation.

To approximate √125.04 using linear approximation and the function f(x) = ³√x, first find the linearization of f(x) at x = 125 in the form y = mx + b. Calculate f(125) and f'(x).Calculate f'(125): Use the point-slope form of the equation
1: Calculate f(125) and f'(x).
f(125) = ³√125 = 5
f'(x) = (1/3)x^(-2/3)
2: Calculate f'(125).
f'(125) = (1/3)(125)^(-2/3) = 1/15
3: Use the point-slope form of the equation y - y1 = m(x - x1) to find the equation of the tangent line.
y - 5 = (1/15)(x - 125)
4: Rearrange to find y in terms of x.
y = (1/15)(x - 125) + 5
5: Determine the values of m and b.
m = 1/15
b = (1/15)(-125) + 5
6: Plug in x = 125.04 to approximate √125.04.
²√125.04 ≈ (1/15)(125.04 - 125) + 5
The linearization of f(x) at x = 125 is y = (1/15)x + b, with m = 1/15 and b = (1/15)(-125) + 5. Using these values, the approximation of √125.04 is (1/15)(125.04 - 125) + 5.

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To construct a square and a regular pentagon with equal side lengths of 0.5 inch, we need to use basic geometric constructions.

To construct a square and a regular pentagon with equal side length of 0.5 inch, follow these steps:

(a) Construct a Square:

Connect the endpoints of the four line segments to form a square.

(b) Construct a Regular Pentagon:

Repeat this process for all five vertices of the circle to form the regular pentagon.

A geometric construction is a method of drawing a figure using only a straightedge (an unmarked ruler) and a compass.

For the square, we start by drawing a horizontal line segment of length 0.5 inch. We then draw two perpendicular lines of length 0.5 inch each, meeting at the endpoints of the original line segment.

These two new line segments represent the adjacent sides of the square. We then repeat this process to create the remaining two sides of the square, and connect all four endpoints to form the complete square.

For the regular pentagon, we need to construct a circle with a radius of 0.5 inch. This will be the circumcircle of the pentagon, meaning that all five vertices of the pentagon will lie on the circle.

We draw a horizontal line through the center of the circle, and mark the points where the line intersects the circle. These five points will be the vertices of the pentagon.

We then draw line segments connecting adjacent vertices, using a compass to copy the length of each line segment from the previous one.

This process will create all five sides of the pentagon, and the figure will be a regular pentagon with equal side lengths of 0.5 inch.

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The differential equation to describe the situation is:
dP/dt = 0.000508 * P * (1 - P/732)

We can write a logistic differential equation to describe the rate of change of students who heard the rumor. The equation is:

dP/dt = k * P * (1 - P/M)

where dP/dt is the rate of change in the number of students who heard the rumor, k is a constant, P is the number of students who have heard the rumor at a given time, and M is the maximum capacity of the school (732 students).

At 12 PM, P = 227 and dP/dt = 24 students per hour. We can plug these values into the equation:

24 = k * 227 * (1 - 227/732)

Now, solve for k:

k ≈ 0.000508

So, the differential equation to describe the situation is:

dP/dt = 0.000508 * P * (1 - P/732)

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The effective price paid by the company after taking account of hedging would be $63.50, which is the August futures price on July 1. Calculate the profit or loss on the futures contracts and subtract that from the spot price on July 1, to determine the effective.

By entering into futures contracts on March 1, the company was able to lock in the price of $59 for the commodity, when the spot price was $60 and the futures price was $59, the difference between the futures price and the spot price on March 1 was $1 ($60 - $59), so the company had to pay an extra $1 per unit to hedge its purchase.

When the spot price increased to $64 on July 1, the company was still able to purchase the commodity at the lower hedged price of $59, plus the cost of the futures contract, which resulted in an effective price of $63.50. Overall, hedging helped the company mitigate the risk of price volatility and ensured a more predictable cost for the commodity purchase.

Effective price = Spot price - Profit from futures contracts

Effective price = $64 - $0.50(The difference between the futures price and the spot price on July 1 was $0.50 ($64 - $63.50))

Effective price = $63.50 per unit

Therefore, the effective price paid by the company after taking into account hedging was $63.50 per unit.

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Answer:

Step-by-step explanation:

Answer:

76

Step-by-step explanation:

If the mean of 28 numbers is 18 then the sum of those numbers=28×18=504

if one number is added then 29×20=580

the new number therefore=580-504=76

The probability of flipping a coin and getting tails at least once in four flips is 15/16 or approximately 0.94. (option d).

To determine the probability of flipping a coin and getting tails at least once in four flips, we can use a probability table. The table shows all the possible outcomes of flipping a coin four times.

Flip 1 Flip 2 Flip 3 Flip 4

Outcome 1 H H H H

Outcome 2 H H H T

Outcome 3 H H T H

Outcome 4 H H T T

Outcome 5 H T H H

Outcome 6 H T H T

Outcome 7 H T T H

Outcome 8 H T T T

Outcome 9 T H H H

Outcome 10 T H H T

Outcome 11 T H T H

Outcome 12 T H T T

Outcome 13 T T H H

Outcome 14 T T H T

Outcome 15 T T T H

Outcome 16 T T T T

In the table, H represents heads, and T represents tails. There are 16 possible outcomes when flipping a coin four times. We can see that getting tails at least once is possible in 15 of these outcomes: Outcome 2, Outcome 3, Outcome 4, Outcome 6, Outcome 7, Outcome 8, Outcome 10, Outcome 11, Outcome 12, Outcome 14, Outcome 15, and Outcome 16.

Therefore, the probability of flipping a coin and getting tails at least once in four flips is the number of outcomes where tails appear at least once divided by the total number of outcomes, which is 15/16 or approximately 0.94. (option d).

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Rate of change in the temperature of lasagna given by function T(t) = 70 + (200 - 70) × [tex]e^{(-0.0001t)}[/tex] exactly 2 hours after taking it out of oven is -0.013 °F/hour.

Surrounding room temperature is equal to 70°F

Temperature at which lasagna comes out of the oven = 200°F

The temperature T (in °F) of the lasagna at time t (in hours) after taking it out of the oven is equal to,

T(t) = 70 + (200 - 70) × [tex]e^{(-0.0001t)}[/tex]

To find the rate of change in the temperature of the lasagna exactly 2 hours after taking it out of the oven,

Find the derivative of the temperature function with respect to time t.

T'(t) = -0.013[tex]e^{(-0.0001t)}[/tex]

Substituting t = 2 into this expression gives:

T'(2) = -0.013[tex]e^{(-0.0001\times 2)}[/tex]

= -0.013[tex]e^{-0.0002}[/tex]

= -0.013 × 0.99980

= -0.0129974

= -0.013

Therefore, the rate of change in the temperature of the lasagna exactly 2 hours after taking it out of the oven is approximately -0.013 °F/hour.

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The given question is incomplete, I answer the question in general according to my knowledge:

A tray of lasagna comes out of the oven at 200°F and is placed on a table where the surrounding room temperature is 70°F. The temperature T (in °F) of the lasagna is given by the function T(t) = 70 + (200 - 70) × [tex]e^{(-0.0001t)}[/tex] where t is time(in hours) after taking the lasagna out of the oven. What is the rate of change in the temperature of the lasagna exactly 2 hours after taking it out of the oven?

The commission of the salesperson is $750 if he received a commission of 25% of the difference between the auction price and the dealership price.

The salesperson's commission can be calculated by first finding the dealership price, which is 30% more than the auction price of $10,000.

30% of $10,000 = $3,000

Dealership price = $10,000 + $3,000 = $13,000

Next, we need to find the difference between the dealership price and the auction price

$13,000 - $10,000 = $3,000

The salesperson's commission is 25% of this difference

25% of $3,000 = $750

Therefore, the salesperson's commission is $750.

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The cost of purchasing 2 pounds of sliced ham and 3 pounds of sliced turkey from the meat counter is $30.95.

The table provided shows the purchases made by two customers at a meat counter. To determine how much you will pay for 2 pounds of sliced ham and 3 pounds of sliced turkey, you need to first look at the prices listed in the table. For sliced ham, the price per pound is $4.99, and for sliced turkey, the price per pound is $6.99.

To calculate the cost of 2 pounds of sliced ham, you can multiply the price per pound ($4.99) by the number of pounds (2), which gives you a total cost of $9.98. Similarly, to calculate the cost of 3 pounds of sliced turkey, you can multiply the price per pound ($6.99) by the number of pounds (3), which gives you a total cost of $20.97.

Therefore, the total cost for 2 pounds of sliced ham and 3 pounds of sliced turkey would be $9.98 + $20.97 = $30.95.

In conclusion, by using the prices listed in the table, it is possible to determine the cost of purchasing 2 pounds of sliced ham and 3 pounds of sliced turkey from the meat counter. It is important to remember to multiply the price per pound by the number of pounds needed for each item, and then add the costs together to get the total price.

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The set of all points in quadrant I not including the axis's can be represented by the following system of inequalities:

x > 0

y > 0

Inequalities are useful in modeling situations where there are constraints or limitations. For  illustration, in real- life  scripts, there may be limited  coffers or capacity, or certain variables must fall within a specific range. Systems of inequalities are  frequently used to represent these constraints or limitations graphically.   One common  operation of systems of inequalities is in optimization problems, where the  thing is to maximize or minimize a particular function subject to certain constraints.

In these situations, the  doable region, or the set of all points that satisfy the constraints, is  frequently represented as a shadowed region on a graph. The optimal  result is  also  set up by  relating the point( s) within this region that maximize or minimize the function.

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The volume of the solid will be 32/7 √3.

Since base of a solid whose cross-sections is perpendicular to the w-axis are equilateral triangles

Now base of triangle. is f(x) = x³ and the Area of Equilateral Triangle is ✓3/4 base²

The volume of the solid will be:

= ✓3/4 (x^7/7)²

= ✓3/4 (128/7)

= 32/7 √3

Therefore, the volume of the solid will be 32/7 √3.

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According to the distance, it will take the jet 1 hour to overtake the small plane.

Let's first calculate the distance traveled by the small plane in the time it takes for the jet to overtake it. Since the small plane is flying for an extra 30 minutes, its travel time is "t + 0.5" hours. Therefore, the distance traveled by the small plane is:

Distance of small plane = Speed of small plane x Time of small plane

Distance of small plane = 240 x (t + 0.5)

Now, let's calculate the distance traveled by the jet in "t" hours:

Distance of jet = Speed of jet x Time of jet

Distance of jet = 360 x t

Since both planes are at the same point at the time of overtaking, we can set the distances traveled by both planes equal to each other:

240 x (t + 0.5) = 360 x t

We can solve for "t" using algebra:

240t + 120 = 360t

120 = 120t

t = 1

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The volume of the solid obtained by rotating the region bounded about the x-axis is 3π/4 cubic units.

To find the volume of the solid obtained by rotating the region bounded by y = 4x^2, y = 1, and y = 0 about the x-axis, we can use the method of cylindrical shells.

First, we need to find the limits of integration. The region is bounded by y = 4x^2 and y = 1, so we can set up the integral as follows:

V = ∫[0,1] 2πx(1-4x^2)dx

Next, we can simplify the integrand:

V = ∫[0,1] 2πx dx - ∫[0,1] 8πx^3 dx

V = π - 2π/4

V = 3π/4

Therefore, the volume of the solid obtained by rotating the region bounded by y = 4x^2, y = 1, and y = 0 about the x-axis is 3π/4 cubic units.

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the value of the line integral ∫_C 2y dx + 2x dy along the straight line path from (4,3) to (9,6) is 84.

To evaluate the line integral ∫_C 2y dx + 2x dy along the straight line path from (4,3) to (9,6), follow these steps:
Step:1. Parametrize the straight line path: Define a vector-valued function r(t) = (1-t)(4,3) + t(9,6) = (4+5t, 3+3t), where 0 ≤ t ≤ 1. Step:2. Calculate the derivatives: dr/dt = (5,3). Step:3. Substitute the parametric equations into the line integral: 2(3+3t)(5) + 2(4+5t)(3). Step:4. Calculate the line integral: ∫(30+30t + 24+30t) dt, where the integration is from 0 to 1. Step:5. Combine the terms and integrate: ∫(54+60t) dt from 0 to 1 = [54t + 30t^2] from 0 to 1.
Step:6. Evaluate the integral at the limits: (54(1) + 30(1)^2) - (54(0) + 30(0)^2) = 54 + 30 = 84.

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Answer: attach an image

Step-by-step explanation:

To find the principal, we can use the formula for simple interest:

I = P*r*t

where I is the interest, P is the principal, r is the interest rate, and t is the time in years.

We need to convert 82 days to years by dividing it by 365 (the number of days in a year):

t = 82/365

t = 0.2247

Now we can plug in the values we know and solve for P:

67.14 = P*0.0625*0.2247

P = 67.14/(0.0625*0.2247)

P = 1900

Therefore, the principal is $1900.

Answer: B, C, E

Step-by-step explanation: Other dude posted wrong answer.

The perimeter of the track is of 377 m.

The circumference of a circle of radius r is given by the equation presented as follows:

C = 2πr.

Considering the rectangle with area 5500 m² and base 110 m, the height h, representing the diameter d of the circumference, is obtained as follows:

110d = 5500

d = 550/11

d = 50 m.

The radius is half the diameter, hence it is given as follows:

r = 25 m.

The perimeter is given as follows:

Hence it is given as follows:

P = 2 x 110 + 2 x 3.14 x 25

P = 377 m.

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Answer:

Kofta receives $15,250 and Kojo receives $22,750.

Step-by-step explanation:

Let x represent the amount of money that Kofta has.

x + (x +7,500) = 38,000

          - 7,500   - 7,500

___________________

x + x = 30,500

2x = 30,500

÷ 2 = ÷2

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

x = 15,250

Therefore, Kofta has $15,250.

Let k represent the amount of money that Kojo has.

k + 15,250 = 38,00

k = 38,000 - 15,250

k =  $22,750

Therefore, Kojo has $22,750

The only possible value of m that would make the ranges of f(x) and g(x) the same is any positive real number.

The range of a function is the set of all possible output values. In this case, we are given that the ranges of two functions, f(x) and g(x), are the same.

The function f(x) = √(mx) has a domain of x ≥ 0, since the square root of a negative number is not a real number. The function g(x) = m√x has a domain of x ≥ 0 for the same reason.

To find the range of these functions, we need to consider the possible values of the input x. For f(x), as x increases, the output √(mx) also increases, and as x approaches infinity, so does the output. For g(x), as x increases, the output m√x also increases, and as x approaches infinity, so does the output.

Therefore, if the ranges of f(x) and g(x) are the same, this means that they both have the same maximum and minimum values, and these values are achieved at the same inputs.

In particular, if we consider the minimum value of the range, this is achieved when x = 0, since both functions are defined only for non-negative inputs. At x = 0, we have f(0) = g(0) = 0, so the minimum value of the range is 0.

To find the maximum value of the range, we need to consider the behavior of the functions as x approaches infinity. As noted above, both functions increase without bound as x increases, so the maximum value of the range is infinity.

Therefore, the only possible value of m that would make the ranges of f(x) and g(x) the same is any positive real number.

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