The table shows one (x, y) pair for a proportional function. What is the constant of the proportionality?
xy
32

The missing length of the right triangle is 8√3

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

The right triangle

The missing length of the right triangle can be calculated using the following Pythagoras theorem

x² = difference of squares of the other sides

Using the above as a guide, we have the following:

x² = (16)² - (8)²

Evaluate

x² = 192

Take the square roots

x = 8√3

Hence, the missing length of the right triangle is 8√3

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According to the information we can infer that the correct match is screwdriver: 250mm, height of a kitchen worktop: 0.9m, the overall length of a JCB 3cx Backhoe loader digger: 5620mm, and height of a door: 2m

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Answer: 1105 grams of protein

Step-by-step explanation:

If the athlete eats 65 grams of protein per day, we multiply that by 17 for the amount of days.

1105x65=1105

Answer:

Step-by-step explanation:

Mario recibe Q40,000.00, Leticia recibe Q20,000.00, and Beatriz receives Q20,000.00. Therefore, Mario and Leticia receive a total of Q60,000.00.

Answer:

  210

Step-by-step explanation:

You want the product of x and y given similar triangles with side ratios ...

Corresponding sides in similar triangles are proportional. This means ...

  x/10 = 21/y

  xy = 210 . . . . . . . cross multiply

The product of x and y is 210.

__

Additional comment

Sides in the larger triangle are 3/2 times those in the smaller triangle. This means x = 15, y = 14.

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Part A: The mean and median for Bay Side School are 3.8 and 5, respectively, while for Seaside School they are 4.93 and 5.

Part B: The range and IQR for Bay Side School are 8 and 4, respectively, while for Seaside School they are 8 and 2.

Part C: Bay Side School is a better choice for smaller class sizes due to its smaller mean, smaller range, and larger IQR.

Part A: Measures of Center

Bay Side School:

Mean = 3.8 (The sum of all class sizes divided by the number of classes)

Median = 5 (The middle value when class sizes are arranged in ascending order)

Seaside School:

Mean = 4.93 (The sum of all class sizes divided by the number of classes)

Median = 5 (The middle value when class sizes are arranged in ascending order)

Part B: Measures of Variability

Bay Side School:

Range = 8 (The difference between the maximum and minimum class sizes)

Interquartile Range (IQR) = 4 (The difference between the first quartile and third quartile)

Seaside School:

Range = 8 (The difference between the maximum and minimum class sizes)

Interquartile Range (IQR) = 2 (The difference between the first quartile and third quartile)

Part C: Conclusion and Reasoning

If you are interested in a smaller class size, Bay Side School is a better choice. Bay Side School has a smaller mean (3.8) compared to Seaside School (4.93), indicating a generally smaller class size. Additionally, the median class size is the same for both schools, but the range and IQR are smaller for Bay Side School.

This suggests that class sizes at Bay Side School have less variation and are more consistently smaller. Therefore, based on the provided data, Bay Side School is the better choice for smaller class sizes.

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Step-by-step explanation:

To calculate the amount of air used to fill a spherical balloon, we need to determine its volume. The volume of a sphere can be calculated using the formula:

Volume = (4/3) × π × radius³

Given that the radius of the balloon is 165 mm, we can substitute this value into the formula:

Volume = (4/3) × π × (165 mm)³

Now, let's calculate the volume of the balloon:

Volume = (4/3) × 3.14159 × (165 mm)³

Volume ≈ 4.18879 × (165 mm)³

Evaluating this expression will give us the volume of the balloon in cubic millimeters. However, to express it in a more practical unit, such as liters, we need to convert the volume from cubic millimeters to liters.

1 liter = 1000 cubic centimeters (cc)

1 cubic centimeter (cc) = 1 milliliter (ml)

1 milliliter (ml) = 1 cubic millimeter (mm³)

To convert the volume from cubic millimeters to liters, we divide the volume by 1000:

Volume (liters) = Volume (cubic millimeters) / 1000

So, let's calculate the volume of the balloon in liters:

Volume (liters) = (4.18879 × (165 mm)³) / 1000

Using a calculator to evaluate this expression, we find:

Volume (liters) ≈ 7.976 liters

Therefore, approximately 7.976 liters of air were used to fill the spherical balloon.

Answer:

Step-by-step explanation:

if x = -1,

     a-b+c = 0 (eq1)

if x = 1,

    a+b+c = 0(eq2)

from eq 1 and 2;

a+c = 0;

so, a = -c,

if a+c = 0, b = 0

Ans : a=-c, b=0

The value of constants a and b are -2 and 0 respectively.

Given: x=-1 and x=1 are both the solution to equation ax²+bx+c=0.

To find: the constant a and b

Solution:

To obtain this, first, we have to simplify the equation by putting x=1 into the given equation, we get;

a+b=-2

afterward, we have to put x=-1 in the given equation, and we get;

a-b=-2

Now, we will add the two obtained equations, we get;

2a=-4 i.e., a=-2

Now, a+b=0 which means the value of b is 0.

Hence, the constants a and b are -2 and 0 respectively.

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9.  42 chocolate lava cakes and 15 caramel apple tarts were sold.

10.Brian needs to sell approximately 14,286 meals to break even.

9. Let's denote the number of chocolate lava cakes as x and the number of caramel apple tarts as y. We can set up a system of equations based on the given information:

x + y = 57 (sum of the two desserts is equal to 57)

x - y = 27 (difference of the two desserts is 27)

We can solve this system of equations using the method of substitution or elimination. Let's use the method of elimination:

Add the two equations:

(x + y) + (x - y) = 57 + 27

2x = 84

x = 42

Substitute the value of x into one of the equations to find y:

42 - y = 27

y = 42 - 27

y = 15

Therefore, 42 chocolate lava cakes and 15 caramel apple tarts were sold.

10. To determine the number of meals Brian needs to sell to break even, we can calculate the break-even point by comparing the total cost and the total revenue.

Total cost = $250,000 (start-up cost) + ($17.50 per meal) * N (number of meals)

Total revenue = ($35.00 per meal) * N (number of meals)

To break even, the total cost should be equal to the total revenue:

$250,000 + ($17.50 per meal) * N = ($35.00 per meal) * N

Simplifying the equation:

$250,000 = $35.00N - $17.50N

$250,000 = $17.50N

N = $250,000 / $17.50

N ≈ 14,285.71

Since we cannot sell a fraction of a meal, we round the number of meals up to the nearest whole number.

Therefore, Brian needs to sell approximately 14,286 meals to break even.

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Answer:

170, 171, 172

Step-by-step explanation:

x + x + 1 + x + 2 = 513

3x + 3 = 513

3x = 510

x = 170

x + 1 = 171

x + 2 = 172

The polar form of (2cis60°)(5cis30°) is √(75 - 15√3) cis(π/4). The rectangular form of (2cis60°)(5cis30°) is (-5 + 5√3)/2 + (15√3 + 5)i/2.

Polar form:

(2cis60°) and (5cis30°) can be expressed in polar form as follows:

2cis60° = 2(cos60° + isin60°) = 2(1/2 + i√3/2) = 1 + i√3

5cis30° = 5(cos30° + isin30°) = 5(√3/2 + i/2) = (5√3)/2 + (5i)/2

Thus, (2cis60°)(5cis30°) = (1 + i√3)((5√3)/2 + (5i)/2)

To multiply these complex numbers, we can use FOIL method as we do with algebraic expressions to find:

= [(1)(5√3)/2 + (1)(5i)/2 + (i√3)(5√3)/2 + (i√3)(5i)/2]

= [(5√3)/2 + (5i)/2 + (5i√3)/2 − (15/2)]

= [(5√3)/2 − (15/2) + (5i)/2 + (5i√3)/2]

Now, let's determine the magnitude and argument of this complex number:

Magnitude:

|z| = √[(5√3)/2 − (15/2)]² + [(5/2)√3]²

= √[75/4 - 15√3 + 75/4]

= √(75 - 15√3)

Argument:

θ = tan^⁻1((5/2)√3 / [(5√3)/2 − (15/2)])

= tan^⁻1(1)

= π/4

Rectangular form:

To find the rectangular form, we can use the magnitude and argument we just found, and the following formula:

z = r(cosθ + isinθ)

So, we have:

(2cis60°)(5cis30°) = √(75 - 15√3) cis(π/4)

= √(75 - 15√3)(cos(π/4) + isin(π/4))

= (√(75 - 15√3)/2)(1 + i) + (√(75 - 15√3)/2)(√3 - i)

= ([(5√3)/2 − (15/2)] + [(5/2)√3]) + ([(5√3)/2 − (15/2)]√3 + (5/2))

= (-5 + 5√3) + (15√3 + 5)i/2

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Step-by-step explanation:

The rectangular form of a cis equation is

[tex]a + bi[/tex]

Given that we have

[tex]2cis60 \times 5cis30[/tex]

We can just multioly the magnitudes

[tex]2 \times 5 = 10[/tex]

And add the angles

[tex]60 + 30 = 90[/tex]

And that gives us our new equestion

[tex]10cis(90)[/tex]

Remember that

cis is basically

[tex] \cos( \alpha ) + i \sin( \alpha ) [/tex]

So our answer in polar form is

[tex]10( \cos(90) + i \sin(90) )[/tex]

To convert to rectangular form, distribute the 10

[tex]10 \cos(90) = a[/tex]

[tex]10i \sin(90) = bi[/tex]

So

[tex]a = 0[/tex]

[tex]b = 10[/tex]

So, our equation in rectangular form is

[tex]10i[/tex]

The equation of the line that goes through the point (-2, -2) with a slope of -4 is y = -4x - 10.

To find the equation of a line, we need to use the point-slope form of a linear equation:

y - y₁ = m(x - x₁)

Where (x₁, y₁) is the given point on the line, and m is the slope.

Given that the point is (-2, -2) and the slope is -4, we can substitute these values into the equation:

y - (-2) = -4(x - (-2))

Simplifying:

y + 2 = -4(x + 2)

Expanding the brackets:

y + 2 = -4x - 8

Now, let's rearrange the equation to get it into slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept:

y = -4x - 8 - 2

y = -4x - 10

Therefore, the equation of the line that goes through the point (-2, -2) with a slope of -4 is y = -4x - 10.

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The total cost of 21 bottles of cola at £1.59 each is £33.39.

1. Determine the cost per bottle: Each bottle of cola costs £1.59.

2. Multiply the cost per bottle by the number of bottles: £1.59 x 21 = £33.39.

3. Therefore, the total cost of 21 bottles of cola is £33.39.

In the given scenario, each bottle of cola is priced at £1.59. To calculate the total cost of 21 bottles, we need to multiply the cost per bottle by the number of bottles. By multiplying £1.59 by 21, we get £33.39 as the final answer.

Certainly! Here's an expanded explanation in 100 additional words:

To calculate the total cost of 21 bottles of cola, we need to multiply the cost per bottle by the number of bottles. In this case, each bottle costs £1.59. By multiplying £1.59 by 21, we can determine the total cost.

To perform the calculation, we can use the distributive property of multiplication. We multiply the cost per bottle (£1.59) by the number of bottles (21) individually, and then sum up the results.

First, we multiply the ones place: 9 multiplied by 21, which equals 189. Since 9 multiplied by 1 equals 9, we write down the 9 in the ones place and carry over the 18 to the tens place.

Next, we multiply the tens place: 5 multiplied by 21, which equals 105. Adding the carried-over 18, we get 123. We write down the 3 in the tens place and carry over the 12 to the hundreds place.

Finally, we multiply the hundreds place: 1 multiplied by 21, which equals 21. Adding the carried-over 12, we get 33.

Combining the results, we have £33.39 as the total cost of 21 bottles of cola.

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The correct answer is option e. The joint probability of two statistical dependent events Y and Z can be written as P(Y and Z) = P(Y) * P(Z|Y).

The concept of joint probability is utilized in probability theory, a statistical tool that investigates the probability of two events happening simultaneously. It is an estimation of the probability of two or more events occurring simultaneously. Joint probability is a method of defining the probability of two events happening at the same time.The joint probability of two statistical dependent events Y and Z can be written as P(Y and Z) = P(Y) * P(Z|Y).This implies that the possibility of Z happening given Y has happened multiplied by the probability of Y happening is the joint probability of Y and Z. P(Z|Y) refers to the probability of Z occurring when Y is true or has already occurred. It is conditional probability and is often used to calculate joint probabilities.The formula for conditional probability is P(A|B) = P(A and B)/P(B), which can be simplified to P(A and B) = P(A|B) * P(B). This formula is applicable to P(Y and Z), where P(Y and Z) = P(Z|Y) * P(Y).

The correct answer is option e.

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Answer:

The correct answer is A. triangles.

Step-by-step explanation:

Answer:

A)triangles

Step-by-step explanation:

CPCTC is a theorem that states that if two triangles are congruent, then their corresponding parts are also congruent. This means that if you have two triangles that are identical in shape and size, you can conclude that their corresponding angles and sides are equal.

In this triangle, the product of sin B and tan C is[tex]b^2/(ac)[/tex]  , and the product of sin C and tan B is [tex]c^2/(ab)[/tex].

In a right-angled triangle ABC, where angle A is 90 degrees, we have the following side lengths:

AB = c (base)

BC = a (hypotenuse)

AC = b (perpendicular)

We need to calculate the products of sin B and tan C, and sin C and tan B.

First, let's calculate sin B and tan C:

sin(B) = opposite/hypotenuse = AC/BC = b/a

tan(C) = opposite/adjacent = AC/AB = b/c

The product of sin B and tan C is sin(B) * tan(C) = (b/a) * (b/c) = [tex]b^2[/tex]/(ac).

Next, let's calculate sin C and tan B:

sin(C) = opposite/hypotenuse = AB/BC = c/a

tan(B) = opposite/adjacent = AB/AC = c/b

The product of sin C and tan B is sin(C) * tan(B) = (c/a) * (c/b) = [tex]c^2[/tex]/(ab).

Therefore, in the given right-angled triangle ABC, the product of sin B and tan C is[tex]b^2[/tex]/(ac), and the product of sin C and tan B is [tex]c^2[/tex]/(ab).

These formulas hold true for any right-angled triangle, where the base is AB, the hypotenuse is BC, and the perpendicular is AC.

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The question probable may be:

In this triangle, the product of sin B and tan C is _____ , and the product of sin C and tan B is _______.

The measure that changes the most when the value of 82° is changed to 94° is the Median, which becomes 84°.

To determine which measure changes the most when the value of 82° is changed to 94°, we need to calculate the measures for both the original and new data.

Given the average high temperatures: 58, 61, 71, 77, 91, 100, 105, 102, 95, 82, 66, 57.

First, let's calculate the original measures:

Mean: The mean is the average of all the values.

Mean = (58 + 61 + 71 + 77 + 91 + 100 + 105 + 102 + 95 + 82 + 66 + 57) / 12 = 81.4°.

Median: The median is the middle value when the data is arranged in ascending order.

Arranging the data in ascending order: 57, 58, 61, 66, 71, 77, 82, 91, 95, 100, 102, 105.

The median is the 6th value, which is 82°.

Range: The range is the difference between the highest and lowest values.

Range = Highest value - Lowest value

= 105 - 57

= 48°.

IQR (Interquartile Range): The IQR is the difference between the third quartile (Q3) and the first quartile (Q1).

Calculating Q1 and Q3:

Q1 = 66, Q3 = 95.

IQR = Q3 - Q1

= 95 - 66

= 29°.

Now, let's calculate the new measures after changing the value of 82° to 94°:

Mean: The mean is unaffected by changing a single value.

Median: Since we only changed one value, the new median will still be the same as the original median, which is 82°.

Range: The range is unaffected by changing a single value.

IQR (Interquartile Range): The IQR is unaffected by changing a single value.

Therefore, the measure that changes the most when the value of 82° is changed to 94° is the Median, which becomes 84°.

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The calculated value of y when x = -10 is -30

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

y varies directly with x

This means that

y = kx

Where

k = constant of variation

So, we have

24 = 8k

k = 3

This also means that

y = 3x

When x = -10, we have

y = 3 * -10

Evaluate

y = -30

Hence, the value is -30

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Question

Suppose y varies directly with x and y=24 when x=8.

What is the value of y when x = -10?

Answer:The equation that can be paired with x + 2y = 5 to create an inconsistent system is given by:

2x + 4y = 3.

Step-by-step explanation:

A system of equations is when two or more variables are related, and equations are built to find the values of each variable.

An inconsistent system is a system that has no solutions, and it happens when the coefficients of x and y are multiples, and the independent terms are not.

Hence:

x + 2y = 5.

2x + 4y = 3.

Is inconsistent, as the variables are each multiplied by 2, but the independent term is not.

The measure of line MN in the secant line using the Intersecting theorem is approximately 19.5.

Intersecting secants theorem states that " If two secant line segments are drawn to a circle from an exterior point, then the product of the measures of one  of secant line segment and its external secant line segment is the same or equal to the product of the measures of the other secant line segment and its external line secant segment.

From the figure:

Sectant line segment QO = 48

internal line segement of QO = QP = 29

External line segement of QO =  PO = 48 - 29 = 19

Second sectant line segment MO = NO + MN = ( 22 + x )

External line segement of MO = NO = 22

Using the Intersecting secants theorem:

22( 22 + x ) = 19 × 48

Solve for x:

484 + 22x = 912

22x = 912 - 484

22x = 428

x = 428/22

x = 19.5

Therefore, line MN is approximately 19.5.

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To calculate the present value of the cost savings, we can use the formula for the present value of a growing perpetuity. The formula is:

K = C / (r - g)

Where:

K = Present value of the growing perpetuity

C = Cash flow in the first year

r = Interest rate

g = Growth rate

In this case, C = $16,000, r = 8% (or 0.08), and g = -3% (or -0.03). We use a negative sign for the growth rate since the cost savings are decreasing each year.

Now we can substitute the values into the formula and calculate the present value:

K = 16000 / (0.08 - (-0.03))

K = 16000 / 0.11

K ≈ $145,454.55

Therefore, the present value of these cost savings is approximately $145,454.55.

The rational expression that is undefined when x = 0 is (8x + 6)/(-x^2 + 1).

To determine which rational expression is undefined when x = 0, we need to find the values of x that make the denominator equal to zero. When the denominator of a rational expression is zero, the expression is undefined because division by zero is not possible.

Let's analyze each of the given expressions:

(5x^2 - 4)/(3x - 3):

To check if this expression is undefined when x = 0, we substitute x = 0 into the denominator:

3(0) - 3 = -3

Since the denominator is not zero, this expression is defined when x = 0.

(6x^2)/(9x - 3):

Similarly, we substitute x = 0 into the denominator:

9(0) - 3 = -3

Again, the denominator is not zero, so this expression is defined when x = 0.

(8x + 6)/(-x^2 + 1):

For this expression, we substitute x = 0 into the denominator:

-(0)^2 + 1 = -0 + 1 = 1

The denominator is not zero, indicating that this expression is defined when x = 0.

(4x - 8)/(-2x^2):

When we substitute x = 0 into the denominator:

-2(0)^2 = -2(0) = 0

The denominator becomes zero when x = 0. Therefore, this expression is undefined when x = 0.

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The probability that a randomly chosen member of the Student Government Board is a graduate student or lives in on-campus housing is 8/25 or 0.32 (rounded to the nearest millionth).

1. Calculate the total number of students on the Student Government Board by summing up the numbers in the table:

  Total Students = 2 + 2 + 2 + 4 + 0 + 3 + 4 + 2 = 19

2. Calculate the total number of graduate students on the Student Government Board:

  Total Graduate Students = 2 + 0 = 2

3. Calculate the total number of students living in on-campus housing:

  Total On-Campus Housing = 2 + 2 + 0 + 4 + 2 = 10

4. Calculate the probability of selecting a graduate student from the Student Government Board by dividing the total number of graduate students by the total number of students:

  Probability of Graduate Student = Total Graduate Students / Total Students = 2 / 19

5. Calculate the probability of selecting a student living in on-campus housing by dividing the total number of students in on-campus housing by the total number of students:

  Probability of On-Campus Housing = Total On-Campus Housing / Total Students = 10 / 19

6. Calculate the probability that a randomly chosen member of the Student Government Board is a graduate student or lives in on-campus housing by summing up the probabilities from steps 4 and 5:

  Probability = Probability of Graduate Student + Probability of On-Campus Housing = 2 / 19 + 10 / 19

7. Simplify the fraction if necessary. In this case, the fraction cannot be simplified further, so the final probability is 2 / 19 + 10 / 19 = 12 / 19.

8. Convert the fraction to a decimal by dividing the numerator by the denominator: 12 / 19 ≈ 0.631578947, which rounds to 0.632 (rounded to the nearest thousandth).

9. Finally, express the probability as a fraction in lowest terms: 12 / 19 is already in lowest terms.

Therefore, the probability that a randomly chosen member of the Student Government Board is a graduate student or lives in on-campus housing is 12/19 or approximately 0.632 (rounded to the nearest thousandth).

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To solve the question, we need to consider the transformation described and determine the function that represents the stretched graph.

The absolute value parent function is [tex]\displaystyle\sf f(x) = |x|[/tex]. According to the question, this parent function is horizontally stretched by a factor of 5 to create the graph of [tex]\displaystyle\sf g(x)[/tex].

A horizontal stretch by a factor of 5 means that the x-values on the graph are multiplied by 5. In other words, the transformation can be represented as [tex]\displaystyle\sf g(x) = |5x|[/tex].

Therefore, the correct function that represents the graph of [tex]\displaystyle\sf g(x)[/tex], the stretched version of [tex]\displaystyle\sf f(x) = |x|[/tex], is [tex]\displaystyle\sf g(x) = |5x|[/tex].

The correct answer is option D: [tex]\displaystyle\sf g(x) = |5x|[/tex].

Algebra is introduced to primary school learners in a basic form to provide them with a foundation for mathematical thinking and problem-solving.

Here are some reasons why algebra is taught to primary school learners and the importance of teaching it at an early age:

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The range of g(x) is [8, ∞).Thus, the function g(x) = -2√x + 8 has the same range as f(x) = -2√(x - 3) + 8.

Let f (x) = -2√(x - 3) + 8. The function g(x) has the same range as f (x) if the range of f(x) is equal to the range of g(x). Let's determine the range of f(x).Range of f(x):The domain of f(x) is all non-negative real numbers because the square root of a negative number is not a real number. In other words, x - 3 ≥ 0 ⇒ x ≥ 3. This implies that f(x) is a decreasing function because as x increases, the value of f(x) decreases.

Furthermore, the minimum value that f(x) can take is 8, which occurs when x = 3. As x approaches infinity, the value of f(x) approaches negative infinity, but never quite reaches it. Therefore, the range of f(x) is [8, ∞).In order for g(x) to have the same range as f(x), its range must also be [8, ∞). Therefore, g(x) must also be a decreasing function that is bounded from below by 8.

A possible function that satisfies these conditions is g(x) = -2√x + 8. Let's check the range of g(x):Range of g(x):As with f(x), the domain of g(x) is all non-negative real numbers. g(x) is also a decreasing function because as x increases, the value of g(x) decreases. Furthermore, the minimum value that g(x) can take is 8, which occurs when x = 0. As x approaches infinity, the value of g(x) approaches negative infinity, but never quite reaches it

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