Find three consecutive odd
three times the
integers such that the sum of one times the first,
two times the second, and
third is 334. List the numbers in order from smallest to in largest.

Answer:

1

Step-by-step explanation:

according to hooke's law and Pythagorean theorem your answer is should be most definitely will be or is it honestly I don't know. but the answer should be too is in the number not the figure of speech.

Length of missing side is 8/9x² - 5/9

The steps for this process are as follows

Step 1: Add the two equations together to form a new equation.

10z² + 42 - 3 + 82² + 6x - 7 = 8/9x² - 5/9

Step 2: Simplify the expression on the left side of the equation.

92² + 48x - 10 = 8/9x² - 5/9

Step 3: Subtract 8/9x² from both sides of the equation.

92² + 48x - 10 - 8/9x² = - 5/9

Step 4: Simplify the left side of the equation.

92² + 40x - 10 = -5/9

Step 5: Add 10 to both sides of the equation.

92² + 40x = 5/9

Step 6: Divide both sides of the equation by 40.

92²/40 + x = 5/9

Step 7: Simplify the left side of the equation.

x = 5/9 - 92²/40

Step 8: The length of the missing side is 8/9x² - 5/9 = 8/9 (5/9 - 92²/40) - 5/9.

To learn more about the length of the missing side refer :

brainly.com/question/30373320

#SPJ1

Using the given equation, Length of missing side is 8/9x² - 5/9

The steps for this process are as follows

Step 1: Add the two equations together to form a new equation.

10z² + 42 - 3 + 82² + 6x - 7 = 8/9x² - 5/9

Step 2: Simplify the expression on the left side of the equation.

92² + 48x - 10 = 8/9x² - 5/9

Step 3: Subtract 8/9x² from both sides of the equation.

92² + 48x - 10 - 8/9x² = - 5/9

Step 4: Simplify the left side of the equation.

92² + 40x - 10 = -5/9

Step 5: Add 10 to both sides of the equation.

92² + 40x = 5/9

Step 6: Divide both sides of the equation by 40.

92²/40 + x = 5/9

Step 7: Simplify the left side of the equation.

x = 5/9 - 92²/40

Step 8: The length of the missing side is 8/9x² - 5/9 = 8/9 (5/9 - 92²/40) - 5/9.

To learn more about the equations refer :

https://brainly.com/question/22688504

#SPJ1

The closest to the average(arithmetic mean) number of seeds per apple is 6.

What is average ?

An average, also known as a mean, is a measure of central tendency that represents the typical value of a set of data. To calculate the average of a set of numbers, you add them up and then divide by the number of items in the set. For example, the average of the numbers 2, 4, 6, and 8 is (2 + 4 + 6).

Average =  Total number of seeds/12

⇒ Average = 2×3+4×5+1×6+2×7+3×9/12

​⇒ Average = 73/12

⇒ Average ≈6

The closest to the average(arithmetic mean) number of seeds per apple is 6.

To learn more about average visit : brainly.com/question/24057012

#SPJ4

The items in A or the elements included in set A are A = 1, 2, 3, 4.

What is an element in a set?

In mathematics, any one of the unique things that make up a set is referred to as an element (or member). A distinct object belonging to a set is referred to as an element (or member) of a set.

The items in A and B, or the elements included in both sets, make up its constituent parts.

Example The elements of the set A B should be listed if A = 1, 2, 3, 4 and B = 2, 4, 6, 8.

A little letter, like x, y, or z, is typically used to represent a set element.

By listing all of a set's components inside braces, a set can be explained.

A = 2, 4, 6, 8 for instance, if Set A is composed of the integers 2, 4, 6, and 8.

Hence, The items in A, or the elements included in set A are A = 1, 2, 3, 4.

To learn more about elements in a set visit,

brainly.com/question/24462379

#SPJ1

Answer:

[tex]x=\frac{26}{3}[/tex]

Step-by-step explanation:

Lets solve for [tex]x[/tex].

Subtract [tex]3x[/tex] from both sides of the equation.

[tex]9x-20-3x=32[/tex]

Add 20 to both sides of the equation.

[tex]9x-3x=32+20[/tex]

Simplify both sides.

[tex]6x=52[/tex]

Divide both sides of the equation by 6.

[tex]x=\frac{52}{6}[/tex]

Write the fraction in simplest form.

[tex]52 \div 2 = 26\\6 \div 2 = 3[/tex]

[tex]x=\frac{26}{3}[/tex]

Answer:

Step-by-step explanation:

9x-20=3x+32

9x-20+20=3x+32+20

9x=3x+52

9x-3x=3x+52-3x

6x=52

6x/6=52/6

x=26/3

The probability that the first MP3 song heard is the fifth song played is:

[tex]\frac{C^9^0_4.C^1^0_1}{C^1^0^0_5}[/tex] =  [tex]\frac{2,555,190. 10}{75,287,520}[/tex] ≈ 0.3394

The probability that there are exactly two MP3 songs in the first five songs played is:

[tex]\frac{C^1^0_2.C^9^0_3}{C^1^0^0_5}[/tex] [tex]= \frac{45.117,480}{75,287,520}[/tex] ≈ 0.0702

Now, According to the question:

The 1st MP3 player songs heard is the 5th song played means that first 4 songs are not MP3 songs (there are 100 - 10 = 90 such songs) and the 5th song is MP3 song (there are 10 such songs). Hence, the probability that the first MP3 song heard is the fifth song played is:

[tex]\frac{C^9^0_4.C^1^0_1}{C^1^0^0_5}[/tex] =  [tex]\frac{2,555,190. 10}{75,287,520}[/tex] ≈ 0.3394

There are exactly two MP3 songs in the first five songs played means there are 2 MP3 songs and 3 not MP3 songs (the number when these songs are played do not matter). Hence, the probability that there are exactly two MP3 songs in the first five songs played is:

[tex]\frac{C^1^0_2.C^9^0_3}{C^1^0^0_5}[/tex] [tex]= \frac{45.117,480}{75,287,520}[/tex] ≈ 0.0702

Learn more about Probability at:

https://brainly.com/question/11234923

#SPJ4

Answer:

Yes

Step-by-step explanation:

This only works for positive integers. For example, 11/0.5 equals 22, but if you were to negate the 11 then the answer would be -22 which is less than negative 11 so the answer is yes it is true for whole positive integers.

The height is 114.83 ft and the distance of the ø if someone is observing below is 84.8.

The branch of mathematics that sets up a relationship between the sides and the angles of the right-angle triangle is termed trigonometry.

The trigonometric functions, also known as a circular, angle, or goniometric functions in mathematics, are real functions that link the angle of a right-angled triangle to the ratios of its two side lengths.

Given that at a given point of 67.7ft at the ground level of a covered court, the angle of elevation of a roof worker from the bottom is 38.6⁰ and the angle of elevation of the roof from the top is 42.4 degrees.

The base will be calculated as:-

tan(38.6) = P / B

tan(38.6) = 67.7 / B

B = 67.7 / tan(38.6)

B = 54.8 ft

The height will be calculated as:-

Cos(42.4) = B / H

H = 84.8 / Cos(42.4)

H = 114.83 ft

Therefore, If someone were to look down from above, they would be 114.83 feet from the object and 84.8 feet away.

To know more about Trigonometry follow

brainly.com/question/24349828

#SPJ1

The equation of the linear function in this problem is given as follows:

y = 0.5x + 4.

The slope-intercept definition of a linear function is given as follows:

y = mx + b.

In which:

When x = 0, y = 4, hence the intercept b is given as follows:

b = 4.

When x increases by 8, y increases by 4, hence the slope m is given as follows:

m = 4/8 = 0.5.

The linear function goes through the points (-8,0) and (0,4).

More can be learned about linear functions at https://brainly.com/question/24808124

#SPJ1

The situation that can be described as dependent events is c.) Renata randomly chooses a chocolate chip cookie from a cookie jar. She eats the cookie and chooses again. This time she gets a sugar cookie.

Events that are dependent upon prior events are called dependent events. The results of earlier events have an impact on these current ones. For example, two or more occurrences that are interdependent are referred to as dependent events. An event generally qualifies as being reliant if it offers details about another event. If an event provides no information about other events, it is considered independent.

The dependent event would be Renata picking a sugar cookie the second time she chooses another cookie because she ate from the first cookie and so was able to avoid it the second time.

Find out more on dependent events at https://brainly.com/question/27941720

#SPJ1

The probability of selecting a 9 and the coin landing on tails is 1/26.

Probability can be defined as the ratio of the number of favourable outcomes to the total number of outcomes of an event.

We know that, probability of an event = Number of favourable outcomes/Total number of outcomes

Here,

The probability of selecting a card 9 is

Total number of outcomes =52

Number of favourable outcomes =4

Now, probability = 4/52

= 1/13

The probability of coin landing on tails is

1/2

Probability of an event = 1/13 × 1/2

= 1/26

Therefore, the probability of selecting a 9 and the coin landing on tails is 1/26.

To learn more about the probability visit:

https://brainly.com/question/11234923.

#SPJ1

Answer:

4x−3y+4=0

⟹x=

4

−4+3y

​

Let, y=4, then x=

4

−4+3(4)

​

=2

Similarly, when y=0,x=−1

4x+3y−20=0

⟹x=

4

20−3y

​

Let, y=4, then x=

4

20−3(4)

​

=2

Similarly, when y=0,x=5

The function f(x) can shift for log (w ± dx) in any of the four directions

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

log(DWX)

Express properly

log(w ± dx)

Where w and d are constants and x is the variable

The above function is a logarithmic function, and the parent function is

log(x)

The direction the function shifts is dependent on the values of w and d i.e.

Read more about transformation at

https://brainly.com/question/1548871

#SPJ1

The standard normal distribution is used when calculating the confidence interval for the mean using a sample size of 30 or greater with a known population standard deviation.

The Z-distribution (also known as the standard normal distribution) is a normal distribution with a mean of 0 and a standard deviation of 1. When the population standard deviation is known, the sample mean follows a normal distribution and the standard error is given by the population standard deviation divided by the square root of the sample size. So, the Z-score is given by (x - μ) / (σ / √(n)) where x is the sample mean, μ is the population mean and σ is the population standard deviation and n is the sample size.

With a sample size of 30 or greater, the central limit theorem states that the sample mean will be approximately normally distributed, regardless of the shape of the population distribution. So, a Z-score can be used to find the probability that a sample mean falls within a certain range of the population mean, which is commonly used to construct a confidence interval.

Learn more about normal distribution here: https://brainly.com/question/4079902

#SPJ4

Answer:

70.824

Step-by-step explanation:

to solve, multiply the amount of pounds by 0.454.

EX:

156 * 0.454 = 70.824

what it actually equals:

70.7604 (answ on web)

vs

70.824 (answ with formula)

not bad eh?

hope this helps :)

Number of students who are expected to choose hip hop or rock as their favorite type of music is 240.

Percentage is defined as the parts of a number per fraction of 100. We have to divide a number with it's whole and then multiply with 100 to calculate the percentage of any number.

So the percentage actually means a part per 100.

Percentage is usually denoted by the symbol '%'.

From the graph,

Number of students who choose rock music = 20

Number of students who choose hip hop music = 40

Number of students who choose country music = 30

Total number of students = 20 + 40 + 30 = 90

Percentage of students who choose rock or hip hop = (20+ 40) / 90

                                                                                        = (60 / 90)

                                                                                        = 66.67%

Out of 360 students, 66.67% of students will chose hip hop or rock.

360 × (60/90) = 240

Hence 240 students choose either hip hop or rock as their favorite music.

Learn more about Percentage here :

https://brainly.com/question/28269290

#SPJ1

In the triangles value of x is 10.39 and z is 2.6.

We can see from the diagram that both triangles are right triangles. As a result, we can use Pythagorean theorems. The longest side of a right triangle is the hypotenuse. In the diagram, those are sides with lengths of 4 and 12. The formula for this is:

c² = (a²+ b²), where c is the hypotenuse and the other shorter sides are a and b.

c = [tex]\sqrt{(a^{2} + b^{2} )}[/tex]

To calculate z,

By substituting the values,

4² = (z² + 3²)

16 =  (z² + 9)

z² = 16 - 9

z² = 7

z = √7

z =  2.6

To find x,

use the following formula: 12² = (x² + 6²)

144 = x² + 36

x² = 144 - 36 = 10.39

x = 10.39

To learn more about right angle triangle refer to :

https://brainly.com/question/1248322

#SPJ1

The z-score for a backyard structure costing $2300 is -0.67.

The standard score in statistics is the amount of standard deviations that a raw score deviates from or exceeds the mean of the phenomenon being observed or measured. Standard scores are positive for raw scores above the mean and negative for raw scores below the mean.

The Z-score quantifies the deviation between a given value and the standard deviation. The Z-score, also known as the standard score, indicates how many standard deviations a given data point deviates from the mean. In essence, standard deviation represents the degree of variability present in a given data set.

The z score will be:

= (2300 - 3100) / 1200

= -0.67.

Learn more about z score on:

https://brainly.com/question/25638875

#SPJ1

The high costs in the California real estate market have caused families who cannot afford to buy bigger homes to consider backyard sheds as an alternative form of housing expansion. Many are using the backyard structures for home offices, art studios, and hobby areas as well as for additional storage. The mean price of a customized wooden, shingled backyard structure is $3100 (Newsweek, September 29, 2003). Assume that the standard deviation is $1200.

a. What is the z-score for a backyard structure costing $2300?

The area of triangle ABC is 0.5cm² where base and height are given.

Triangle area is used to measure the size of a region on a surface. The area of a form or planar lamina is referred to as the plane region or plane area, as opposed to surface area, which denotes the area of an open surface or the boundary of a three-dimensional object.

The area of a triangle can be compared to the volume of material at a specific thickness needed to make a model of the form or the amount of paint needed to cover a surface in a single coat. It is equivalent to a solid's volume or a curve's length in two dimensions (a three-dimensional concept).

From the given figure, we can say that

Height = AD = 1cm

Base = BC = 1cm

Area of Triangle = Base × Height/2

Area = 1 × 1 /2

Area = 0.5 cm²

To learn more about Area of Triangle from the given link

brainly.com/question/19305981

#SPJ1

The quartic function y = 0.5 · x⁴ - x³ - 15.5 · x² + 30 · x + 21 contains roots 4 - √2 and - 3 + √6 and intercept - 21.

Quartic functions are polynomials of grade 4, whose factor form is introduced below:

y = a · (x - r₁) · (x - r₂) · (x - r₃) · (x - r₄)

Where:

According to statement, the quartic equation has an intercept of - 21 and two real roots: 4 - √2, - 3 + √6. The two remaining roots can be found based on features exhibited by quadratic formula and that quartic functions are the product of two quadratic functions: 4 + √2, - 3 - √6

Now we proceed to expand the quartic function:

y = a · (x - 4 + √2) · (x - 4 - √2) · (x + 3 - √6) · (x + 3 + √6)

y = a · (x² - 8 · x + 14) · (x² + 6 · x + 3)

y = a · [x² · (x² - 8 · x + 14) + 6 · x · (x² - 8 · x + 14) + 3 · (x² - 8 · x + 14)]

y = a · (x⁴ - 8 · x³ + 14 · x² + 6 · x³ - 48 · x² + 84 · x + 3 · x² - 24 · x + 42)

y = a · (x⁴ - 2 · x³ - 31 · x² + 60 · x + 42)

And we find the lead coefficient:

- 21 = 42 · a

a = - 0.5

And we obtain the quartic function in standard form:

y = 0.5 · x⁴ - x³ - 15.5 · x² + 30 · x + 21

To learn more on quartic functions: https://brainly.com/question/28593533

#SPJ1

The correct option regarding the first quartile of a data-set is given as follows:

One-quarter of the data lies below Q1 and three-quarters of the data lies above Q1.

The first quartile Q1 of a data-set is the 25th percentile, meaning that:

This means that the correct option is given by the first option.

More can be learned about the quartiles of a data-set at https://brainly.com/question/3514929

#SPJ1

The equation is as follows: Q = a - bP^2.Nonlinear demand curve like this can be found in luxury goods and Veblen goods. Where the luxury goods are the goods whose demand increases as the price increases and Veblen goods are the goods whose demand increases with an increase in the consumer's income.

The equation is as follows: Q = a - bP^2.

To derive a demand schedule, we can plug in different values of P (price) and solve for Q (quantity demanded). For example:

Q = a - bP^2

Q = a - (b * 10^2)

Q = a - 1000b

Q = a - (b * 9^2)

Q = a - 810b

Q = a - (b * 8^2)

Q = a - 640b

and so on.

To derive a demand curve, we can plot the different points from the demand schedule on a graph, with P on the x-axis and Q on the y-axis. The demand curve will be a downward-sloping parabola.

Nonlinear demand curve like this can be found in luxury goods and Veblen goods. Where the luxury goods are the goods whose demand increases as the price increases and Veblen goods are the goods whose demand increases with an increase in the consumer's income.

Learn more about equation here

https://brainly.com/question/29657992

#SPJ4

A projectile have a minimum speed at the highest point of its trajectory. This is because horizontal speed of the projectile remains constant.

Since the only force exerted on the projectile is gravity, which is

9.8 ms ^2  acting downwards and has no effects on the horizontal velocity.The maximum height the pumpkin reaches occurs at 62.5 horizontal meters from its launching spot.

The calculation is as follows;

The equation is y = -0.008x^2 +x

that corresponds to a curve presented by a parabola.

Now we can determine the maximum of this parabola with arms pointing down requesting the derivative slope of the tangent line to

the curve to be zero

So,= 62.5m

Learn more about trajectory here :

https://brainly.com/question/28874076

#SPJ4

Answer:

  0.70

Step-by-step explanation:

You want p(A+B) given that p(A) = 0.60, p(B) = 0.25, and p(A·B) = 0.15.

The relevant formula is ...

  p(A+B) = p(A) +p(B) -p(A·B)

  p(A+B) = 0.60 +0.25 -0.15

  p(A+B) = 0.70

__

Additional comment

In the attached Venn diagram, the sum of the numbers inside the circle labeled A is p(A) = 0.45 +0.15 = 0.60. Likewise, the sum of the numbers inside circle B is p(B) = 0.15 +0.10 = 0.25.

As the attachment shows, the event A includes the event 'A and B'. Likewise, event B includes the event 'A and B'. Simply adding the probabilities of A and B will cause p(A and B) to be counted twice.

Answer:

Step-by-step explanation:

Given that akua is 3 years older than yaboah and the sum of their ages is 23, you want the age of yaboah and the product of their ages.

Let 'a' and 'y' represent the ages of akua and yaboah, respectively. The given relations are ...

  a + y = 23

  a - y = 3

Adding these equations gives ...

  2a = 26

  a = 13

Using this in the second equation, we find ...

  y = a -3 = 13 -3 = 10

The product of their ages is ...

  a·y = 13·10 = 130

Yaboah is 10 and the product of their ages is 130.

Answer:

P = ₹4,000

F = ₹14,400

r = 10% = 0.10

m = 4 (because interest is compounded quarterly)

So, we can plug in these values into the formula:

n = (log(14400 / 4000)) / (log(1 + 0.10 / 4))

n = (log(3.6)) / (log(1.025))

n = 2.59

Therefore, it will take approximately 2.59 years for the investment of ₹4,000 to grow to ₹14,400 at a 10% annual interest rate compounded quarterly.

So, the correct ordering from smallest to largest is 17/15, 7/6, 28/25.

When comparing fractions, one way to determine which fraction is larger or smaller is by finding a common denominator and then comparing the numerators. A common denominator is a multiple of both denominators. The least common denominator (LCD) is the smallest common denominator.

In this case, the least common denominator for the fractions 7/6, 17/15, 28/25 is 30.

This is because 30 is the smallest multiple of 6 and 15 that is also a multiple of 25

So, 7/6 = 35/30, 17/15 = 17/15 and 28/25 = 84/75

The numerators of the fractions are 35, 17, and 84 respectively. Since 17 is the smallest numerator and 84 is the largest numerator, we can conclude that 17/15 is the smallest fraction and 84/75 is the largest fraction.

Therefore, the correct ordering from smallest to largest is 17/15, 7/6, 28/25.

In summary, to compare fractions, we can find a common denominator, then compare the numerators. In this case, we used the least common denominator 30, and then compared the numerators 35, 17, 84, which allowed us to order the fractions in an increasing order.

To learn more about Fractions

Visit; brainly.com/question/10354322

#SPJ4

The papers graded exponentially at the end of 5 days are 134, and it will take approximately around 12 days to complete 6000 papers.

A function is considered to be exponentially growing if its value rises faster than any polynomial. The function e^x, serves as a good illustration.

The exponential growth is given by the following formula:

[tex]A = Pe^{rt}[/tex]

where, A is the final amount, P is the initial amount, r is the rate and t is the time.

For t = 0, the number of papers graded are, 9, hence the value of P = 9.

To calculate the rate at which the work is done, substitute the value of P = 9, A = 27 and t = 2 in the equation:

[tex]27 = 9 e^{2r}\\3 = e^{2r}[/tex]

Taking ln on both sides of the equation:

[tex]ln (3) = ln(e^{2r})\\\\ln(3) = 2r[/tex]

1.09 = 2r

r = 0.54

To calculate the number of papers graded in five days, substitute the value of P = 9, t = 5 and r = 0.54.

[tex]A = 9 e^{(0.54)(5)}[/tex]

A = 134

Hence, the papers graded at the end of 5 days are 134.

To complete grading 6000 papers, the time taken will be:

6000 = 9e^(0.54)(t)

666.66 = e^(0.54)(t)

Taking ln on both sides of the equation:

ln (666.66) = ln(e^(0.54)(t))

6.50 = (0.54)(t)

t = 12.04

Hence, it will take approximately around 12 days to complete 6000 papers.

Learn more about exponential function here:

https://brainly.com/question/14355665

#SPJ1

Based on the ratio of the largest angle of the triangle with 180°, the degrees of the largest angle is 77°.

The ratio is the proportion of one value to the whole.

Ratios compare the relative sizes of components.

Ratios are computed as the quotients resulting from division operations.

The ratio of the measures of the angels of a triangle = 5:7:9

The sum of ratios = 21

The ratio of the largest angle = 9/21

We know that the total angles of a triangle = 180°

The angle (degrees) of the largest angle = 77° (180° x 9/21)

Thus, the degree of the largest angle is 77°.

Learn more about ratios at https://brainly.com/question/12024093

#SPJ1

Answer:

Question 33

a) x-int (2, 0).  y-int (0, 2/3)

b) Vertical asymptotes: x = -1 and x = 3

   Horizontal asymptote: y = 0

c) f(x) → -∞, as x → -1⁻

   f(x) → +∞, as x → -1⁺

   f(x) → -∞, as x → 3⁻

   f(x) → +∞, as x → 3⁺

d) Domain: (-∞, -1) ∪ (-1, 3) ∪ (3, ∞)

   Range: (-∞, ∞)

Question 34

a)  x-int (-1, 0) and (2,0).  y-int (0, 2/3)

b) Vertical asymptote: x = 3

c) f(x) → -∞, as x → 3⁻

   f(x) → +∞, as x → 3⁺

d) Domain: (-∞, 3) ∪ (3, ∞)

   Range: (-∞, 1] ∪ [9, ∞)

Question 35

a) No axis intercepts

b) Vertical asymptotes: x = -2 and x = 2.  Hole at (0, ¹/₄).

c) f(x) → -∞, as x → -2⁻

   f(x) → +∞, as x → -2⁺

   f(x) → +∞, as x → 2⁻

   f(x) → -∞, as x → 2⁺

d) Domain: (-∞, -2) ∪ (-2, 0) ∪ (0, 2) ∪ (2, ∞)

   Range: (-∞, 0) ∪ (¹/₄, ∞)

Step-by-step explanation:

Definitions

Given rational function:

[tex]g(x) =\dfrac{x - 2}{x^2 - 2x - 3}[/tex]

x-intercept

[tex]\begin{aligned}y=0 \implies \dfrac{x - 2}{x^2 - 2x - 3}&=0\\x - 2&=0\\x&=2\end{aligned}[/tex]

y-intercept

[tex]x=0 \implies \dfrac{0 - 2}{0^2 - 2(0) - 3}=\dfrac{2}{3}[/tex]

Vertical asymptotes

Set the denominator to zero and solve for x:

[tex]\implies x^2-2x-3=0[/tex]

[tex]\implies x^2-3x+x-3=0[/tex]

[tex]\implies x(x-3)+1(x-3)=0[/tex]

[tex]\implies (x+1)(x-3)=0[/tex]

[tex]\implies x+1=0 \implies x=-1[/tex]

[tex]\implies x-3=0 \implies x=3[/tex]

Horizontal asymptote

As the degree of the numerator is less than the degree of the denominator, there is a horizontal asymptote at y = 0.  

Holes

Factor the denominator:

[tex]g(x) =\dfrac{x - 2}{(x+1)(x-3)}[/tex]

There are no holes as there is no common factor in the numerator and the denominator.

End behaviour near the vertical asymptotes

f(x) → -∞, as x → -1⁻

f(x) → +∞, as x → -1⁺

f(x) → -∞, as x → 3⁻

f(x) → +∞, as x → 3⁺

Domain

As there are vertical asymptotes at x = -1 and x = 3, the domain is restricted:

Range

The range is unrestricted:

Given rational function:

[tex]k(x)= \dfrac{x^2 - x - 2}{x - 3}[/tex]

x-intercepts

[tex]\begin{aligned}y=0 \implies \dfrac{x^2 - x - 2}{x - 3}&=0\\x^2 - x - 2&=0\\x^2-2x+x-2&=0\\x(x-2)+1(x-2)&=0\\(x+1)(x-2)&=0\\\\x+1&=0 \implies x=-1\\x-2&=0 \implies x=2\end{aligned}[/tex]

y-intercept

[tex]x=0 \implies \dfrac{(0)^2 - 0 - 2}{0 - 3}=\dfrac{2}{3}[/tex]

Vertical asymptotes

Set the denominator to zero and solve for x:

[tex]\implies x-3=0[/tex]

[tex]\implies x=3[/tex]

Holes

Factor the numerator:

[tex]k(x)= \dfrac{(x+1)(x-2)}{x - 3}[/tex]

There are no holes as there is no common factor in the numerator and the denominator.

End behaviour near the vertical asymptote

f(x) → -∞, as x → 3⁻

f(x) → +∞, as x → 3⁺

Domain

As there is vertical asymptote at x = 3, the domain is rest:

Range

The extreme points are (1, 1) and (5, 9).  Therefore, the range is restricted:

Given rational function:

[tex]f(x)= -\dfrac{x}{x^3 - 4x}[/tex]

x-intercepts

As the degree of the numerator is less than the degree of the denominator, there is a horizontal asymptote at y = 0.  Therefore, there are no x-intercepts.

y-intercept

Substitute x = 0 and solve for y:

[tex]x=0 \implies -\dfrac{0}{0(x+2)(x-2)}=\dfrac{0}{0}=\textsf{unde\:\!fined}[/tex]

Therefore, there is no y-intercept.

Factor the denominator:

[tex]f(x)= -\dfrac{x}{x(x+2)(x-2)}[/tex]

Cancel the common factor x:

[tex]f(x)= -\dfrac{1}{(x+2)(x-2)}[/tex]

Vertical asymptotes

Set the denominator to zero and solve for x:

[tex]\implies (x+2)(x-2)=0[/tex]

[tex]\implies x+2=0 \implies x=-2[/tex]

[tex]\implies x-2=0 \implies x=2[/tex]

Holes

Factor the denominator:

[tex]f(x)= -\dfrac{x}{x(x+2)(x-2)}[/tex]

As the numerator and denominator have a common factor of x, the function is not defined when x = 0. Therefore, there is a hole when x = 0.

To find the y-value of the hole, cancel the common factor x and input x = 0 into the resulting function:

[tex]x=0 \implies -\dfrac{1}{(0+2)(0-2)}=\dfrac{1}{4}[/tex]

Therefore, there is a hole at (0, ¹/₄).

End behaviour near the vertical asymptotes

f(x) → -∞, as x → -2⁻

f(x) → +∞, as x → -2⁺

f(x) → +∞, as x → 2⁻

f(x) → -∞, as x → 2⁺

Domain

As there are vertical asymptotes at x = -2 and x = 2, and a hole at (0, ¹/₄), the domain is restricted:

Range

As there is a hole at (0, ¹/₄) and a horizontal asymptote at y = 0, the range is restricted: