s for each matrix below, apply the given row operation(s) to the matrix in order, and give the resulting matrix.

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

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

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.

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

The value of x in the equation,  X^3= -125 is -5.

To find x in the above polynomial equation, we need to collect like terms and equate the expression to 0. To begin we can arrange the expression this way:

x^3 + 125 = 0

This means that we should find the figure which when multiplied three times would equate to 125. This figure is -5. So, when solved, we have:

-5³ + 125 = 0

-125 +125 = 0

So, the value for x in the equation above is -5.

Learn more about polynomials here:

https://brainly.com/question/4142886

#SPJ1

The change in the function per unit over a specific period of time is the average rate of change. Between x is 5 and x is 7, the average rate of change is -4.

The y = f graph

The points on f(x) that connect x = 5 to x = 7 are in the attachment. A green line serves as the line segment's representation.

The following values can be used to calculate the line segment's average rate of change.

If x = 5, then y = 8.

Y Equals 0 when x = 7

Consequently, the typical rate of change (m) is

(Delta*y/(Delta*x)) = m

Where:

Delta*y equals y 2 - y 1.

Sigma*y = 0 to 8

Delta*y equals -8

Delta*x equals x 2 - x 1.

x = 7 - 5 for delta.

x*Delta = 2

We therefore have:

(Delta*y/(Delta*x)) = m

m = - 8/2 m = - 4

Hence, the average rate of change from x = 5 to 7 is -4

To learn more about graph refer to:

https://brainly.com/question/19153277

#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

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.

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

A's x- intercept is 3 and y-intercept is 5. Both are favourable.

To learn more about y intercepts refer to:

https://brainly.com/question/25722412

#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 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

all the mentioned angles have been found.

What is are of triangle?

The territory included by a triangle's sides is referred to as its area. Depending on the length of the sides and the internal angles, a triangle's area changes from one triangle to another. Square units like m2, cm2, and in2 are used to express the area of a triangle.

The sum of all angles of triangle = 180

∠ICA =180-20 =160

∠EAF = 20

∠ICB =180

∠GAF = 120 -20 =100

∠CAB= 180-80=100

∠HAG=180-120 =60

∠CBA=180-120=60

∠HAC.= 20

Hence all the mentioned angles have been found.

Learn more about triangle, by the following link

brainly.com/question/17335144

#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

True, When assessing schedule feasibility, a systems analyst must consider the interaction between time and costs.

When assessing the viability of scheduling a project, an organisation determines the anticipated completion time. The feasibility analysis assists in identifying any potential obstacles the proposed project may encounter, such as: Internal project constraints, including those related to technology, finances, and resources

The expansion and acceptance of project management training have undergone significant changes over the past few years, and it is projected that these developments will continue and accelerate. And as project management becomes more popular, feasibility studies become more important. It may be exhilarating to begin a difficult, extensive project that will have a significant influence on your organization.

To learn more about schedule here:

https://brainly.com/question/12516618

#SPJ4

The unit vector orthogonal to the plane can be found by taking the cross product of the vectors p-q and p-r. The vector should have a positive y-component if the cross product points in the positive y direction.

To find a unit vector orthogonal to the plane through the points p, q, and r, with a positive y-component, we will use the cross product of two of the vectors. The cross product of two vectors is a vector perpendicular to both of them. First, we will create vector p-q, which is a vector from point p to point q. We can do this by subtracting the coordinates of point q from the coordinates of point p. Next, we will create vector p-r, which is a vector from point p to point r. We can do this by subtracting the coordinates of point r from the coordinates of point p. Finally, we will take the cross product of the two vectors. This will give us a vector orthogonal to the plane through the three points. If the vector has a positive y-component, then it is the unit vector orthogonal to the plane that we are looking for.

Learn more about orthogonal here

https://brainly.com/question/29580789

#SPJ4

The average of given set of integers is -17.

In maths, the average value in a set of numbers is the middle value, calculated by dividing the total of all the values by the number of values. When we need to find the average of a set of data, we add up all the values and then divide this total by the number of values.

The given set of integers are -20, -54, -93, 37 and 45.

Here, average = Sum of all the observations/Number of observations

= (-20-54-93+37+45)/5

= -85/5

= -17

Therefore, the average of given set of integers is -17.

To learn more about an average visit:

https://brainly.com/question/11195029.

#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 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 value of x which holds true for the given system of equations as required is; x = 4 / 5.

It follows from the task content that the value of x which holds true for the given system of equations be determined.

Since the given system is;

2x - 3y = 7........(i)

-4x + y = -5........(ii)

The value of x which holds true for the given system of equations can be determined by making y the subject of the formula in (ii) and substitution into (i) as follows;

y = 4x - 5.

Therefore;

2x - 3 (4x - 5) = 7

2x - 12x + 15 = 7

-10x = -8

x = 4/5

Ultimately, the value of x which holds true for the given system of equations is; x = 4 / 5.

Read more on system of equations;

https://brainly.com/question/25976025

#SPJ1

2 x – 3 y = 7 –4 x + y = –5 Wh

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

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

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 equation of the word sentence, "5 is one-fourth of a number c" is: 5 = 1/4c.

To write a word sentence as an equation, you need to identify the variables and the operations being described in the sentence, and then use mathematical symbols to represent them.

For example, we are given the sentence "5 is one-fourth of a number c,", in this case, the variable is "c" and the operation is "one-fourth of," which is represented by the fraction 1/4 or the multiplication by 1/4.

Therefore, the equation would be:

5 = 1/4c.

Learn more about word sentence as an equation on:

https://brainly.com/question/23384508

#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 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

One of the steps in statistical inference, to check the claim is hypothesis testing. In this method, we determine whether to reject or accept the null hypothesis.

Hypothesis testing is the procedure in statistical inference that helps to decide whether the result of the research study supports a particular hypothesis when applied to a population. This helps to test the relationship between two statistical variables.

This includes two types null and alternate hypotheses. If the results say that there is no relationship seen between the two variables, then the null hypothesis is accepted. But if the results so there is a relationship seen between two variables, then an alternate hypothesis is accepted. So to check our claim, we do hypothesis testing.

The complete question is -

The following question is from the four steps of statistical inference. If you respond with a summary of all four steps, you will receive no credit on this problem:

What do we do to check a claim?

To know more about alternate hypothesis:

https://brainly.com/question/29708875

#SPJ4

The number of notepads per department used in the Southeast location is given as follows:

6.

The number of notepads per department is obtained applying the proportions in the context of this problem, considering the graph.

The number of departments of the Southeast location is given as follows:

6.

Which is given from the table given in the second image for this problem.

More can be learned about proportions at https://brainly.com/question/24372153

#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: