find any points of discontinuity for each rational function. Sketch a graph. Describe any verticals or horizontal asympyoyes and any holes
(X^2-4)/(X+2)(X^2-49)
Answers
y = (x^2 -4)/((x +2)(x^2 -49))
the numerator factors to (x -2)(x +2), so the factor of (x +2) will cancel with that in the denominator, leaving
y = (x -2)/(x^2 -49)
There are points of discontinuity at the hole, x=-2, and at each of the vertical asymptotes, at x=-7, +7.
The horizontal asymptote is y=0.
To find the points of discontinuity and sketch the graph of the rational function (x^2-4)/(x+2)(x^2-49), we need to determine the values of x that make the denominator equal to zero. We find that x = -2, x = 7, and x = -7 are the points of discontinuity. We can then sketch the graph by plotting the vertical asymptotes at x = -2, x = 7, and x = -7, and determining the behavior of the function on each side of those points.
Explanation:To find the points of discontinuity for the rational function (x^2-4)/(x+2)(x^2-49), we need to determine where the denominator is equal to zero. The denominator has two factors, (x+2) and (x^2-49). Setting each factor equal to zero, we find that x = -2, x = 7, and x = -7. These are the points where the function has vertical asymptotes or holes.
To sketch the graph, we can start by plotting the vertical asymptotes at x = -2, x = 7, and x = -7. Next, we can examine the sign of the function on each side of the vertical asymptotes to determine the behavior of the graph. Finally, we can plot any holes in the graph at the corresponding x-values.
Additionally, we should determine if there are any horizontal asymptotes. For rational functions, horizontal asymptotes occur when the degree of the numerator is less than or equal to the degree of the denominator. In this case, the degree of the numerator is 2 and the degree of the denominator is 3, so there is no horizontal asymptote.
Related Questions
Complete the equation of the line whose slope is − 2 and y-intercept is ( 0 , 3 )
Answers
This is the equation in slope point form.
ABC Bookstore sells new books according to the demand function P = -Q + 25, where P is the price and Q is the quantity. If the bookstore charges $18 for a new book, how many new books can the store expect to sell?
Answers
Therefore, the store can expect to sell 7 new books if they will charge $18 for the new books.
Answer:
P = -Q + 2518 = -Q + 2518-25 = -Q-7 = -Q-7/-1 = -Q/-1Q = 7
Therefore, the store can expect to sell 7 new books if they will charge $18 for the new books.
During the first stages of an epidemic, the number of sick people increases exponentially with time. Suppose that at = 0 days there are 2 people sick. By the time = 3, 40 people are sick. a) Let be the number of sick people at time . Find an exponential equation = expressing in terms of .
b) How many people will be sick by the time = 6?
Answers
r = rate of growth
by 0 day, or t = 0, there are 2 folks sick,
[tex]\bf \qquad \textit{Amount for Exponential Growth} \\\\ A=P(1 + r)^t\qquad \begin{cases} A=\textit{accumulated amount}\to &2\\ P=\textit{initial amount}\\ r=rate\to r\%\to \frac{r}{100}\\ t=\textit{elapsed time}\to &0\\ \end{cases} \\\\\\ 2=P(1+r)^0\implies 2=P\cdot 1\implies 2=P\qquad \boxed{A=2(1+r)^t}[/tex]
by the third day, t = 3, there are 40 folks sick,
[tex]\bf \qquad \textit{Amount for Exponential Growth} \\\\ A=P(1 + r)^t\qquad \begin{cases} A=\textit{accumulated amount}\to &40\\ P=\textit{initial amount}\to &2\\ r=rate\to r\%\to \frac{r}{100}\\ t=\textit{elapsed time}\to &3\\ \end{cases} \\\\\\ 40=2(1+r)^3\implies 20=(1+r)^3\implies \sqrt[3]{20}=1+r \\\\\\ \sqrt[3]{20}-1=r\implies 1.7\approx r\qquad \boxed{A=2(2.7)^t}[/tex]
how many folks are there sick by t = 6? [tex]\bf \stackrel{that~many}{A=2(2.7)^6}[/tex]
Han has 410,000 in a retirement accounts that earns 15,785 each year find the simple interest rate for this investment
Answers
Interest = principle * rate * time
I=prt
r=I/(pt)=15785/(410,000*1)= .0385
The simple interest rate on this investment is.0385 or 3.85%:
What is Simple Interest?Simple interest is calculated based on a loan's principal or the initial deposit into a savings account. Simple interest doesn't compound, therefore a creditor will only charge interest on the principal sum, and a borrower will never be required to pay further interest on the interest that has already accrued.
Given:
Han has 410,000 in a retirement accounts that earns 15,785 each year.
Using Simple Interest
Interest = principle x rate x time / 100
Rate = Interest / ( principle x time )
Rate = 15785/(410,000x1)
Rate = 0.0385
Rate= 3.85 %
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The scale factor of the dilation of figure ABCD is
Answers
The scale factor is 2.
Use the quadratic formula to solve the equation. If necessary, round to the nearest hundredth. –2y2 + 6y = –2
–0.61, 6.61
–0.3, 3.3
–3.3, 0.3
–11.5, 14.5
Answers
-2y2 + 6y = -2
Rewriting we have:
-2y2 + 6y + 2 = 0
Using the quadratic formula we have:
x = (- b +/- root (b ^ 2 - 4 * a * c)) / (2 * a)
Substituting values:
x = (- (6) +/- root ((6) ^ 2 - 4 * (- 2) * (2))) / (2 * (- 2))
Rewriting:
x = (- 6 +/- root (36 + 16)) / (- 4)
x = (- 6 +/- root (52)) / (- 4)
x = (- 6 +/- root (4 * 13)) / (- 4)
x = (- 6 +/- 2 * root (13)) / (- 4)
x = (- 3 +/- root (13)) / (- 2)
The roots are:
x1 = (- 3 - root (13)) / (- 2)
x2 = (- 3 + root (13)) / (- 2)
Rewriting:
x1 = -0.3
x2 = 3.3
Answer:
–0.3, 3.3
Farmer Ed has 8500 meters of fencing and wants to enclose a rectangular plot that borders on a river. If farmer Ed does not fence the side along the river, what is the largest are that can be enclosed?
Width = 2x
Length = 8,500-2x
Answers
Area (A) = xy m^2
If the side bordering the river is x, then
Circumference (C) = 8500 = x+2y => x = 8500-2y
Substituting for w in the equation for A;
A = (8500-2y)y = 8500y -2y^2
For maximum area, derivative of A with respect to y is 0. That is;
dA/dy = 0 = 8500 -4y => 4y = 8500 => y = 8500/4 = 2125 m
And, x = 8500-2y = 8500 -2*2125 = 4250 m
Therefore,
One side = 4,250 m and second side = 2,125 m.
Largest area, A = 4250*2125 = 9,031,250 m^2
Adair had 6 1/2 cups of tomato juice. He used 4/5 of the juice to make soup. How many cups of tomato juice did Adair use to make soup?
Answers
However, first we must convert 6 1/2 from a mixed number to an improper fraction to simplify the multiplication process. To do this, we have to multiply the whole number (6) by the denominator (2) and then add the numerator (1).
6*2=12, 12+1=13. Therefore, 6 1/2 = 13/2.
Now, we must multiply 4/5 by 13/2 to figure out how much tomato juice Adair used. To do this, we must multiply the two numerators and the two denominators together and simplify the resulting fraction.
4/5 * 13/2 = 4*13/5*2=52/10
To simplify 52/10, we must find the GCF, or greatest common factor of 52 and 10, which is 2.
52/2 = 26, 10/2 = 5, 52/10 = 26/5.
Therefore, Adair used 26/5, or 5 1/5 cups of tomato juice in his soup.
Hope this helps!
To find how many cups of tomato juice Adair used to make soup, we multiply the fraction representing the portion of tomato juice used (4/5) by the total amount of tomato juice Adair had (6 1/2 cups). The answer is 5 1/5 cups.
Explanation:To find how many cups of tomato juice Adair used to make soup, we need to multiply the fraction representing the portion of tomato juice used (4/5) by the total amount of tomato juice Adair had (6 1/2 cups). First, we can convert 6 1/2 cups to an improper fraction: 6 1/2 = 13/2 cups. Next, we multiply the fractions: 4/5 * 13/2 = 52/10 = 5 2/10 = 5 1/5 cups.
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Make a box and whisker plot that represents the data
Number of chairs in a classroom
30,27,32,25,12,22,20,29,35,35,28
Answers
30,27,32,25,12,22,20,29,35,35,28
rearranging it gives us:
12,20,22,25, 27,28,29,30,32,35,35
lowest value is 12
highest value is 35
median=28
lower quartile 23.5
Upper quartile 31
thus the boxplot will be:
Get Brainliest if you answer What was Hammurabi's most important achievement as the ruler of Babylon? A. He built extensive canals, which carried water to many parts of his kingdom. B. He established permanent armies, which he used to control his empire. C. He appointed governors to rule over conquered cities. D. He was responsible for a codification of laws that influenced later societies.
Answers
Answer:
Its D
Step-by-step explanation:
I did the quiz
If there are 4 math courses, 3 psychology courses, and 5 english courses offered in non-overlapping times so that you could select one of each for your schedule, how many different schedules would be possible?
Answers
[tex]4 \times 3 \times 5 = 60[/tex]
Answer:
there are many different answers that is because there are four different ways to answer this one question.
Step-by-step explanation:
4*3*5=60
4-+5+3=12
1) 435 2) 435 3) 435
A researcher published this survey result: "74% of people would be willing to spend 10% more for energy from a non-polluting source". The survey was conducted by posing the question on a national radio show and asking listeners to call in. 1,200 listeners responded. What is wrong with this survey?
correlation and causality
voluntary response
nonresponse
precise numbers
Answers
What is the value of x, if 105% of 90 is equal to 200% of x?
Answers
From the given equation
105% * 90 = 200% * x
105/100 * 90 = 200/100 * x
105*90 = 200*x
x = 105*90/200
X = 21*9/4
X = 189/4
x = 47.25
Value of x = 47.25 or 47 1/4
What is the value of n for the equation 1n+4n+16n2−6n=4n−6?
Answers
To find the value of 'n' for the given equation, we first combine like terms to get 16n^2 - n = 4n - 6. Then, set the equation to zero to solve for 'n'. The resulting quadratic equation, 16n^2 - 5n + 6 = 0, can be solved by factoring, completing the square, or applying the quadratic formula.
Explanation:The student's equation is 1n + 4n + 16n2 − 6n = 4n − 6. To simplify and solve for n, we need to consolidate like terms and find the value of n that satisfies the equation.
First, we start by adding the like n terms. This gives us:
5n + 16n2 − 6n
Next, we combine the like terms:
16n2 − n (since 5n - 6n gives us -n)
Now, we set the equation equal to the right side of the original equation:
16n2 − n = 4n − 6
To find the value of n, we have to set the equation to zero and solve for n:
16n2 − 5n + 6 = 0
The student must now use techniques such as factoring, completing the square, or the quadratic formula to find the value(s) of n that satisfy the quadratic equation.
Final answer:
The value of n varies in different Chemistry contexts, such as in sequence manipulations, when converting to scientific notation, in electron configurations indicating n = 3 for a 3p electron, and when determining the total number of orbitals for n = 4.
Explanation:
The student question involves determining the value of n (the principal quantum number) in different scenarios within the field of Chemistry. Here are examples covering various cases:
For the expression of n terms leading to 2n², it's concluded by manipulating terms within a sequence.When converting numbers like 60.474 × 10-4 to scientific notation, n becomes more positive by 1 if the value of N decreases.The electron configuration of an atom indicates the last electron added is a 3p electron, which implies n = 3.For n = 4, an atom has s, p, d, and f subshells, leading to a total of 16 orbitals.A survey was conducted at Springdale High to find the number of hours students sleep at night. The histogram shows the data collected in this survey. Which statement about the number of hours students sleep is true? Seventeen students sleep between 6 and 8 hours at night. Seventeen students sleep for 6 hours at night. Ten students sleep for 4 hours at night. Six students sleep between 2 and 4 hours at night. Four students sleep for 10 hours at night.
Answers
James bought $175 in accessories for his video game console. He spent $15 on a new power cord and the rest of his money on 5 new video games. Each video game cost the same amount. Write two equations you could use to find the cost of each video game.
Answers
I believe two equations you can make to represent the question are...
(175-15)/5
and
5x+15=175
The first equation, (175-15)/5, makes you subtract 15 from 175 to get 160. You'll then divide 160 by 5 to get 32.
The second equation, 5x+15=175, makes you subtract 15 from 15 and 175, so you'll equation will look like this...
5x=160
You'll then divide 5x and 160 by 5 to get x=32.
So for both equations, you'll get an answer of 32.
I hope this helps!
The solution is, two equations are: (175-15)/5 and, 5x+15=175
the cost of each video game is 32.
What is equation?An equation is a mathematical statement that is made up of two expressions connected by an equal sign. In its simplest form in algebra, the definition of an equation is a mathematical statement that shows that two mathematical expressions are equal.
Here, we have,
two equations we can make to represent the question are...
(175-15)/5
and
5x+15=175
The first equation, (175-15)/5, makes you subtract 15 from 175 to get 160. You'll then divide 160 by 5 to get 32.
The second equation, 5x+15=175, makes you subtract 15 from 15 and 175, so you'll equation will look like this...
5x=160
You'll then divide 5x and 160 by 5 to get x=32.
So for both equations, we'll get an answer of 32.
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Hey can you please help me posted picture of question
Answers
Reflection along x-axis can be obtained by multiplying the function by -1.
So, G(x) reflected along x-axis will be: - x²
Next the function is shifted down by 3 units. So this can be obtained by subtracting 3 from the function.
So, we can write
F(x) = -x² - 3
Thus, option C is the correct answer
In a string of 100 christmas lights, there is
a.015 chance that a given bulb will fail within the first year of use (if one bulb fails, it does not affect the others). find the approximate probability that six or more bulbs will fail within the first year. (round your answer to 4 decimal places.) probability
Answers
[tex]\mathbb P(X\ge6)=1-\mathbb P(X<6)=1-\displaystyle\sum_{x=0}^5\binom{100}x0.015^x(1-0.015)^{100-x}\approx0.0041[/tex]
The approximate probability that six or more bulbs will fail within the first year is approximately 0.1461, rounded to four decimal places.
To find the approximate probability that six or more bulbs will fail within the first year, we can use the binomial probability formula:
P(X ≥ k) = 1 - P(X < k)
where:
P(X ≥ k) is the probability of getting "k" or more successes (bulbs failing).
P(X < k) is the probability of getting less than "k" successes (bulbs failing).
In this case, "k" is 6 (six or more bulbs failing), and the probability of a bulb failing in the first year is 0.015.
Now, let's calculate the probability of getting less than six failures (k < 6):
P(X < 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
Using the binomial probability formula:
P(X = k) = C(n, k) *[tex]p^k * (1 - p)^{n - k}[/tex]
where:
C(n, k) is the binomial coefficient (n choose k).
n is the number of trials (total bulbs, which is 100 in this case).
p is the probability of success (a bulb failing, which is 0.015).
k is the number of successes (bulbs failing).
Now, calculate P(X < 6):
P(X = 0) = C(100, 0) * 0.015⁰ * (1 - 0.015)^(100 - 0)
P(X = 1) = C(100, 1) * 0.015¹ * (1 - 0.015)^(100 - 1)
P(X = 2) = C(100, 2) * 0.015² * (1 - 0.015)^(100 - 2)
P(X = 3) = C(100, 3) * 0.015³ * (1 - 0.015)^(100 - 3)
P(X = 4) = C(100, 4) * 0.015⁴ * (1 - 0.015)^(100 - 4)
P(X = 5) = C(100, 5) * 0.015⁵ * (1 - 0.015)^(100 - 5)
After calculating each term, add them up:
P(X < 6) ≈ 0.8539
Now, find the probability of getting six or more failures:
P(X ≥ 6) = 1 - P(X < 6)
P(X ≥ 6) ≈ 1 - 0.8539 ≈ 0.1461
So, the approximate probability that six or more bulbs will fail within the first year is approximately 0.1461, rounded to four decimal places.
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What is the output of the function f(p) = 3p – 2 when the input is 2?
Answers
f(2) = 3×2 -2
f(2) = 6 -2 = 4
The output of the function is 4 when the input is 2.
What position in the distribution corresponds to a z-score of z = +2.00?
Answers
A z-score of +200 corresponds to a position that is 2 standard deviations above the mean, and signifies a value within the top 2.5% of a normally distributed data set.
The z-score of z = +2.00 in a distribution indicates that the corresponding value is exactly 2 standard deviations above the mean of the distribution. According to the empirical rule (also known as the 68-95-99.7 rule), approximately 95% of the data falls within two standard deviations of the mean. Therefore, a z-score of +2.00 means the position is within the top 2.5% of the distribution, since it is above the mean and within 95% coverage of the data towards the higher end. This concept is fundamental when assessing data in relation to the normal distribution, and it's widely used to determine how far a particular data point is from the average value in terms of standard deviations.
A bag of rice weights 3.18 lbs find its mass in kilograms
Answers
kaitlin is participating in a 5-day cross country biking challenge. She biked for 55,67,56,and 49 miles on the first four days. How many miles does she need to bike on the last day so that her average (mean) is 55 miles per day?
Answers
To find out how many miles Kaitlin needs to bike on the last day to average 55 miles per day over a 5-day biking challenge, we calculate the total intended miles (55*5 = 275 miles) and subtract the total miles already biked (275 - (55+67+56+49) = 48 miles). Therefore, Kaitlin needs to bike 48 miles on the last day.
Explanation:In the content-loaded problem, Kaitlin is participating in a 5-day cross country biking challenge. She's has already completed biking for four days with distances of 55, 67, 56, and 49 miles respectively. To find out how many miles she needs to bike on the last day for her to average 55 miles per day, we first need to find the total miles she should cover in 5 days.
The total miles intended to be covered in 5 days should be 55 miles/day * 5 days = 275 miles
Then we subtract the total miles covered in the first four days from this. So, 275 miles - (55 miles + 67 miles + 56 miles + 49 miles) = 48 miles
So, Kaitlin needs to bike 48 miles on the last day to achieve an average of 55 miles per day.
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9.
Leola Hain purchases 220 shares of Circuit City stock at $42.75 per share.
Find the annual dividend if the dividend per share is $0.0625.
$43
$55
$9,460
$9,405
Answers
$55 is the answer
The median number of business trips for 6 employees in a certain quarter is 10. The range of number of business trips for those employees that quarter is 9. The fewest business trips could be 0. True, false, or not enough information?
Answers
Answer:
False
Step-by-step explanation:
You want to know if it is true that the fewest number of business trips could be zero if the median of 6 values is 10 and their range is 9.
MedianThe median of 6 numbers is the average of the middle two when they are sorted into increasing order. That means the smallest number that can be the upper end of the range is 10. (In that case, 4 of the 6 values would be 10.)
Since the lower end of the range is 9 less than the upper end, the fewest number of business trips cannot be less than 10-9 = 1.
It is False that the fewest number could be 0.
__
Additional comment
The numbers of trips could be ...
1, 1, 10, 10, 10, 10
This 6-element data set has a median of 10 and a range of 9.
what is the positive solution to the equation 0=⅓x²-3
Answers
3 = (1/3)x²
9 = x²
√9 = 3 = x
The positive solution to the equation is x = 3.
The area of an artist's square canvas can hold 113 square inches of paint. What is the approximate length of one side of the canvas? (Approximate to the nearest hundredth inch.)
10.62 inches
10.63 inches
10.64 inches
10.65 inches
Answers
You'll need a calculator to find this exactly.
Answer:
The approximate length of one side of the canvas is 10.63 inches
Step-by-step explanation:
Given :The area of an artist's square canvas can hold 113 square inches of paint.
To Find :What is the approximate length of one side of the canvas?
Solution:
Canvas is in the shape of square
Area of square = [tex]Side^2[/tex]
Let the side of the canvas be x
Area of canvas = [tex]x^2[/tex]
Since we are given that The area of an artist's square canvas can hold 113 square inches
So, [tex]113=x^2[/tex]
[tex]\sqrt{113}=x[/tex]
[tex]10.630=x[/tex]
So, the approximate length of one side of the canvas is 10.63 inches
Hence Option B is correct.
two cylindrical candles have a scale factor of 3:5. The larger candle has a volume of 40(3.14) cubic inches. Fund the volume of thel the smaller candle. Write your answer in term of (3.14)
Answers
[tex]5 = 40\pi \\ 3 =( (40 \div 5) \times 3)\pi \\ = 24\pi[/tex]
The volume of the smaller candle is 8.64π cubic inches.
How to find the volume
The volume of a cylinder is given by the formula
V = π r² h,
where
r is the radius and
h is the height.
The scale factor is given as 3:5, which means the ratio of the volumes is (3³ : 5³)
Using the ratio, let volume of the the smaller candle. be x
3³ : 5³
x : 40π
cross multiply
x = 40π * 3³ / 5³
x = 40π * 27 / 125
x = 8.64π cubic inches.
he actual weights of 10 sample “1 lb packages” of meat are found to be: sample 1 2 3 4 5 6 7 8 9 10 weight 1.01 1.03 0.98 0.99 1.01 1.02 0.98 1.05 0.97 0.98 What is the mean of the actual weights?
Answers
1.01+1.03+.98+.99+1.01+1.02+.98+1.05+.97+.98
10.02
Now divide by 10 (the amount of samples)
10.02/10
1.002
The Mean of the Data Set is 1.002
Hope this helps!
10.02 / 10= 1.002
Answer: 1.002
What is the quotient when 0.75 is divided by 5?
Answers
(its just 0.75 ÷ 5)
Consider the following sets: R = {x | x is the set of rectangles} P = {x | x is the set of parallelograms} T = {x | x is the set of triangles} I = {x | x is the set of isosceles triangles} E = {x | x is the set of equilateral triangles} S = {x | x is the set of scalene triangles} Which statements are correct? Check all that apply. T is a subset of P. E is a subset of I. S is a subset of T. I ⊂ E T ⊂ E R ⊂ P
Answers
Not true since triangle has three sides but parallelogram has four sides.
"E is a subset of I"
True since equilateral triangles are isosceles triangles with all angles equal.
"S is a subset of T"
True since scalene triangles are still triangle.
"I ⊂ E"
False since there are isosceles triangles those are not equilateral triangles. Namely triangle with angles 20°, 20°, 140°
"T ⊂ E"
False since not all triangles are equilateral. Scalene triangle is one of counterexamples.
"R ⊂ P"
True since rectangles are parallelograms with right angles.
Final answer: E is a subset of I, S is a subset of T, and R ⊂ P.
Hope this helps.
Answer:
The answers are 2, 3, 6
Step-by-step explanation: