a rational function with at least one vertical asymptote, and a horizontal asymptote. You will describe the characteristics of your function, and give a numerical example of the function at one of the vertical asymptotes.
Answers
ANSWERS
Function:
[tex]h(x)=\frac{x+1}{(x-1)(x+3)}[/tex]Key features:
• Asymptotes: x = 1, x = -3, (vertical), and ,y = 0, (horizontal).
,• Intercepts: (0, -1/3), (-1, 0)
,• Symmetry: none
,• Example of end behavior: as x → 1⁺, y → infinity,, while, as x → 1⁻, y → -infinity
EXPLANATION
The vertical asymptotes of a rational function are given by the zeros of the denominator. For example, the function,
[tex]h(x)=\frac{x+1}{(x-1)(x+3)}[/tex]Has two vertical asymptotes: one at x = 1, and one at x = -3.
The horizontal asymptote is determined by the degrees of the numerator (n) and denominator (m):
• If n < m then the horizontal asymptote is the x-axis
,• If n = m then the horizontal asymptote is y = a/b, where a and b are the leading coefficients of the numerator and denominator, respectively.
,• If n > m then there is no horizontal asymptote
In the given example, the degree of the numerator is 1, while the degree of the denominator is 2. Thus, function h(x) has a horizontal asymptote that is the x-axis.
Now, we have to find the key features for this function:
• y-intercept:, occurs when x = 0
[tex]h(0)=\frac{0+1}{(0-1)(0+3)}=\frac{1}{-3}=-\frac{1}{3}[/tex]Hence, the y-intercept is (0, -1/3)
• x-intercepts:, the x-intercepts are the x-intercepts of the numerator: ,(-1, 0),.
,• The ,asymptotes, are the ones mentioned above: ,x = 1, x = -3, (vertical), and ,y = 0, (horizontal).
,• The ,symmetry, is determined as follows:
[tex]\begin{gathered} Even\text{ }functions:f(-x)=f(x) \\ Odd\text{ }functions:f(-x)=-f(x) \end{gathered}[/tex]If we replace x with -x in function h(x), we will find that the result is neither h(x) nor -h(x) and, therefore this function is neither even nor odd.
Finally, an example of the end behavior of this function around the asymptote x = 1 is that as x → 1⁺, y → infinity, while as x → 1⁻¹, y → -infinity.
If we take values of x greater than x = 1 - so we will approach 1 from the right, we will get values of the function that show an increasing behavior. Let's find h(x) for x = 4, x = 3, and x = 2,
[tex]\begin{gathered} h(4)=\frac{4+1}{(4-1)(4+3)}=\frac{5}{3\cdot7}=\frac{5}{21} \\ \\ h(3)=\frac{3+1}{(3-1)(3+3)}=\frac{4}{2\times6}=\frac{4}{12}=\frac{1}{3} \\ \\ h(2)=\frac{2+1}{(2-1)(2+3)}=\frac{3}{1\times5}=\frac{3}{5} \end{gathered}[/tex]And we will find the opposite behavior for values that are less than 1 - but greater than -1 because there the function has a zero and its behavior could change,
[tex]\begin{gathered} h(0)=\frac{0+1}{(0-1)(0+3)}=\frac{1}{-1\times3}=-\frac{1}{3} \\ \\ h(\frac{1}{2})=\frac{\frac{1}{2}+1}{(\frac{1}{2}-1)(\frac{1}{2}+3)}=\frac{\frac{3}{2}}{-\frac{1}{2}\times\frac{7}{2}}=\frac{\frac{3}{2}}{-\frac{7}{4}}=-\frac{3\cdot4}{2\cdot7}=-\frac{6}{7} \end{gathered}[/tex]The graph of the function and its key features is,
Related Questions
DEF~△VXW.244FED122WXVWhat is the similarity ratio of △DEF to △VXW?Simplify your answer and write it as a proper fraction, improper fraction, or whole number.
Answers
We are given two triangles. We notice that each corresponding angle is equal, therefore, by Angle Angle Angle (AAA) theorem the triangles are similar. This means that each corresponding side is at the same ratio. That ratio is called the similarity ratio and it is obtained by finding the quotient between any two corresponding sides, like this:
[tex]r=\frac{DF}{VW}=\frac{DE}{VX}=\frac{FE}{WX}[/tex]Where "r" is the similarity ratio. Now, we substitute the sides:
[tex]r=\frac{4}{2}=\frac{4}{2}=\frac{2}{1}=2[/tex]Therefore, the similarity ratio is 2.
ok heres my problem,on average, a refrigerator door is opened 68 times each day.Len has 2 refrigerators in his house.based on this average,about how many times ina 1 week period are the refrigerator doors opened?
Answers
68 times / day
ok
If he open only one refrigerator per day
68 x 7 = 476
He opens the refrigerator 476 times per week
But he has 2 refrigerators
476 x 2 = 952 times
Result, Len open the doors of both refreigerators 952 times per week
Done
Do you have any question?
the table below shows the length of time a plumber takes on a job and price he charges. what does the y-intercept represent?
Answers
The data that represents the y-intercept in a table is always the output/result. In the given table, it shows the cost that the plumber charges after the given hours of work.
Thus, the y-intercept in the data is the total cost of repairs.
The answer is letter B.
5 centimeters by 3 centimetres and a height of 2 centimetres
Answers
firstly you have to calculate the volume of the rectangular prism
volume = base area x height
since it is a rectangular prism
then the area = length x breath
length = 5cm , breath = 3cm
therefore
[tex]\begin{gathered} \text{Area = l }\times b \\ =\text{ 5 }\times3 \\ =15cm^2 \end{gathered}[/tex]volume = base area x height
volume = 15 x 2
[tex]\text{volume = 15}\times2=30cm^3[/tex][tex]^{}\text{thus, the cameron fills }\frac{1}{2}cm^3[/tex]so the amount of the cameron fills that can fill up the prism is
[tex]\frac{30}{\frac{1}{2}}\text{ = 30 }\times\frac{2}{1}=60cm^3[/tex]the answer is D
Quadrilateral BCDE is similar to quadrilateral FGHI. Find the measure of side HI. Round your answer to the nearest tenth if necessary
Answers
we have:
[tex]\frac{HI}{DE}=\frac{IF}{EB}[/tex]so
[tex]\begin{gathered} \frac{HI}{23}=\frac{59}{14} \\ 14HI=59\cdot23 \\ 14\cdot HI=1357 \\ HI=\frac{1357}{14} \\ HI=96.9 \end{gathered}[/tex]answer: HI = 96.9
Subtract. x^2−x+3/x^2+2x−8 − x^2−3x−5/x^2+2x−8
Answers
The value after subtraction will be;
⇒ 2 / (x - 2)
What is mean by Subtraction?
Subtraction in mathematics means that is taking something away from a group or number of objects. When you subtract, what is left in the group becomes less.
Given that;
The expression is,
⇒ (x² - 3x - 5) / (x² + 2x - 8) - (x² - 3x - 5) / (x² + 2x - 8)
Now,
Subtract the expression as;
The expression is,
⇒ (x² - x + 3) / (x² + 2x - 8) - (x² - 3x - 5) / (x² + 2x - 8)
⇒ (x² - x + 3) - (x² - 3x - 5) / (x² + 2x - 8)
⇒ x² - x + 3 - x² + 3x + 5 / (x² + 2x - 8)
⇒ 2x + 8 / (x² + (4-2)x - 8)
⇒ 2x + 8 / (x² + 4x - 2x - 8)
⇒ 2x + 8 / (x (x + 4) - 2 (x + 4))
⇒ 2(x + 4) / (x + 4) (x- 2)
⇒ 2 / (x - 2)
Thus, The value after subtraction will be;
⇒ 2 / (x - 2)
Learn more about the subtraction visit:
https://brainly.com/question/28467694
#SPJ1
Triangle - Interior Angles Find the measure of the indicated angle in each triangle. 3 27 311>P 26 A ma m2Q= 1 s 스 minta mothed
Answers
Answer: We are goinf to sovle triangle Number-03:
[tex]m\angle Q=?[/tex]We know that the sum of angles in a triangle is 180 degrees, therefore we can do the following:
[tex]\begin{gathered} m\angle Q+30+31=180^{\circ} \\ \therefore\rightarrow \\ m\angle Q=180^{\circ}-61=119^{\circ} \\ \therefore\rightarrow \\ m\angle Q=119^{\circ} \end{gathered}[/tex]This is the unknown angle that we were interested in.!
Solve forx: 3x - 5 = 2x + 6.1-111-11
Answers
To solve the equation we need to isolate the "x" variable on the left side. This is done step-by-step below:
3x - 5 = 2x + 6
3x - 2x -5 = 6
x = 6 + 5
x = 11
You have $500,000 saved for retirement. Your account earns 8% interest. How much will you be able to pullout each month, if you want to be able to take withdrawals for 15 years?$
Answers
The rule of the payout annuity is
[tex]P=\frac{d(1-(1+\frac{r}{n})^{-nt})}{\frac{r}{n}}[/tex]P is the initial amount
d is regular withdrawals
r is the annual rate in decimal
n is the number of periods in a year
t is the time
Since you have $500 000 saved, then
P = 500000
Since the interest is 8%, then
r = 8/100 = 0.08
Since the time is 15 years, then
t = 15
Since you want the monthly amount, then
n = 12
Substitute them in the rule to find d
[tex]\begin{gathered} 500000=\frac{d(1-(1+\frac{0.08}{12})^{-12(15)})}{\frac{0.08}{12}} \\ 500000(\frac{0.08}{12})=d(1-(\frac{151}{150})^{-180}) \\ \frac{10000}{3}=d(1-(\frac{151}{150})^{-180}) \\ \frac{\frac{10000}{3}}{(1-(\frac{151}{150})^{-180})}=d \\ 4778.260422=d \end{gathered}[/tex]Then you will be able to pull $4778.260422 each month
Hello,Can you please help me with question# 11 ? it is express each sum using summation notation. Use 'i' as the index of th sum
Answers
Given:
The sum of terms
[tex]4^3+5^3+6^3..........+13^3[/tex]Required:
Find sum.
Explanation:
We know sum of cube of first n terms of natural numbers
[tex]\sum_{n\mathop{=}1}^{\infty}n^3=[\frac{n(n+1)}{2}]^2[/tex]Now,
[tex]\begin{gathered} =(1^3+2^3+....+13^3)-(1^3+2^3+3^3) \\ =[\frac{13(13+1)}{2}]^2-36 \\ =8281-36 \\ =8245 \end{gathered}[/tex]Answer:
The sum of terms is 8245.
There was a survey taken to see which types of pets people prefer. Out of 11 participents, 5 said they prefer dogs, 4 said they prefer cats, and 3 said they prefer birds. What is the percentages of people that prefer dogs, cats, and birds?
Answers
To find the percentages of people that prefer dogs, cats or birds, divide the corresponding amount of people that likes each pet by the total amount of people in the survey, and then multiply that quantity by 100.
Since there are 11 participants, we should divide each quantity by 11.
Dogs:
There are 5 people who prefer dogs. The percentage is:
[tex]\frac{5}{11}\times100\text{ \%}=45.4545\ldots\text{ \%}[/tex]Cats:
There are 4 people who prefer cats. The percentage is:
[tex]\frac{4}{11}\times100\text{ \%=36.3636}\ldots\text{ \%}[/tex]Birds:
There are 3 people who prefer birds. The percentage is:
[tex]\frac{3}{11}\times100\text{ \%=27.2727}\ldots\text{ \%}[/tex]1. Find Each Right triangles missing length. If necessary, round to thenearest tenth,5 pointsleg =8 cm, leg= 21 cm
Answers
We have a right triangle, for which we know the two legs.
We can calculate the hypotenuse H by applying the Pythagorean theorem:
[tex]\begin{gathered} H^2=L^2_1+L^2_2=8^2+21^2=64+441=505 \\ H=\sqrt[]{505}\approx22.5 \end{gathered}[/tex]The missing length (the hypotenuse) is approximately 22.5 cm.
Consider the function f defined by:f(x)=4x-12 FIND: The value of f at x=-7The value of f when x=1
Answers
Answer:
• f(-7)=-40
,• f(1)=-8
Explanation:
Given the function f(x) defined below:
[tex]f(x)=4x-12[/tex](a)The value of f at x=-7
When x=-7
Substitute -7 for x:
[tex]\begin{gathered} f(-7)=4(-7)-12 \\ =-28-12 \\ =-40 \end{gathered}[/tex]The value of f at x=-7 is -40.
(b)The value of f when x=1
When x=1
Substitute 1 for x:
[tex]\begin{gathered} f(1)=4(1)-12 \\ =4-12 \\ =-8 \end{gathered}[/tex]The value of f at x=1 is -8.
can you please help me?
Answers
Answer : The linear function of the graph when move down 3 units is y = x - 3
The graph shift vertically because it is down 3 units
The standard equation of a linear function is
y = mx + b
Where b = intercept and m = slope
Move 3 unit down means b = -3
Hence, y = x-3
Answer is y = x - 3
You will complete the following question on your own paper. Make sure to show ALL work including a picture you draw. He 2 A point on the ground is 50 feet from my house. The angle of elevation to the top of the house is 48º. Find the height of the house to the nearest tenth. Finis the following template: "Last Name First Name Assignment
Answers
ANSWER
The height of the house is 55.5 feet
EXPLANATION
Since this forms a right triangle, we can use the tangent of the elevation angle to the top of the house to find its height - because we know the lenght of the adjacent side and we want to know the lenght of the opposite side:
[tex]\begin{gathered} \tan 48º=\frac{h}{50} \\ h=50\tan 48º \\ h=55.53062574\approx55.5\text{ feet} \end{gathered}[/tex]6.f(x) = 2x + 3x +1g(x)=7X - 2x+7XFind h(x) = f(x) - gix)A.h(x) = -7% -5€ -5x-1B.h --- 5 -- |318
Answers
rearrange the terms and simplify
[tex]\begin{gathered} h(x)=-7x^3-7x^2+2x^2+3x+2x+1 \\ =-7x^3-5x^2+5x+1 \end{gathered}[/tex]The right option is A
3 Step Problem: Erik is building a cubby bookshelf, that is, a bookshelf divided into storage holes (cubbies) instead of shelves. He wants the height of the bookshelf to be x^2 - 2x - 3 and the width to be x^2 + 4x + 3. Each cubby hole in the bookshelf will have a height of x + 3 and width of x - 3.STEP 1 of 3: Write a rational expression to determine how many cubbies high the book shelf will be.
Answers
Step 1:
In order to determine the number of cubbies high, we just need to divide the total height x² - 2x - 3 by the height of one cubby x + 3:
x² divided by x: x
x multiplied by (x + 3): x² + 3x
x² - 2x - 3 minus x² + 3x: -5x - 3
-5x divided by x: -5
-5 multiplied by (x + 3): -5x - 15
-5x - 3 minus -5x - 15: 12
The division doesn't have remainder 0, so let's write the division as a fraction:
[tex]\text{number of cubbies}=\frac{x^2-2x-3}{x+3}[/tex]3.5 feet3 feet3 feet2 feet4.5 feet
Answers
The shape in the image is a trapezium. The area of a trapezium is given by:
[tex]A_{trapezium}=\text{ }\frac{1}{2}\text{ (a + b)h}[/tex]From the image, a= 3.5 feet, b= 4.5 feet and h = 2 feet.
Thus, we have:
[tex]\begin{gathered} A_{trapezium}=\text{ }\frac{1}{2}\text{ ( 3.5 + 4.5) 2} \\ A_{trapezium}=8ft^2 \end{gathered}[/tex]Hence, the area of the figure is 8 square feet
1 point It takes 500 packing peanuts to fill a box that is 3 inches x 4 inches x 5 inches. How many peanuts would it take to fill a box that is 6 inches x 8 inches x 10 inches? 1,000 packing peanuts 2,000 packing peanuts 4,000 packing peanuts 8,000 packing peanuts
Answers
Well, this is a volume and ratio problem. In a box the volume is given by width times length times height, so the volume of the first box is 3*4*5=60 cubic inches.
In the second box, the volume is 6*8*10=480 cubic inches.
[tex]\frac{480}{60}=\frac{8}{1}=8[/tex]This means that in the big box you could place 8 peanuts for every peanut placed in the small box. If you need 500 peanuts to fill the small box, then for the big box:
[tex]500\cdot8=4000[/tex]So, you need
what is the slope and y-intercept of negative 3x + 5y equals -15
Answers
-3x + 5y = -15
The general form of a line using slope and y intercept is: y = mx + b
where m is the slope and b is the y intercept
So we have to write the original equation in this form:
-3x + 5y = -15
5y = 3x - 15
y = (3/5)x - 15/5
y = (3/5)x - 3
In this case, m= 3/5 and b = -3
Therefore m = 3/5
Therefore the y intercept (when x = 0) is -3
Answer:
slope is 3/5
y intercept -3
A rectangular paperboard measuring 32 in long and 24 in wide has a semicircle cut out of it, as shown below.Find the area of the paperboard that remains. Use the value 3.14 for π, and do not round your answer. Be sure to include the correct unit in your answer.
Answers
We need to subtract the area of the semicircle that was cut out of the paperboard from its original area.
The original area of the paperboard was the area of a rectangle 32in long and 24 in wide:
[tex]32in\cdot24in=768in^{2}[/tex]And the area of the semicircle, noticing that its radius r is 24in/2, is given by:
[tex]\frac{\pi r^2}{2}=\frac{3.14\cdot(12in)^{2}}{2}=\frac{3.14\cdot144in^2}{2}=226.08in^2[/tex]Thus, the area of the paperboard that remains is:
[tex]768in^2-226.08in^2=(768-226.08)in^{2}=541.92in^{2}[/tex]Therefore, the answer is 541.92 in².
What is the solution to the equation 3^x = 10?
Answers
Start by applying the log on both sides wi
Write the quadratic equation in Vertex form with vertex (4 8) and passing through the origin.
Answers
Vertex = (4,8)
Passing through the origin = (0,0)
Vertex form:
y= a (x-h)^2+k
(h,k) is the vertex:
y= a (x-4)^2+8
Replace the (x,y ) by the origin coordinates
Solve for a
0= a (0-4)^2+8
-8 = a(-4)^2
-8 = a 16
-8/16 = a
-1/2 = a
y=-1/2 (x-4)^2+8
Multiply4V3 * 10V12 * V6*V2Enter your answer, in simplest radical form, in the box.
Answers
Given:
[tex]4\sqrt[]{3}\cdot10\sqrt[]{12}\cdot\sqrt[]{6}\cdot\sqrt[]{2}[/tex]Simplify the expression.
[tex]\begin{gathered} 4\sqrt[]{3}\cdot10\sqrt[]{12}\cdot\sqrt[]{6}\cdot\sqrt[]{2} \\ =4\sqrt[]{3}\cdot10\sqrt[]{4\times3}\cdot\sqrt[]{3\times2}\cdot\sqrt[]{2} \\ =4\sqrt[]{3}\cdot10(\sqrt[]{2^2})\sqrt[]{3}\cdot\sqrt[]{2}\cdot\sqrt[]{3}\cdot\sqrt[]{2} \\ =4\sqrt[]{3}\cdot10(2)\sqrt[]{3}\cdot2\sqrt[]{3} \\ =(4\times20\times2)(\sqrt[]{3})^2\cdot\sqrt[]{3} \\ =480\sqrt[]{3} \end{gathered}[/tex]Answer:
[tex]4\sqrt[]{3}\cdot10\sqrt[]{12}\cdot\sqrt[]{6}\cdot\sqrt[]{2}=480\sqrt[]{3}[/tex]Joshua is going to invest $9,000 and leave it in an account for 5 years. Assuming theinterest is compounded continuously, what interest rate, to the nearest tenth of apercent, would be required in order for Joshua to end up with $12,500?
Answers
Let r be the percent annual interest rate of the account. Since $9000 are left for 5 years, for an outcome of $12,500, then:
[tex]9000\times(1+\frac{r}{100})^5=12,500[/tex]Divide both sides by 9000:
[tex](1+\frac{r}{100})^5=\frac{12500}{9000}=\frac{25}{18}[/tex]Take the 5th root to both sides:
[tex]\begin{gathered} 1+\frac{r}{100}=\sqrt[5]{\frac{25}{18}} \\ \Rightarrow\frac{r}{100}=\sqrt[5]{\frac{25}{18}}-1 \\ \Rightarrow r=100(\sqrt[5]{\frac{25}{18}}-1) \end{gathered}[/tex]Use a calculator to find the decimal expression for r:
[tex]r=6.790716585\ldots[/tex]Therefore, to the nearest tenth:
[tex]r=6.8[/tex]This means that Joshua would need to invest his money on a 6.8% annual interest account.
Yolanda is preparing a liquid fertilizer that she will use on her lawn. She mixes 4 tablespoons of liquid fertilizer with 6 gallons of water.Using this ratio, how many gallons of water should be mixed with each tablespoon of liquid fertilizer?
Answers
Given data:
The given amount of liquid fertilizer is f=4.
The given water is w=6 gallons.
4 tablespoons= 6 gallons of water
1 tablespoon= 1.5 gallons of water.
Thus, 1.5 gallons of water is mix with each tablespoon.
I need help to solve this I don't quite understand how to do these
Answers
Recall the definition of the sine of an angle in a right triangle:
[tex]\sin (A)=\frac{\text{Side opposite to A}}{\text{ Hypotenuse}}[/tex]In the given diagram, the length of the side opposite to the angle of 53° has a length of 10, while the length of the hypotenuse is x. Then:
[tex]\sin (53)=\frac{10}{x}[/tex]Isolate x from the equation:
[tex]x=\frac{10}{\sin (53)}[/tex]Use a calculator to find the value of the expression for x:
[tex]\begin{gathered} x=12.52135658\ldots \\ \Rightarrow x\approx12.5 \end{gathered}[/tex]1 Evaluate 0.1m + 8 – 12n when m = 30 and n = 1/4.
Answers
Answer:
8
Step-by-step explanation:
0.1(30)+8-12(0.25)
3+8-3
=8
a basketball court is 94 ft long and 50 ft wide Ryan used long steps to estimate the length of the Court as 93 ft and a width as 48 what is the percent error of Ryan's measure area round to your nearest hundred
Answers
5.02%
Explanation:Percentage error formula = |(approximate value - Exact value)|/(exact value) × 100
Area of the court = length × width
length = 94 ft
width = 50 ft
Exact area = 94ft × 50 ft
Exact area = 4700 ft²
length = 93 ft
width = 48 ft
Approximate area = 93 ft × 48 ft
Approximate area = 4464 ft²
[tex]\begin{gathered} \text{percent error = }\frac{|4464\text{ - 4700|}}{4700}\times\text{ 100} \\ \text{percent error = }\frac{|-236|}{4700}\times100 \\ \text{percent error = }\frac{236}{4700}\times100 \end{gathered}[/tex][tex]\begin{gathered} \text{Percent error = }0.0502\text{ }\times\text{ 100} \\ \text{Percent error = 5.02\%} \end{gathered}[/tex]The angle between 0 degrees and 60 degrees that is coterminal with the 1993 angle is degrees.Please show work neatly
Answers
Given:
[tex]1993^0[/tex]To Determine: The coterminal angle of the given angle
Solution
Coterminal angles are angles in standard position (angles with the initial side on the positive x -axis) that have a common terminal side. For example 30° , −330° and 390° are all coterminal
[tex]\begin{gathered} 1993^0-360^0=1633^0 \\ 1633^0-360^0=1273^0 \\ 1273^0-360^0=913^0 \\ 913^0-360^0=553^0 \\ 553^0-360^0=193^0 \end{gathered}[/tex]Hence, the angle that coterminal with 1993 degrees is 193⁰