Determine the domain and the range of the function.
f = {(2,4),(29,-7),(44, 4), (54, 2)}

The positive intercept y is [tex]\sqrt{ }(3 / 2)[/tex], the negative  y intercept is y= [tex]-\sqrt{ }\left(\frac{3}{2}\right)[/tex], the equation of the tangent line at positive intercept is y=-0.48989x+1.2247,  the equation of the tangent line negative intercept  is y=0.48989x-1.2247.

(a). Here we have to find the positive y intercept

So letting x=0 in the given equation [tex]\frac{1}{(\mathrm{x}-1)^2+\mathrm{y}^2}+\frac{4}{(\mathrm{x}+1)^2+\mathrm{y}^2}=2[/tex]

x=0 implies

[tex]$$\begin{aligned}\frac{1}{(0-1)^2+y^2}+\frac{4}{(0+1)^2+y^2} & =2 \\\frac{1}{1+y^2}+\frac{4}{1+y^2} & =2 \\\frac{5}{1+y^2} & =2 \\5 & =2+2 y^2 \\5-2 & =2 y^2 \\\frac{3}{2} & =y^2 \\\mathrm{y} & =\sqrt{ }\left(\frac{3}{2}\right)\end{aligned}$$[/tex]

Therefore, the positive intercept y is [tex]\sqrt{ }\left(\frac{3}{2}\right)[/tex].

(b). Next we have to find the negative y intercept , the negative y intercept is y= [tex]-\sqrt{ }\left(\frac{3}{2}\right)[/tex]

(c). Next we have to find the equation of the tangent line at [tex]$(0, \sqrt{ }(3 / 2))$[/tex]using implicit differentiation Here

[tex]\frac{1}{(x-1)^2+y^2}+\frac{4}{(x+1)^2+y^2}=2 \\[/tex]

[tex]\left((x-1)^2+y^2\right)^{-1}+4 \times\left((x+1)^2+y^2\right)^{-1}=2[/tex]

Then differentiating using power rule

[tex](-1) \times\left((\mathrm{x}-1)^2+\mathrm{y}^2\right)^{-1-1} \times\left(2(\mathrm{x}-1)+2 \mathrm{y} \times \mathrm{y}^{\prime}\right)+4 \times(-1) \times\left((\mathrm{x}+1)^2+\mathrm{y}^2\right)^{-1-1} \times\left(2(\mathrm{x}+1)+2 \mathrm{yy} \mathrm{y}^{\prime}\right)=0 \\[/tex]

[tex](-1) \times\left((\mathrm{x}-1)^2+\mathrm{y}^2\right)^{-2} \times\left(2(\mathrm{x}-1)+2 \mathrm{y} \times \mathrm{y}^{\prime}\right)+(-4) \times\left((\mathrm{x}+1)^2+\mathrm{y}^2\right)^{-2} \times\left(2(\mathrm{x}+1)+2 \mathrm{yy} \mathrm{y}^{\prime}\right)=0 \\[/tex]

[tex](-1) \times\left((\mathrm{x}-1)^2+\mathrm{y}^2\right)^{-2} \times\left((\mathrm{x}-1)+\mathrm{yy} \mathrm{y}^{\prime}\right)+(-4) \times\left((\mathrm{x}+1)^2+\mathrm{y}^2\right)^{-2} \times((\mathrm{x}+1)+\mathrm{yy})=0[/tex]

Then Substituting x=0 and y= [tex]\sqrt{ }(3 / 2)$[/tex], then

[tex](-1) \times\left(1+\left(\frac{3}{2}\right)\right)^{-2} \times\left(-1+\sqrt{ }\left(\frac{3}{2}\right) \mathrm{y}^{\prime}\right. & =4\left(1+\left(\frac{3}{2}\right)\right)^{-2} \times\left(1+\sqrt{ }\left(\frac{3}{2}\right) \mathrm{y}^{\prime}\right) \\[/tex][tex]\left.-1 \times\left(\frac{2}{5}\right)^2\right) \times\left(-1+\sqrt{ }\left(\frac{3}{2}\right) \mathrm{y}^{\prime}\right) & =4\left(\frac{2}{5}\right)^2 \times\left(1+\sqrt{ }\left(\frac{3}{2}\right) \mathrm{y}^{\prime}\right) \\[/tex]

[tex]-\left(-1+\sqrt{ }\left(\frac{3}{2}\right) \mathrm{y}^{\prime}\right) & =4 \times\left(1+\sqrt{ }\left(\frac{3}{2}\right) \mathrm{y}^{\prime}\right) \\[/tex]

[tex]1-\sqrt{ }\left(\frac{3}{2}\right) \mathrm{y}^{\prime} & =4+4 \sqrt{ }\left(\frac{3}{2}\right) \mathrm{y}^{\prime} \\[/tex][tex]-3 & =5 \sqrt{ }\left(\frac{3}{2}\right) \mathrm{y}^{\prime} \\[/tex]

[tex]\mathrm{y}^{\prime} & =-0.48989[/tex]

Then the equation of the tangent line is y-y1=m(x-x1)

Here X1=0

[tex]& \mathrm{y} 1=\sqrt{ }(3 / 2)=1.2247 \\[/tex]

m=-0.48989

y=-1.2247=-0.48989x

y=-0.48989x+1.2247

(d). Next we have to find the equation of the tangent line at[tex]$(0,-\sqrt{ }(3 / 2))$[/tex] using implicit differentiation

Here [tex]\left((x-1)^2+y^2\right)^{-1}+4 \times\left((x+1)^2+y^2\right)^{-1}=2[/tex]

Then differentiating using power rule

[tex](-1) \times\left((\mathrm{x}-1)^2+\mathrm{y}^2\right)^{-2} \times\left((\mathrm{x}-1)+\mathrm{yy^{ \prime } )}+(-4) \times\left((\mathrm{x}+1)^2+\mathrm{y}^2\right)^{-2} \times\left((\mathrm{x}+1)+\mathrm{y} \mathrm{y}^{\prime}\right)=0\right.[/tex]

Then Substituting x=0 and y=[tex]-\sqrt{ }(3 / 2)$[/tex], then

[tex](-1) \times\left(1+\left(\frac{3}{2}\right)\right)^{-2} \times\left(-1-\sqrt{ }\left(\frac{3}{2}\right) y^{\prime}\right) & =4\left(1+\left(\frac{3}{2}\right)\right)^{-2} \times\left(1-\sqrt{ }\left(\frac{3}{2}\right) y^{\prime}\right) \\[/tex]

[tex](-1) \times\left(-1-\sqrt{ }\left(\frac{3}{2}\right) y^{\prime}\right) & =4 \times\left(1-\sqrt{ }\left(\frac{3}{2}\right) y^{\prime}\right) \\[/tex]

[tex]\left(1+\sqrt{ }\left(\frac{3}{2}\right) y^{\prime}\right) & =4-4 \sqrt{ }\left(\frac{3}{2}\right) y^{\prime} \\[/tex]

[tex]-3 & =-5 \sqrt{ }\left(\frac{3}{2}\right) y^{\prime} \\y^{\prime} & =\frac{3}{5 \sqrt{ }\left(\frac{3}{2}\right)}=0.48989[/tex]

Then the equation of the tangent line is

y+1.2247 =0.48989x

y=0.48989x-1.2247.

For more such questions on intercept

https://brainly.com/question/29113022

#SPJ4

Answer:

{x = -2, y = -1

Step-by-step explanation:

Solve the following system:

{-3 y + 8 x = -13

5 y - 3 x = 1

Choose an equation and a variable to solve for.

In the first equation, look to solve for x:

{-3 y + 8 x = -13

5 y - 3 x = 1

Isolate terms with x to the left-hand side.

Add 3 y to both sides:

{8 x = 3 y - 13

5 y - 3 x = 1

Hint: | Solve for x.

Divide both sides by 8:

{x = (3 y)/8 - 13/8

5 y - 3 x = 1

Perform a substitution.

Substitute x = (3 y)/8 - 13/8 into the second equation:

{x = (3 y)/8 - 13/8

5 y - 3 ((3 y)/8 - 13/8) = 1

Expand the left-hand side of the equation 5 y - 3 ((3 y)/8 - 13/8) = 1.

5 y - 3 ((3 y)/8 - 13/8) = 5 y + (-(9 y)/8 + 39/8) = (31 y)/8 + 39/8:

{x = (3 y)/8 - 13/8

((31 y)/8 + 39/8) = 1

Choose an equation and a variable to solve for.

In the second equation, look to solve for y:

{x = (3 y)/8 - 13/8

(31 y)/8 + 39/8 = 1

Isolate terms with y to the left-hand side.

Subtract 39/8 from both sides:

{x = (3 y)/8 - 13/8

(31 y)/8 = -31/8

Solve for y.

Multiply both sides by 8/31:

{x = (3 y)/8 - 13/8

y = -1

Hint: | Perform a back substitution.

Substitute y = -1 into the first equation:

Answer: {x = -2, y = -1

Answer:

43.96

Step-by-step explanation:

C = 2πr

C = 2 × 3.14 × 7

C = 43.96

Step-by-step explanation:

given in fig :

radius = 7

so,

circumference of the given figure= 2πr

=2×3.14×7

= 43.96

Answer:

-14/3

Step-by-step explanation:

this is a geometric series with r = -7/8

The infinite sum of geometric series with |r| < 1 is

We already have r, but we need to find a1 which is the first term of the series.

To find it, only plug in 1 for n, 10(-7/8) and this = -35/4

Now plug in r and a1 in the formula above:

[tex]\frac{-35/4}{1+7/8}[/tex]

after simplifying it, you will get the infinite sum = -14/3

To answer this question, we need to first multiply the left side of the given equation and write it as the difference of two squares. This can be done by using the difference of squares formula, which states that the difference of two squares can be written as the product of the square of the sum and the square of the difference.

The left side of the given equation can be written as:

(csc0+1)(csc0-1)

We can then apply the difference of squares formula to this expression to get:

(csc0+1)(csc0-1) = (csc0+1)(csc0-1)

Now, we can see that this expression is equivalent to cot 0 using the Pythagorean Identity. This identity states that the sum of the squares of the cosecant and cotangent of an angle is equal to 1. In this case, since (csc0+1)(csc0-1) = cot 0, we can use the Pythagorean Identity to rewrite the left side of the equation as (csc0^2 + cot0^2) = 1, which is equivalent to cot 0.

Therefore, the correct answer is C. Pythagorean Identity

In Gallup political polls, respondents are asked which political party they identify with: independent, democratic, or republican. Nominal level of measurement is used to measure party

Nominal level of measurement is used when assigning a name, label, or category to a particular attribute of an individual, group, or phenomenon. In the case of party affiliation, the Gallup Political Polls ask people to identify themselves as Democrat, Republican, or Independent. These labels are mutually exclusive, and do not imply any particular order. Therefore, the level of measurement used to measure party is nominal.

Learn more about nominal level here

https://brainly.com/question/4823744

#SPJ4

H>50 means H is larger than 50

The crocodile eats the bigger # is a way to remember it.

If the width of a rectangle is two feet less than the length and the perimeter is 52 feet, then the length and width of the rectangle is 14 feet and 12 feet respectively

The width of a rectangle is two feet less than the length

Consider the length of the rectangle as x

Then the width of the rectangle = x - 2

The perimeter of the rectangle = 52 feet

The perimeter of the rectangle = 2(l + w)

Where l is the length and w id the width

Substitute the values in the equation

2(x + x - 2) = 52

2(2x - 2) = 52

2x - 2 = 52/2

2x - 2 = 26

2x = 28

x = 14 feet

Width of the rectangle = 14 - 2

= 12 feet

Therefore, the length and width of the rectangle is 14 feet and 12 feet respectively

Learn more about perimeter here

brainly.com/question/15287805

#SPJ4

π/32 is the area enclosed by the curve r= sin(8θ)

The given curve is polar curve and hence the area of the polar curve is given by:

Let A be the area of the curve so,

A = [tex]\int\limits^a_b {\frac{1}{2} r^2 } \, d\theta[/tex]

where a and b is the boundary at which r=0

so after equation r=0

sin(8θ) =0

=> sin(8θ) =0

=> 8θ = 0,π

=> θ = 0, π/8

so a=0 , b= π/8

now  

   A = [tex]\int\limits^a_b {\frac{1}{2} r^2 } \, d\theta[/tex]    ------(i)

 so   [tex]r^2[/tex] = (sin(8θ))^2

=>  [tex]sin^2[/tex] ( 8θ )

ans we know that

cos(2α) = 1 -  2[tex]sin^2[/tex] α

so   [tex]r^2[/tex]  = (1- cos(16θ) )/2

putting the value of r in the equation (i) we get :-

A = [tex]\int\limits^a_b {\frac{1}{4} *(1-cos(16\alpha ) } \, d\alpha[/tex]

=> 1/4* [tex]\int\limits^a_b {(1-cos(16\alpha ) } \, d\alpha[/tex]

here a=0 and b=π/8

after putting the value and solving the integral

A = π/32

so A is the area enclosed by r=sin(8θ) is π/32

To know more about polar curve click on below link:

https://brainly.com/question/1094340#

#SPJ4

Answer:

1/5

Step-by-step explanation:

of: *

is: =

1/7 * x = 1/35

Multiply by 7 on both sides.

7(1/7x) = 7(1/35)

x = 1/5

Answer:

v=116

u=137

Step-by-step explanation:

hope it's helpful for you

If there are ten five and six year old attending a birthday party when a 30 year old mother walks in to the room with an infant in her arm then the Mean of the ages in the room changes.

The term average refers to any of the measures of central tendency. The arithmetic mean of a set of observed data is defined as being equal to the sum of the numerical values of each and every observation, divided by the total number of observations. suppose there are ten five and six year old attending a birthday party when a 30 year old mother walks into the room with an infant in her arms. The mean of a dataset exists as the sum of all values divided by the total number of values. It's the most generally utilized measure of central tendency and exists often referred to as the “average.”

To learn more about Arithmetic mean please visit:

https://brainly.com/question/23706022

#SPJ4

The number of free throws that adam can expect to make if he attempts 20 free throws is 14.

Probability is a metric used to express the possibility or chance that a particular event will occur. Both proportions ranging from 0 to 1 and percentages ranging from 0% to 100% can be used to describe probabilities.

Given the probability that the person making a free throw is 70%. The question asks to estimate how many of his 20 free throw attempts will be successful.

Then, the number of free throws is calculated as follows,

[tex]\begin{aligned}\text{70\% of 20}&=\frac{70\times20}{100}\\&=14\end{aligned}[/tex]

Then, the number of successful free throws is 14.

To know more about probability:

https://brainly.com/question/28146099

#SPJ4

Answer:

1 minute

because the first elevator has traveled 200 meters and the second elevator goes 200 meters in one minute so they will catch up with the first group in 1 minute.

Answer: The answers is, 1.77777, repeated 7s

Step-by-step explanation: The explanation if you need to show your work is: Start with the goal of (48 books). Subtract the number of books already read (16 books read).  

(48 - 16) = 32 books left to read.

You must then take the other given of 18 weeks left to complete said goal of (48 books) and divide the number of books left (32 books to read) by the amount of time left (18) weeks to complete the desired goal.

(32 / 18) = 1.77777

Therefore, the answer to your question is that Gracie must read 1.777 books per week to complete her goal on time.    

Good Luck, Hope this helps.

Answer:

Below

Step-by-step explanation:

4%  /12  = .33 % per period

  periods per year = 12

  money is compounded each month for 12 years = 144 times

Answer: 12 cups of sugar

Step-by-step explanation:

you can set it up as a ratio

2:4 and this can be simplified to 1:2, so for every cup of flour there is 2 cups of sugar, you could count it by hand but with a ratio of 1:2 you can just multiply by 2 so 6x2=12

Answer:

Step-by-step explanation:

Latisha needs to pay first month's rent ($675), last month's rent ($675), and the security deposit ($325), which totals $1,325.

Latisha needs to pay a total of $1,325 to move into her new apartment. This includes the first month's rent of $675, the last month's rent of $675, and the security deposit of $325. This total amount must be paid before Latisha can move in and enjoy her new place. In addition to the up-front costs for move-in, Latisha will also need to be prepared to pay her rent each month and maintain her security deposit in case there are any damages to the apartment. Paying the move-in costs is the first step to ensure that Latisha can enjoy her new home.

Learn more about deposit here

https://brainly.com/question/11432827

#SPJ4

Answer:

The answer is, rounded to the nearest degree, 51°

The expression 6x-8 represents the perimeter of a shape in terms of the length of one of its sides, while the expression 4(1.5x-2) represents the perimeter of the same shape in terms of the length of its diagonals. Therefore, the expression 4(1.5x-2) provides additional information about the shape that is not contained in the expression 6x-8. Specifically, the expression 4(1.5x-2) tells us that the length of each diagonal of the shape is equal to 1.5 times the length of one of its sides minus 2. This information allows us to determine the lengths of the diagonals of the shape, which is not possible with the information provided by the expression 6x-8.

The measure of m∠XBC=(3p-6)°=(3·47-6)°=135°.

Given data :

m∠XBC = m∠BAC + m∠BCA,

Now if : m∠XBC=(3p-6)°: m∠BAC=(p+4)°: m∠BCA=84°,

then we get the equation :

3p-6=p+4+84

(option 2), 3p-6=p+88

(option 3), 3p-p-6=p-p+88,2p-6=88

(option 4), 2p-6+6=88+6,2p=94

(option 5), p=47.

Then m∠XBC=(3p-6)°=(3·47-6)°=135°.

Learn more about geometric mean theorem refer to :

https://brainly.com/question/10612854

#SPJ1

Answer:

135
answer in picture

Step-by-step explanation:

Answer:

y = 5

Step-by-step explanation:

A line with a slope of 0 is a horizontal line parallel to the x- axis with equation

y = c ( c is the value of the y- coordinates the line passes through )

the line passes through (- 4, 5 ) with y- coordinate 5 , then

y = 5 ← equation of line

Answer: 371.28

I hope that this helps! :)

Answer: 371.28

Step-by-step explanation: hope it helps mark brainlist if helpful

The standard form of the linear equation of a straight line is given as  

2x - y + 17 = 0

What in mathematics is a linear equation?

A linear equation is a first-order (linear) term plus a constant in the algebraic form y=mx+b, where m is the slope and b is the y-intercept.    

                           Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables.

y=mx+c; where y and x stand for the respective axis's; 'm' represents the slope while 'c' represents the constant.

Let the point through which the line passes be P

Then, P= (-6,5)

m= 2

The equation of a straight line as a single point form can be found using the formula:

y-y1=m(x-x1)+c

y-(5) = 2(x-(-6))+c

y -5= 2(x+ 6)+c

Taking LCM

y - 5 =2(x+6)

on cross multiplying we get:

y - 5 = 2x + 12

 y  = 2x + 12 + 5

  2x - y + 17 = 0

Thus, Equation of the line comes out to be    2x - y + 17 = 0

Learn more about  linear equation

brainly.com/question/11897796

#SPJ4

The number of scholars who voted for the play named Wicked will be equal to 48.

This problem can be solved by using the concept of Ratio. It is given that the total number of students are 240. Now, the ratio of Annie to Wicked is equal to 5:2 and ratio of Wicked to Wizard Oz is equal to 2:3. If we assume that the number of people who voted for Annie is 'x', the number of people who voted for Wicked is 'y' and number of people who voted for Wizard Oz is 'z'. Now, we write the expressions as

x/y = 5:2; from this we get x = 5y/2                              ....(1)

y/z = 2:3; from this we get z = 3y/2                              ....(2)

x + y + z = 240                                                               ....(3)

We use the above values of equation (1) and (2) in equation (3) as

5y/2 + y + 3y/2 = 240

4y + y = 240

5y = 240

y = 48

So, the number of students who voted for Wicked will be 48.

Learn more about Ratio at:

brainly.com/question/2914376

#SPJ4

He will make $697.5 in a work week if he sold $4500 worth of merchandise.

Mea Ling earn a weekly salary of $360 which means salary per week is $360.

Then plus a 7.5% commission on sales at a gift shop if  she sold $4,500  worth of merchandise, which means additional.

Divide by 100. Most interest rates are quoted and advertised in terms of a percentage.

7.5%= 0.075

Multiplication In mathematics, multiplication (denoted by ‘ × ’) is a method of finding the product of two or more values.

In arithmetic, multiplication of two numbers represents the repeated addition of one number with respect to another.

0.075×4500= 337.5

Then total she make will be

337.5+360

= $697.5

Therefore, he will make $697.5 in a work week.

For such more questions about mathematical operations:

https://brainly.com/question/30007286

#SPJ4

Answer:

The heart weighs 404 ounces and a human heart compared to that is 303

Step-by-step explanation:

4 x 102 = 404 - 101

The greatest common factor of polynomial  64xyz−36xy + 24x  is 4x

The greatest common factor of a polynomial is a monomial that divides all terms of the polynomial.

The given polynomial is:

64xyz−36xy + 24x

Notice that there are 3 terms:

64xyz

−36xy

24x

To find the greatest common factor, notice that each terms contain x.

Next, find the gcf of 64, 36, 24.

Using prime factorization:

64 = 2⁶

36 = 2² × 3²

24 = 2³ × 3

Hence the gcf (64, 36, 24) = 2² = 4

Therefore the gcf of  64xyz−36xy + 24x  is 4x

Learn more about greatest common factor here:

https://brainly.com/question/22290900

#SPJ4

The solution to the quadratic equation 3x²-1 = 5x by formula method is

A quadratic equation is a second-order polynomial equation in a single variable of x . It is expressed as ax²+bx+c=0. with a ≠ 0 . Because it is a second-order polynomial equation, the fundamental theorem of algebra guarantees that it has at least one solution. The solution may be real or complex.

In the equation 3x²-1= 5x

we put the equation in standard form. i.e 3x²- 5x -1 = 0

therefore a = 3 , b = -5 and c = -1

formula method is a method of solving the root of a quadratic equation. It is given by x = (-b±√b²-4ac)/2a

=(-(-5±√-5²-4×3×-1)/2×3

=(5±√25+12)/6

=(5±√37)/12

x = 5+6.1/12 or

x= 5-6.1/12

therefore x = 11.1/12 or -1.1/12

learn more about quadratic equation from

https://brainly.com/question/1214333

#SPJ1

Answer: it's d.

StartFraction 5 + StartRoot 37 EndRoot Over 6 EndFraction, StartFraction 5 minus StartRoot 37 EndRoot Over 6 EndFraction just took the test