Find the inverse for each relation: 4 points each



1. {(1,‐2), (2, 3),(3, ‐3),(4, 2)}



2. {(4,2),(5,1),(6,0),(7,‐1)}



Find an equation for the inverse for each of the following relations.



3. Y=-8x+3



4. Y=2/3x-5



5. Y=1/2x+10



6. Y=(x-3)^2



Verify that f and g are inverse functions.



7. F(x)=5x+2;g(x)=(x-2)/5



8. F(x)=1/2x-7;g(x)=2x+14

Answer:

Step-by-step explanation:

Let's use the method of setting up a system of equations to solve this problem.

Let x be the number of pints of the first type of fruit drink (30% pure), and y be the number of pints of the second type of fruit drink (55% pure). We want to find the values of x and y that will produce 60 pints of a mixture that is 40% pure.

We can start by setting up two equations based on the information given:

Equation 1: x + y = 60 (since we want to produce 60 pints of the mixture)

Equation 2: 0.3x + 0.55y = 0.4(60) (since we want the mixture to be 40% pure)

Simplifying Equation 2, we get:

0.3x + 0.55y = 24

Now we have a system of two equations with two unknowns:

x + y = 60

0.3x + 0.55y = 24

We can solve this system using substitution or elimination. Here, we'll use substitution:

Solving Equation 1 for x, we get x = 60 - y. Substituting this expression for x in Equation 2, we get:

0.3(60 - y) + 0.55y = 24

Expanding and simplifying, we get:

18 - 0.3y + 0.55y = 24

Combining like terms, we get:

0.25y = 6

Dividing by 0.25, we get:

y = 24

Substituting this value of y back into x + y = 60, we get:

x + 24 = 60

Solving for x, we get:

x = 36

Therefore, we need 36 pints of the 30% pure fruit drink and 24 pints of the 55% pure fruit drink to make 60 pints of a mixture that is 40% pure.

a = 5

b = 4

c= 0

Therefore, the equation for graph C is Y = a ^b + c

Y = 5 ^4 + 0

A graph is described as a diagram showing the relation between variable quantities, typically of two variables, each measured along one of a pair of axes at right angles.

Graphs are a popular tool for graphically illuminating data relationships. A graph serves the purpose of presenting data that are either too numerous or complex to be properly described in the text while taking up less room.

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2x²t + 7xy

    To simplify, we will combine like terms.

    Given:

5xy - x²t + 2xy + 3x²t

    Reorder like terms:

5xy + 2xy + 3x²t - x²t

    Combine like terms:

    ➜ 5 + 2 = 7

    ➜ 3 - 1 = 2

7xy + 2x²t

    Reorder by degree:

2x²t + 7xy

The theoretical probability of rolling a one or two is 3/5 or 0.6 as a decimal.

The theoretical probability of an event happening is the number of favorable outcomes divided by the total number of possible outcomes.

In this case, Alexandria rolled the number cube 60 times and recorded her results in the table.

Looking at the table, we can see that the number 1 came up 16 times and the number 2 came up 20 times.

So the number of favorable outcomes is 16 + 20 = 36.

And the total number of possible outcomes is 60.

Therefore, the theoretical probability of rolling a one or two is:

P(1 or 2) = favorable outcomes/total outcomes = 36/60

Simplifying the fraction, we get:

P(1 or 2) = 3/5

So the theoretical probability of rolling a one or two is 3/5 or 0.6 as a decimal.

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Answer:

y = 1 + 2√7 or y = 1 - 2√7

Step-by-step explanation:

To complete the square, you need to add and subtract the square of half of the coefficient of the y term.

First, you can factor out a 1 from the y^2 - 2y term:

y^2 - 2y - 27 = 0

y^2 - 2y = 27

Next, take half of the coefficient of the y term (-2/2 = -1) and square it (1):

y^2 - 2y + 1 - 1 = 27

The "+1 -1" doesn't change the value of the equation, it's just a way to add 0 to the equation so we can complete the square.

Now you can rewrite the left side as a perfect square:

(y - 1)^2 - 28 = 0

Add 28 to both sides:

(y - 1)^2 = 28

Take the square root of both sides (remembering to include both positive and negative square roots):

y - 1 = ±√28

y = 1 ± 2√7

So the solutions are:

y = 1 + 2√7

y = 1 - 2√7

Answer: y=± 2√7+1

6.29

Step-by-step explanation: Use the formula

(b/2)^2  in order to create a new term. Solve for y by using this term to complete the square.

If f'(x) = 8x³ + 12x + 2 and f(1) = -4, f(-1) is equal to -18.

Given that f'(x) = 8x³ + 12x + 2, we can find the original function f(x) by integrating f'(x) with respect to x:

f(x) = 2x⁴ + 6x² + 2x + C, where C is an arbitrary constant.

We can then use the given initial condition f(1) = -4 to solve for C:

f(1) = 2(1)⁴ + 6(1)² + 2(1) + C = -4

Simplifying, we get:

C = -16

Therefore, the function f(x) is:

f(x) = 2x⁴ + 6x² + 2x - 16

To find f(-1), we substitute x = -1 into the expression for f(x):

f(-1) = 2(-1)⁴ + 6(-1)² + 2(-1) - 16 = -18

Thus, f(-1) equals -18.

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The position of the particle at time t = 1 is x = 4 units.

To find the position of the particle at time t = 1, we need to integrate the given velocity function v(t) with respect to time from 0 to 1:

x(t) = ∫v(t)dt (from t = 0 to t = 1)

= ∫([tex]3t^2[/tex]+ 6t)dt (from t = 0 to t = 1)

= ([tex]t^3 + 3t^2[/tex]) (from t = 0 to t = 1)

[tex]= (1^3 + 3(1^2)) - (0^3 + 3(0^2))[/tex]

= 4

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The new price per share is $111.48.

Stock price is the current market value of a share of a publicly traded company's stock. It represents the price at which buyers and sellers agree to trade shares in the open market. Stock prices are determined by a variety of factors, including the company's financial performance, industry trends, and overall market conditions. The stock price is often used as an indicator of investor sentiment about the company and its future prospects.

To find the new price per share, follow these steps:

1. Note the initial stock price: $109.99
2. Note the increase in stock price: $1.49
3. Add the increase to the initial stock price: $109.99 + $1.49 = $111.48

The new price per share is $111.48.

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Answer:

Step-by-step explanation:

(d) Probability of at least one major earthquake in the next 49 years is 0.9884, and (e) the probability of no major earthquakes in the next 49 years is 0.0116.

We will use the Poisson distribution to compute the probabilities. First, we need to find the value of λ (average number of earthquakes in a given time period).

(d) Compute the probability of at least one major earthquake in the next 49 years:

1. Calculate λ for 49 years:

(2.0 earthquakes / 22 years) * 49 years = 4.45 (rounded to the nearest hundredth)

2. Compute the probability of no major earthquakes (P(0)) in the next 49 years using Poisson distribution formula:

P(0) = (e^(-λ) * (λ^0)) / 0! = (e^(-4.45) * (4.45^0)) / 1 = 0.0116 (rounded to 4 decimal places)

3. Compute the probability of at least one major earthquake:

P(at least 1) = 1 - P(0) = 1 - 0.0116 = 0.9884

So, the probability of at least one major earthquake in the next 49 years is 0.9884 (rounded to 4 decimal places).

(e) Compute the probability of no major earthquakes in the next 49 years:

As calculated in step (d), the probability of no major earthquakes in the next 49 years is 0.0116 (rounded to 4 decimal places).

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Adding another 12 to the data set would increase the sum of the values by 12, resulting in a new sum of 239. To find the new mean, we divide the new sum by the total number of values in the set, which is now 8. So the new mean would be 29.875, which is slightly lower than the original mean of 31.6.

To find the new median, we first need to rearrange the values in ascending order: 12, 18, 26, 32, 41, 46, 46, 12. Since there are now an even number of values, we take the average of the middle two, which in this case is (26 + 32) / 2 = 29. So the new median would be 29, which is lower than the original median of 32.

In summary, adding another 12 to the data set would slightly decrease the mean and lower the median.

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The amount of interest paid for missing the payment in the eight month would be C. $87.28

First, find the total cost of the computer including the sales price :

= 875 x ( 1 + 0. 05 )

= 875 x 1. 05

= $ 918. 75

Then, find the monthly rate of interest :

= 14. 25 / 12

= 1. 1875 %

This leads to an eight month interest rate of :

= 8 x 1. 1875 %

= 9. 5 %

The interest to be paid:

= 9. 5 % x 918. 75

= $ 87. 28

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Full question is:

You purchase a computer for $875.00 plus 5% sales tax. You decide to finance it through the store's 0% program for 12 months. The terms state that you must pay $100.00/month and that if you miss a payment, you will be assessed a late fee of $39.00 plus the interest accrued to that point at a 14.25% APR. If you miss a payment in the eighth month, how much interest will you be charged?. A. $83.13. B. $16.63. C. $87.28. D. 58.19

According to the information, we can infer that the correct order for these number is -1.5, -0.75, -0.75, 1.25, 3.25.

To plot the numbers on a number line, we need to arrange them in increasing order. Here is the organized list of the numbers:

On the number line, we can mark -1.5 first and then move to the right to mark -0.75 twice (since it appears twice in the list), then 1.25, and finally 3.25 . On the number line, we can see that -1.5 is the smallest number and 3.25 is the largest number.

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x²y' - 2xy' + (x² + 2)y = 0 is the associated homogeneous equation with non-homogeneous ODE x²yⁿ - 2xy' + (x² +2)y = 3x³.

It should be noted that the equation is same as the variable coefficient linear second order non-homogeneous ODE but just the right side zero.

The complementary solutions or homogeneous solutions to this homogeneous equation serve as the foundation for the space of all solutions to the non-homogeneous equation.

By assuming that y has the form y(x) = xr and substituting this into the homogeneous equation to create a characteristic equation, we can determine the complementary solutions. We may find the values of r that correspond to solutions of the type y(x) = xr by looking at the characteristic equation's roots.

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The Rounding to nearest hundredth, we get S₅ ≈ -12.25

The formula for a Riemann sum with n subintervals is:

[tex]S_n[/tex]= ∑ᵢ₌₁ⁿ f(cᵢ) Δx,

where Δx = (b - a)/n is the width of each subinterval and cᵢ is a point in the i-th subinterval. The value of cᵢ can be chosen arbitrarily, but here we are given specific values for c₁, c₂, c₃, c₄, and c₅.

In this problem, we have:

f(x) = 64 - x²

[a, b] = (-8, -3]

n = 5

Δx = (b - a)/n = (-3 - (-8))/5 = 1

Therefore, the width of each subinterval is 1.

The Riemann sum S₅ is:

S₅ = f(c₁) Δx + f(c₂) Δx + f(c₃) Δx + f(c₄) Δx + f(c₅) Δx

Substituting the given values for c₁, c₂, c₃, c₄, and c₅, we get:

S₅ = f(-7.5) + f(-6.5) + f(-5.5) + f(-4.5) + f(-3.5)

where f(x) = 64 - x².

Evaluating each term, we get:

f(-7.5) = 64 - (-7.5)² = 17.75

f(-6.5) = 64 - (-6.5)² = 5.75

f(-5.5) = 64 - (-5.5)² = -2.75

f(-4.5) = 64 - (-4.5)² = -12.25

f(-3.5) = 64 - (-3.5)² = -20.75

Therefore,

S₅ = 17.75(1) + 5.75(1) - 2.75(1) - 12.25(1) - 20.75(1) = -12.25.

Rounding to the nearest hundredth, we get S₅ ≈ -12.25.

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The osmotic pressure for a 4.50% by mass aqueous solution of glucose (C6H12O6) at 300 K is 0.616 atm.

We can use the equation:

π = MRT

where:

π = osmotic pressure

M = molarity of the solution

R = gas constant

T = temperature in Kelvin

We must translate the proportion by mass to molarity in order to determine the molarity of the glucose solution. Glucose (C6H12O6) has a molecular weight of 180 g/mol.

So, for a 4.50% by mass solution of glucose, we have:

4.50 g glucose / 100 g solution = (4.50 g glucose / 180 g/mol) / (Molarity of solution)

Solving for molarity, we get:

Molarity of solution = 0.025 mol/L

Now we can plug in the values into the equation for osmotic pressure:

π = (0.025 mol/L) * (0.0821 L atm / mol K) * (300 K)

π = 0.616 atm

Therefore, the osmotic pressure for a 4.50% by mass aqueous solution of glucose (C6H12O6) at 300 K is 0.616 atm.

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Answer:

238 ft²

Step-by-step explanation:

Base area is l*w and there are two of these so 2(7*7) = 98

Then there 4 faces that have dimensions 4(7*5)=140

98+140=238

Translation, rotation, and reflection, By following these transformations in sequence, you can map triangle ABC to triangle A'B'C'.

To map triangle ABC to triangle A'B'C', you would need to follow a specific sequence of transformations, which may include translation, rotation, and reflection. Here's a step-by-step explanation:

Step 1: Translation
Translate triangle ABC by a specific vector (x, y) so that point A moves to point A'. The same vector will also move points B and C to their corresponding new positions.

Step 2: Rotation
If triangle A'B'C' is rotated compared to the translated triangle, rotate the translated triangle around point A' by a specific angle, either clockwise or counterclockwise, until point B aligns with point B'.

Step 3: Reflection
If triangle A'B'C' is a mirror image of the rotated triangle, reflect the rotated triangle across a line of symmetry (usually a line passing through A'). This will change the orientation of the triangle and align point C with point C'.

By following these transformations in sequence, you can map triangle ABC to triangle A'B'C'. Keep in mind that the specific details of translation, rotation, and reflection will depend on the coordinates and orientation of the given triangles.

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The probability that both events will occur is 0.22.

It is the chance of an event to occur from a total number of outcomes.

The formula for probability is given as:

Probability = Number of required events / Total number of outcomes.

Example:

The probability of getting a head in tossing a coin.

P(H) = 1/2

We have,

The probability of Event A is 2/6 or 1/3

(since there are two ways to get a 1 or 2 on a six-sided die).

The probability of Event B is 4/6 or 2/3

(since there are four ways to get a number 4 or less on a six-sided die).

Using the formula for the probability of the intersection of two independent events.

P(A and B)

= P(A) x P(B)

= (1/3) x (2/3)

= 2/9

Rounded to the nearest hundredth,

The probability that both events will occur is 0.22.

Thus,

The probability that both events will occur is 0.22.

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The perimeter of a rectangle is 34 units.

To find the perimeter of the rectangle with vertices (3,2), (13,2), (13,9), and (3,9), you need to determine the lengths of its sides.

The distance between (3,2) and (13,2) is the length of the base: 13 - 3 = 10 units.
The distance between (3,2) and (3,9) is the height: 9 - 2 = 7 units.

The perimeter of a rectangle is given by the formula P = 2*(length + width). In this case, P = 2*(10 + 7) = 2*17 = 34 units.

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Distribute

50+30(2x+10)

50+30×2x+30×10

50+60×+300

add the number

50+300+60×

350+60x

60x+350

common factors

10(6x+35)

A reasonable prediction for the number of times the spinner will land on the number 3 is 3 times.

Since the spinner has 8 congruent sections and is spun 24 times, we can use probability to make a reasonable prediction for the number of times it will land on the number 3.

1. Calculate the probability of landing on the number 3 for a single spin:
Since there are 8 congruent sections, the probability of landing on the number 3 is 1/8.

2. Determine the expected number of times the spinner will land on the number 3:
To do this, multiply the probability of landing on the number 3 (1/8) by the total number of spins (24).
Expected number of times = (1/8) * 24

3. Simplify the expression:
Expected number of times = 3

So, a reasonable prediction for the number of times the spinner will land on the number 3 is 3 times.

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Ordered pairs must be placed in the following order: (6, 1), (8, 8), (12, 22). (20, 50). The right answer is D.

According to the assertions, it deals with a number of operations in mathematics.

For the first series, let the number be x, and for the second, let it be y.

The first rule is to multiply by 2, then, beginning with 6, remove 4.

The sequence can be written as 6 is the first number: = 2x - 4

next number = 2(6) - 4

                    = 12-4

                    =  8

The following phrase is thus 8. The following two terms can be written as 12 and 20.

Values of x = 6, 8, 12, 20

The second rule is to add three and then multiply by two, counting backward from one.

= (y + 3)2

First term = 1

Next term = (1 + 3)2

               =  8

The next two words can be acquired in a similar manner.

y = 1, 8, 22, 50

Therefore, ordered pairs must be placed in the following order: (6, 1), (8, 8), and (12, 22). (20, 50). The right answer is D.

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Correct question:

The first rule is to multiply by 2, then subtract 4 starting from 6. The second rule is to add 3, then multiply by 2 starting from 1. What are the first four ordered pairs using the two sequences?

A. (0, 0) (6, 1), (8, 2), (12, 22)

B. (6, 1), (10, 8), (14,22), (22, 50)

C. (6, 1), (8, 8), (12, 20), (20, 45)

D. (6, 1), (8, 8), (12, 22), (20, 50)

1. x-3y -3=0, 3x-9y-2=0 has no solution

2. 2x+y=5,3x+2y=8 has a unique solution

3. 3x-5y=20,6x-10y=40 has infinitely many solution

4.  x-3y-7 =0,3x-3y-15=0 has No solution

5. 8x+5y=9,3x+2y=4 has a unique solution

(i) To solve using substitution method, we can rearrange the first equation to x=3y+3 and substitute it into the second equation:

3(3y+3) - 9y - 2 = 0

9y + 9 - 9y - 2 = 0

7 = 0

This is a contradiction, so the pair of equations has no solution.

(ii) To solve using elimination method, we can multiply the first equation by 2 and subtract it from the second equation:

3x + 2y = 8

(4x + 2y = 10)

-x = -2

So, x = 2. Substituting this value into the first equation, we get:

2x + y = 5

2(2) + y = 5

y = 1

Therefore, the unique solution is (x,y) = (2,1).

(iii) To solve using elimination method, we can multiply the first equation by 2 and subtract it from the second equation:

6x - 10y = 40

(6x - 10y = 40)

0 = 0

This equation is true for any value of x and y, so the pair of equations has infinitely many solutions.

(iv) To solve using elimination method, we can subtract the first equation from the second equation:

3x - 3y - 15 - (x - 3y - 7) = 0

2x - 22 = 0

x = 11

Substituting this value into the first equation, we get:

11 - 3y - 7 = 0

-3y = -4

y = 4/3

Therefore, the unique solution is (x,y) = (11,4/3).

(v) To solve using elimination method, we can multiply the first equation by 2 and subtract it from the second equation:

3x + 2y = 4

(16x + 10y = 18)

-29x - 18y = -14

Solving for y, we get:

y = (29/18)x + (7/9)

Substituting this expression for y into the first equation, we get:

8x + 5((29/18)x + (7/9)) = 9

(143/18)x = 2/9

x = 2/13

Substituting this value into the expression for y, we get:

y = (29/18)(2/13) + (7/9) = 41/117

Therefore, the unique solution is (x,y) = (2/13,41/117).

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Therefore, the perimeter of the rectangle is 38 units that is option C.

A coordinate is a set of numbers or values that specifies the position or location of a point or object in a space. In mathematics, coordinates are used to describe the position of a point in a plane, a space or a higher-dimensional object. Coordinates can be represented by ordered pairs or tuples, where the first value corresponds to the position on the horizontal axis, and the second value corresponds to the position on the vertical axis.

Here,

First, we need to find the distance between points A and B to determine the length of the rectangle:

Distance AB = |yB - yA|

= |8 - 2|

= 6

Next, we need to find the distance between points B and C to determine the width of the rectangle:

Distance BC = |xC - xB|

= |6 - (-7)|

= 13

Since opposite sides of a rectangle are equal in length, we know that the distance between points A and D is also 13, and the distance between points C and D is also 6.

Therefore, the perimeter of the rectangle is:

Perimeter = 2 * (length + width)

Perimeter = 2 * (13 + 6)

Perimeter = 2 * 19

Perimeter = 38

So the answer is (C) 38.

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Answer:Each of the shaded sections represents 25% of the total number of students in the class, and since we have evenly distributed the 12 students among them, each shaded section now represents 30% of the students (i.e., 25% + 5% = 30%).

Step-by-step explanation: To model this situation, we need to determine the total number of students in the class. We can use the given percentage to set up a proportion:

40% = 12 students / x total students

To solve for x, we can cross-multiply and simplify:

0.4x = 12

x = 12 / 0.4

x = 30

Therefore, there are 30 students in the class.

To evenly distribute the 12 students in the 4 shaded sections, we can divide 12 by 4 to get the number of students for each section:

12 students / 4 sections = 3 students per section

Answer:

Step-by-step explanation:

The value of 2s in 43,290 is 2,000, while the value of 2s in 32,865 is 20.

B. 200 is 1/20 the value of 2,000.

This statement is correct, as 200 is 1/10 of 2,000, and there are two 0s in the value of 2s in 43,290 compared to one 0 in the value of 2s in 32,865.