A. The given relation is a function, but is not one-to-one. A possible domain for the inverse is given as follows: x ≥ 4.
B. The inverse function is given as follows: [tex]f^{-1}(x) = \sqrt{0.5(x + 8)} + 4[/tex]
C. The composition of the function with it's inverse results in x, hence they are inverse.
How to obtain the features of the function?The quadratic function for this problem is defined as follows:
y = 2(x - 4)² - 8.
The classifications regarding function and one-to-one are given as follows:
The relation is a function, as each value of x is mapped to a single value of y.The relation is not one-to-one, as values of y, may be mapped to multiple values of x.The inverse can only be obtained when the function is one-to-one, hence a possible domain is given as follows:
x ≥ 4.
(as the vertex is at x = 4).
To obtain the inverse, first we must exchange the variables x and y, hence:
x = 2(y - 4)² - 8.
Now we must isolate the variable y, as follows:
2(y - 4)² = x + 8
(y - 4)² = 0.5(x + 8)
[tex]\sqrt{(y - 4)^2} = \sqrt{0.5(x + 8)}[/tex]
[tex]y - 4 = \sqrt{0.5(x + 8)}[/tex]
[tex]f^{-1}(x) = \sqrt{0.5(x + 8)} + 4[/tex]
The composition of the two functions is given as follows:
[tex](f \circ f^{-1}(x)) = 2(\sqrt{0.5(x + 8)} + 4 - 4)^2 - 8)[/tex]
[tex](f \circ f^{-1}(x)) = 2(\sqrt{0.5(x + 8)})^2 - 8[/tex]
[tex](f \circ f^{-1}(x)) = x + 8 - 8[/tex]
[tex](f \circ f^{-1}(x)) = x[/tex]
As the composition results in x, they are inverses.
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Can someone answer this question
Answer:
The correct answer is (d): (x - 5) is a factor of f(x).
Step-by-step explanation:
Solve and graph. |3x+6/9|>=2
The solution to the inequality expression is -8/9 ≤ x ≥ 4/9
How to solve and graph the inequality expressionFrom the question, we have the following parameters that can be used in our computation:
|3x + 6/9| ≥ 2
Expand the inequality expression
So, we have
-2 ≤ 3x + 6/9 ≥ 2
Subtract 6/9 from both sides
So, we have
-24/9 ≤ 3x ≥ 12/9
Divide through the equation by 3
-8/9 ≤ x ≥ 4/9
Hence, the solution to the inequality expression is -8/9 ≤ x ≥ 4/9
The graph is attached
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Find the average rate of change of f(x)=x²+2x+1 from x=4 to x=6.
The average rate of change of the function over the interval is 12
Finding the average rate of changeFrom the question, we have the following parameters that can be used in our computation:
f(x) = x² + 2x + 1
The interval is given as
From x = 4 to x = 6
The function is a quadratic function
This means that it does not have a constant average rate of change
So, we have
f(4) = 4² + 2(4) + 1 = 25
f(6) = 6² + 2(6) + 1 = 49
Next, we have
Rate = (49 - 25)/(6 - 4)
Evaluate
Rate = 12
Hence, the rate is 12
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100 Points! Algebra question. Photo attached. Graph the function. Thank you!
The graph of the function is attached and the amplitude and the period are 5 and 2π
How to determine the amplitude and period of the functionFrom the question, we have the following parameters that can be used in our computation:
f(x) = 5cos(θ)
A sinusoidal function is represented as
f(x) = Acos(B(x + C)) + D
Where
Amplitude = APeriod = 2π/BSo, we have
A = 5
Period = 2π/1
Evaluate
A = 5
Period = 2π
Hence, the amplitude is 5 and the period is 2π
The graph of the function is attached
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Can someone answer this question
Answer:3rd one
Step-by-step explanation:
Using the image below, answer the following question: you are asked to pick 2
marbles out of the bag, what is the probability of picking a blue marble and then a
green marble, without replacing the blue one?
Please
The probability of picking a blue marble and then a green marble without replacing the blue one is 2/15.
From the figure we can write
Number of Blue marbles = 2
Number of Green marbles = 6
Number of Purple marbles = 2
Total number of marbles = 2 + 6 + 2 = 10
Now, Probability of picking a blue marble
= Number of Blue marbles / Total number of marbles
= 2 / 10
= 1/5
After removing the blue marble, the total number of marbles becomes 9
Probability of picking a green marble
= 6 / 9
= 2/3
So, the probability of picking a blue marble and then a green marble
Probability = (1/5) x (2/3) = 2/15.
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find the area of a triangle whose base is 8 inches and whose height is 12 in.
(a) 96in.²
(b) 10in ²
(c) 20in.²
(d) 48in.²
Answer:
48in.²
Step-by-step explanation:
The formula for the area of a triangle is 1/2 x b x h
Since the b = 8 and the h = 12, substitute 8 and 12 for b and h.
1/2 x 8 x 12
4 x 12
48in.²
Answer:
d
Step-by-step explanation:
area= ½ × b × h
= ½ × 8 × 12
= ½ × 96
= ⁹⁶/²
= 48in²
PLEASE HELP
Given f(x)=4x^2 - 1 and g (x) = 2x- 1 , find each function
1. ( f+g )( x )
2. ( fg ) ( x )
3. ( f^n ) ( x )
The values of the composite functions are
(1) (f + g)(x) = 4x² + 2x - 2
(2) (fg)(x) = 8x³ - 4x² - 2x + 1
(3) (fⁿ)(x) = (4x² - 1)ⁿ
How to evaluate the composite functionsFrom the question, we have the following functions that can be used in our computation:
f(x) = 4x² - 1
g(x) = 2x - 1
Using the above as a guide, we have the following:
(f + g)(x) = f(x) + g(x)
(f + g)(x) = 4x² - 1 + 2x - 1
Evaluate the like terms
(f + g)(x) = 4x² + 2x - 2
(fg)(x) = f(x) * g(x)
(fg)(x) = (4x² - 1) * (2x - 1)
Evaluate the products
(fg)(x) = 8x³ - 4x² - 2x + 1
(fⁿ)(x) = (f(x))ⁿ
So, we have
(fⁿ)(x) = (4x² - 1)ⁿ
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Find the sum of the finite series:
10
Σ2· (3)n-1
n=1
Answer: 59,048
Step-by-step explanation:
To find the sum of the given finite series:
Σ2· (3)n-1
n=1
where n ranges from 1 to 10, we can use the formula for the sum of a geometric series.
The formula for the sum of a geometric series is:
S = a * (r^n - 1) / (r - 1)
In this case, a = 2 (the first term of the series) and r = 3 (the common ratio).
Using the formula, we can calculate the sum:
S = 2 * (3^10 - 1) / (3 - 1)
= 2 * (59049 - 1) / 2
= (118098 - 2) / 2
= 118096 / 2
= 59048
Therefore, the sum of the given finite series is 59,048.
What is the probability that either event will occur?
Now, find the probability of event A and event B.
A
B
6
6
20
20
P(A and B) = [?]
Enter as a decimal rounded to the nearest hundredth.
Enter
The probability that either event will occur is 0.62
What is the probability that either event will occur?From the question, we have the following parameters that can be used in our computation:
Event A = 6 + 6 = 12
Event B = 20 + 6 = 26
Both A and B = 6
Other Events = 20
Using the above as a guide, we have the following:
Total = A + B + C + Others - Both
So, we have
Total = 12 + 26 - 6 + 20
Evaluate
Total = 52
So, we have
P(A) = 12/52
P(B) = 26/52
Both A and B = 6/52
For either events, we have
P(A or B) = (12 + 26 - 6)/52
Evaluate
P(A or B) = 0.62
Hence, the probability that either event will occur is 0.62
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how do you find out the area of a tent, floor included
To find the area of a tent, including the floor, we need to measure or determine the dimensions of both the tent's floor and any additional areas such as vestibules or extensions.
Floor Area: Measure the length and width of the tent's floor in the same unit of measurement (e.g., feet or meters). Multiply the length by the width to calculate the floor area. For example, if the length is 8 feet and the width is 6 feet, the floor area would be 8 feet * 6 feet = 48 square feet.Additional Areas: If the tent has vestibules, extensions, or any other separate areas, measure each area's dimensions and calculate their individual areas separately.Total Area: Once you have calculated the individual areas of the floor and any additional areas, simply add them together to find the total area of the tent, including the floor. For example, if the floor area is 48 square feet and there is an additional vestibule area of 10 square feet, the total area would be 48 square feet + 10 square feet = 58 square feet.Learn more about the area of the tent here:
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I’m am so lost please help thank you all
Answer:
Step-by-step explanation:
148 people
What are the side lengths of triangle Graph shows a triangle plotted on a coordinate plane. The triangle is at A(minus 7, 3), B(minus 3, 6), and C(5, 0). Type the correct answer in each box. If necessary, round any decimal to the nearest tenth. units units units
With the coordinate provided, the lengths of the triangle are Side AB = 5 units, Side BC = 10 units and Side CA = 12.4 units.
How do we find the sides of triangles using coordinates?To find the side lengths of the triangle using the coordinates provided, we use the distance formul.
The distance between two points (x1, y1) and (x2, y2) is given by √((x2-x1)² + (y2-y1)²).
Side AB: Distance between A(-7, 3) and B(-3, 6)
= √((-3 - -7)² + (6 - 3)²)
= √((4)² + (3)²)
= √(16 + 9)
= √25
= 5 units
Side BC: Distance between B(-3, 6) and C(5, 0)
= √((5 - -3)² + (0 - 6)²)
= √((8)² + (-6)²)
= √(64 + 36)
= √100
= 10 units
Side CA: Distance between C(5, 0) and A(-7, 3)
= √((-7 - 5)² + (3 - 0)²)
= √((-12)² + 3²)
= √(144 + 9)
= √153
= 12.4 units
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Question 5
TABLE 4 below shows the distribution of revenue among the different spheres of
government in South Africa from the 2017/2018 to 2021/2022 financial year. Some values
have been omitted.
TABLE 4: Distribution of Revenue among the different government spheres for
the financial years 2017/2018 to 2021/2022.
R(billion)
2019/20
546.1
471,4
Government
BHOJDIN
National
Provincial
Local
TOTAL
2017/18
453.4
410.6
**
2018/19
490,00
439,5
87.6
1 017,1
98.3
1:15.8
2020/21
557,5
500,4
103.3
1 161.2
2021/22
1 240.5
(adapted from DBE 2014 MLQP)
Use the table above to answer questions that follows
When the revenue of the local government sector was compared to the provincial sec
2017/18, it was found to be 20, 12% of the provincial sector.
Calculate
the missing value E, the total revenue for the period 2017/18.
ont sector always receives a larger share than t
The missing value E, the total revenue for the period 2017/18, is approximately 82.61 billion.
How to calculate the valueLocal government sector's revenue in 2017/18 (L): Missing value (to be calculated)
Provincial sector's revenue in 2017/18 (P): 410.6 (billion)
According to the information provided, L is 20.12% of P. Mathematically, we can write this as:
L = 20.12/100 * P
Substituting the known value of P, we get:
L = 20.12/100 * 410.6
L ≈ 82.61 (rounded to two decimal places)
Therefore, the missing value E, the total revenue for the period 2017/18, is approximately 82.61 billion.
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Can someone give me the answer
Between 40 and 60 minutes the target heart rate strictly increasing then strictly decreasing. Therefore, option B is the correct answer.
From the given graph, in the first curve between the interval 0-10 speed is increasing, between the interval 10-30 speed is constant and between the interval 30-40 speed is decreasing.
In the second curve between the interval 40-50 speed is increasing, between the interval 50-60 speed is decreasing, between the interval 60-65 speed is increasing and between the interval 65-73 speed is constant.
In the third curve between the interval 73-85 speed is increasing, between the interval 85-92 speed is constant and between the interval 92-100 speed is decreasing.
Therefore, option B is the correct answer.
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Line r
goes through points (−5,2)
and (−3,8).
Line s
goes through points (−6,4)
and (2,12).
Which statement is true about lines r
and s?
Responses
Line r
is steeper than line s.
Line r is steeper than line
Line s
has a negative slope.
Line s has a negative slope.
Line s
is steeper than line r.
Line s is steeper than line
Line r
has a negative slope.
Answer: To determine which statement is true about lines r and s, we need to compare their slopes.
The slope of a line can be calculated using the formula:
slope = (change in y-coordinates) / (change in x-coordinates)
For line r, using the points (-5, 2) and (-3, 8):
slope_r = (8 - 2) / (-3 - (-5))
= 6 / 2
= 3
For line s, using the points (-6, 4) and (2, 12):
slope_s = (12 - 4) / (2 - (-6))
= 8 / 8
= 1
Now, let's analyze the statements:
Line r is steeper than line s. (False)
Since the slope of line r (3) is greater than the slope of line s (1), this statement is true.
Line r is steeper than line (Incomplete statement)
This statement is incomplete.
Line s has a negative slope. (False)
The slope of line s (1) is positive, not negative. So, this statement is false.
Line s is steeper than line r. (False)
Since the slope of line s (1) is less than the slope of line r (3), this statement is false.
Line r has a negative slope. (False)
The slope of line r (3) is positive, not negative. So, this statement is false.
Based on the analysis, the true statement about lines r and s is:
Line r is steeper than line s.
List all numbers from the given set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, and f. real numbers.
Answer:
See attached
Step-by-step explanation:
You want the given numbers classified as to Natural, Whole, Integer, Rational, Irrational, and/or Real.
RealAll the given numbers are real.
IrrationalThe number √2 is irrational. All the others are rational.
IntegerThe values √16 = 4, 0, and -5 are integer values. Whole numbers are non-negative integers, so exclude -5. Natural numbers are positive integers, so exclude 0 and -5.
The Xs indicate the categories each number belongs to.
Mount Saint Helens, a volcano, erupted on May 18, 1980. Before eruption, Mount St. Helens was 2.95 kilometers high. use the bar diagram to find the difference in height of mount st. helens before and after the eruptions in meters PLEASE I NEED HELP :(((
The difference in height is 400 metres.
What is the difference in height of Mount St. Helens?Height refers to vertical distance from the top to the object's base. Occasionally, it is also labeled as an altitude which measures from down to top of a surface.
To get difference in height, we will subtract the height after the eruption from the height before the eruption:
Given:
Height before eruption = 2.95 kilometers
Height after eruption = 2.55 kilometers
Difference in height = Height before eruption - Height after eruption
Difference in height = 2.95 km - 2.55 km
Difference in height = 0.4 kilometers
0.4 kilometers to metres will be:
= 0.4 * 1000 metres
= 400 metres.
Full question:
Mount St. Helens, located in Washington, erupted on May 18, 1980. Before the eruption, the volcano was 2.95 kilometers high. After the eruption, the volcano was 2.55 kilometers high. Find the difference in height of Mount St. Helens before and after the eruption.
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Question Find the distance between the two points. (12, 5), (−12, −2)
Answer:
Distance between points is 25 Units.
Step-by-step explanation:
To find the distance between two points, we can use the distance formula. The formula is:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Given the two points: (12, 5) and (-12, -2), we can assign the values as follows:
(x1, y1) = (12, 5)
(x2, y2) = (-12, -2)
Plugging the values into the distance formula, we get:
d = sqrt((-12 - 12)^2 + (-2 - 5)^2)
= sqrt((-24)^2 + (-7)^2)
= sqrt(576 + 49)
= sqrt(625)
= 25
Therefore, the distance between the points (12, 5) and (-12, -2) is 25 units.
48 as a tree diagram prime factor
Answer:
48 prime factors are: 2 x 2 x 2 x 2 x 3
A rectangle has 2 sides that are each 6 centimeters lo[ng. The perimeter is 22 centimeters. How long are the other sides?
Answer:
The other sides = 5 centimeters.
Step-by-step explanation:
Let's assume the length of the other two sides of the rectangle is "x" centimeters.
A rectangle has two pairs of equal sides. Therefore, we can set up the equation for the perimeter of the rectangle:
Perimeter = 2(length + width)
Given:
Length = 6 centimeters
Perimeter = 22 centimeters
Using the equation, we can substitute the given values:
22 = 2(6 + x)
Simplifying further:
22 = 12 + 2x
Subtracting 12 from both sides:
10 = 2x
Dividing both sides by 2:
5 = x
Therefore, the length of the other two sides of the rectangle is 5 centimeters.
1. ¬{[Q → (¬S ∧ R)] ∨ ¬P} → (¬P ↔ P) ∴ ¬(Q ∧ P) ∨ (T ∧ R)
2. [(T ∨ Q) → (T ∨ S)] ∨ [(T ∨ Q) → (T ∨ S)]
Given statement solution is :- Regardless of the values of T, Q, and S, the whole Valid Statements & Implications is always true.
Let's analyze the two given statements:
¬{[Q → (¬S ∧ R)] ∨ ¬P} → (¬P ↔ P) ∴ ¬(Q ∧ P) ∨ (T ∧ R)
To prove this statement, we can use logical equivalences and deductions:
¬{[Q → (¬S ∧ R)] ∨ ¬P} → (¬P ↔ P) (Given)
¬{[¬Q ∨ (¬S ∧ R)] ∨ ¬P} → (¬P ↔ P) (Implication equivalence)
¬{[(¬Q ∨ ¬S) ∧ (¬Q ∨ R)] ∨ ¬P} → (¬P ↔ P) (Implication equivalence)
¬{(¬Q ∨ ¬S ∨ ¬P) ∧ (¬Q ∨ R ∨ ¬P)} → (¬P ↔ P) (De Morgan's Law)
(Q ∧ S ∧ P) ∨ (¬Q ∧ P) → (¬P ↔ P) (De Morgan's Law)
At this point, the formula ¬P ↔ P is a contradiction. The left side (¬P) states that P is false, while the right side (P) states that P is true. Therefore, the whole statement is always true, regardless of the values of Q, S, and P.
Since the statement is always true, we can conclude ¬(Q ∧ P) ∨ (T ∧ R) is true as well, making the second part of the question valid.
[(T ∨ Q) → (T ∨ S)] ∨ [(T ∨ Q) → (T ∨ S)]
In this statement, we have two identical sub-statements connected by a logical OR operator.
If we assume that (T ∨ Q) → (T ∨ S) is true, then the whole statement is true, regardless of the values of T, Q, and S.
If we assume that (T ∨ Q) → (T ∨ S) is false, then we need to check the other part of the statement.
Assuming (T ∨ Q) → (T ∨ S) is false means that (T ∨ Q) is true, but (T ∨ S) is false. In this case, the second part of the statement [(T ∨ Q) → (T ∨ S)] is true.
Therefore, regardless of the values of T, Q, and S, the whole Valid Statements & Implications is always true.
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area of a rectangular piece of cloth 3 1/4m by 25 cm
Answer:
8125 cm² (or 0.8125 m²)
Step-by-step explanation:
area of a rectangular piece of cloth 3 1/4m by 25 cm
the area of the rectangle is found by making the base by the height, we convert the meters into centimeters
3 1/4 m = 3.25m = 325 cm
we find the area
325 × 25 = 8125 cm² (or 0.8125 m²)
how many triangles?(urgent)
There are 75 triangles that can be formed using these 9 vertices.
Here we have,
We have 9 vertices arranged in a pattern where 5 vertices are above and 4 vertices are below in a parallel direction.
And we want to know how many triangles can be formed using these vertices.
In order to this,
Count the number of ways to choose 3 vertices from the 9 vertices.
This can be done using the combination formula, which is:
[tex]^{9}C_{3}[/tex] = 9! / (3! * (9-3)!)
= 84.
Since,
A degenerate triangle is one where all three vertices lie on the same line. To count the number of degenerate triangles,
We need to count how many ways we can choose 3 vertices that lie on the same line.
There are 5 lines with 3 points each (the 5 lines with the upper vertices). So the number of degenerate triangles is:
5 [tex]^{3}C_{3}[/tex] + 4 [tex]^{3}C_{3}[/tex] = 9.
Subtract the number of degenerate triangles from the total number of triangles to get the number of non-degenerate triangles.
Therefore, the number of non-degenerate triangles is:
84 - 9 = 75.
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a librarian has weekends off and has 10 paid vacation days per year, including holidays that fall on weekends. If his salary is 32,500 per year, what is his pay per workday?
his pay per workday will be $127.45.
How to calculate interest?Thus, the formula for calculating simple interest is J = C i t, where J is interest, C is principal, i is interest rate, and t is time. That is, J represents the amount added to the initial value, the capital is the calculation basis, the interest rate is the percentage applied on the capital and time is the capitalization period.
Knowing that:
10 paid vacation days per yearsalary is 32,500 per yearThe total weekends in a year:
52 * 2 = 104 weekend days.
Total paid vacation days, including holidays that fall on weekends:
10 - 4 = 6 paid vacation days.
Total non-working days:
104 + 6 = 110 non-working days.
Total working days in a year:
365 - 110 = 255 working days.
Pay per workday = Annual salary / Total working days
= $32,500 / 255= $127.45
Therefore, the librarian's pay per workday is approximately $127.45.
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Read the description of g g below, and then use the drop-down menus to complete an explanation of why g g is or is not a function. g g relates a student to each course the student takes in a school year.
g is a function because each student is uniquely mapped to each course the student takes in a school year.
What is a function?In Mathematics and Geometry, a function refers to any mathematical equation which is typically used to define and represent a relationship that exists between two or more variables such as an ordered pair, points on a graph or table.
Based on the description of g, we can reasonably infer and logically deduce that the relation g represent a function because the input values are uniquely mapped to the output values.
In this context, we can conclude that the description of g represents a function because a student represent the input values that is being to each course (output value) the student takes in a school year.
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100 Points! Algebra question. Photo attached. Please show as much work as possible. Thank you!
The probability of the digestive tract disease, given a positive test result, is 45%.
A. Two-Way Frequency Table:
Let's create a two-way frequency table to represent the situation:
Test Positive Test Negative Total
Disease Present A B 5,000
Disease Not Present C D 95,000
Total 5,000 95,000 100,000
In this table:
A represents the number of sheep that have the disease and test positive.B represents the number of sheep that have the disease but test negative.C represents the number of sheep that do not have the disease but test positive.D represents the number of sheep that do not have the disease and test negative.B. Probability of Disease Given Positive Test Result:
P(Disease | Positive)
= (P(Positive | Disease) x P(Disease)) / P(Positive)
= 0.94 x 0.05 / P(Positive)
To calculate P(Positive), we can use the law of total probability:
P(Positive) = P(Positive | Disease) x P(Disease) + P(Positive | No Disease) x P(No Disease)
So, P(Positive | No Disease) = 1 - P(Test Negative | No Disease)
= 1 - 0.94 = 0.06
P(No Disease) = 1 - P(Disease) = 1 - 0.05 = 0.95
Now, P(Positive) = (0.94 x 0.05) + (0.06 x 0.95)
= 0.047 + 0.057
= 0.104
Finally, we can calculate P(Disease | Positive):
P(Disease | Positive) = (0.94 x 0.05) / 0.104
= 0.047 / 0.104
≈ 0.452
Therefore, the probability of the digestive tract disease, given a positive test result, is 45%.
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Use the graph to answer the question.
Graph of polygon ABCD with vertices at negative 2 comma negative 1, 0 comma negative 4, 4 comma negative 4, 2 comma negative 1. A second polygon A prime B prime C prime D prime with vertices at 5 comma negative 1, 7 comma negative 4, 11 comma negative 4, 9 comma negative 1.
Determine the translation used to create the image.
7 units to the right
3 units to the right
7 units to the left
3 units to the left
The translation used to create the image A'B'C'D', from the pre image ABCD is T(7, 0), which corresponds with the option;
7 units to the right
What is a translation translation transformation?The coordinates of the vertices of the polygon ABCD are;
A(-2, -1), B(0, -4), C(4, -4), D(2, -1)
The coordinates of the vertices of the polygon A'B'C'D' are;
A'(5, -1), B'(7, -4), C'(11, -4), D'(9, -1)
The coordinates of the vertices of the image indicates;
The difference between the coordinates are;
A'(5, -1) - A(-2, -1) = (5 - (-2), -1 - (-1)) = (7, 0)
B'(7, -4) - B(0, -4) = (7 - 0, -4 - (-4)) = (7, 0)
C'(11, -4) - C(4, -4) = (11 - 4, -4 - (-4)) = (7, 0)
D'(9, -1) - D(2, -1) = (9 - 2, -1 - (-1)) = (7, 0)
The translation that is used to create the image is therefore;
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After considering the given interval we reach the conclusion that the coefficient of t² is 1/2(3w² - 1), under the condition that the given expression is [tex](1-2 t w+t^{2})^{-1/2}[/tex] with range of (t<<1)
To evaluate the value for the given expression we have to apply the principles of binomial theorem
Then
[tex](1-2 t w+t^{2})^{-1/2} = (1 + (-2 t w + t^{2})/2 + (-2 t w + t^{2})^{2/8} + (-2 t w + t^{2})^{3/16} + ...)[/tex]
= 1 - t w + 3/8 * t² * w² - 5/16 * t³ * w³ +
The coefficient of t is the coefficient of the first term with a power of t.
Therefore, the coefficient of t is 1/2(3w² - 1).
The binomial theorem refers to the statement regarding any positive integer n, the nth power of the sum of two numbers a and b could be expressed as the sum of n + 1 terms of the form. The binomial theorem is applied in algebra and probability theory.
To learn more about binomial theorem
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Please help me important question in image
The following property must be given to prove that ΔCJG ~ ΔCEA: B. GJ║AE.
What are the properties of similar triangles?In Mathematics and Geometry, two triangles are said to be similar when the ratio of their corresponding side lengths are equal and their corresponding angles are congruent.
Additionally, the lengths of three (3) pairs of corresponding sides or corresponding side lengths are proportional to the lengths of corresponding altitudes when two (2) triangles are similar.
Based on the side, side, side (SSS) similarity theorem, we can logically deduce the following congruent and similar triangles:
ΔCJG ≅ ΔCEA (line segment GJ is parallel to line segment AE)
ΔBIJ ≅ ΔBDF (line segment IJ is parallel to line segment DF).
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