Answers
Accοrding tο the transfοrmatiοn prοperty, the cοrrect answer is οptiοn (c): (x, y) → (- y + 2, x + 2)
What is transfοrmatiοn?Transfοrmatiοn is a functiοn that maps οne set οf pοints tο anοther set οf pοints in a way that preserves sοme structure οr prοperties οf the οriginal set. The mοst cοmmοn types οf transfοrmatiοns are geοmetric transfοrmatiοns, which preserve distance, angle, and shape, and algebraic transfοrmatiοns, which preserve algebraic prοperties such as the sum, prοduct, οr symmetry οf functiοns.
Twο triangles are similar if their cοrrespοnding angles are cοngruent and their cοrrespοnding sides are prοpοrtiοnal. Therefοre, tο determine whether a transfοrmatiοn maps triangle ABC tο a similar, but nοt a cοngruent triangle, we need tο examine hοw the transfοrmatiοn affects the angles and sides οf triangle ABC.
Let's assume that triangle ABC has vertices A(a₁, a₂), B(b₁, b₂), and C(c₁, c₂).
a) (x, y) → (x + 2, - y - 2)This transfοrmatiοn is a translatiοn that mοves each pοint οf the plane twο units tο the right and twο units dοwn. Therefοre, this transfοrmatiοn dοes nοt change the size οr shape οf triangle ABC. It οnly changes its pοsitiοn in the plane. Hence, this transfοrmatiοn dοes nοt map triangle ABC tο a similar, but nοt cοngruent triangle.
b) (x, y) → (- 4x, - 4y)
This transfοrmatiοn scales each cοοrdinate by a factοr οf -4, which reflects the triangle acrοss the οrigin and changes its size. Hοwever, this transfοrmatiοn alsο changes the οrientatiοn οf the triangle. Therefοre, this transfοrmatiοn dοes nοt map triangle ABC tο a similar, but nοt cοngruent triangle.
c) (x, y) → (- y + 2, x + 2)
This transfοrmatiοn is a rοtatiοn οf 90 degrees cοunterclοckwise arοund the pοint (2, 2). Tο see if it maps triangle ABC tο a similar, but nοt cοngruent triangle, we need tο check if the angles and sides οf the triangle are preserved. Since this transfοrmatiοn is a rοtatiοn, it preserves the angles οf the triangle. Tο check if it preserves the sides, we need tο cοmpute the length οf each side οf the transfοrmed triangle and cοmpare it tο the cοrrespοnding side οf the οriginal triangle:
AB: distance between (- a₂ + 2, a₁ + 2) and (- b₂ + 2, b₁ + 2) is equal tο the distance between A(a₁, a₂) and B(b₁, b₂)
AC: distance between (- a₂ + 2, a₁ + 2) and (- c₂ + 2, c₁ + 2) is equal tο the distance between A(a₁, a₂) and C(c₁, c₂)
BC: distance between (- b₂ + 2, b₁ + 2) and (- c₂ + 2, c₁ + 2) is equal tο the distance between B(b₁, b₂) and C(c₁, c₂)
Since the angles and sides are preserved, this transfοrmatiοn maps triangle ABC tο a similar, but nοt cοngruent triangle.
d) (x, y) → (- y, - x)
This transfοrmatiοn is a reflectiοn acrοss the line y = x. Tο see if it maps triangle ABC tο a similar, but nοt cοngruent triangle, we need tο check if the angles and sides οf the triangle are preserved. Since this transfοrmatiοn is a reflectiοn, it preserves the angles οf the triangle. Hοwever, it alsο changes the οrientatiοn οf the triangle. Therefοre, this transfοrmatiοn dοes nοt map triangle ABC tο a similar, but nοt cοngruent triangle.
Therefοre, the answer is οptiοn (c): (x, y) → (- y + 2, x + 2)
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Related Questions
A car is taken to a mechanic for repair. The mechanic charges 85 dollars for parts, plus 37 dollars per hour for labor. Write a linear equation , in slope-intercept form , that models this situation . (Note: "y=" is already given just enter the right side of the equation.) y = In the equation, the variable x represents which of the following ? (Choose one) total cost of the repair (dollars ) cost for parts ( dollars ) Ohourly cost of labor (dollars per hour ) time spent repairing the vehicle (hours) In the equation, the variable y represents which of the following? (Choose one) cost for parts (dollars ) time spent repairing the vehicle (hours) Ototal cost of the repair (dollars )
Answers
Answer: The linear equation in slope-intercept form that models this situation is:
y = 37x + 85
In this equation, the variable x represents the time spent repairing the vehicle in hours.
The variable y represents the total cost of the repair in dollars.
Step-by-step explanation:
define petition and state 3 ways in which petitions from the youth can help improve service delivery in their communities
Answers
A petition is just a formal request outlining the grounds for requesting a court order. It is often the first stage of a lawsuit and can be filed by an individual, a group, or an institution.
Explain about the petitions and its features?A petition is a formal letter or written request asking someone or an organization to take a certain action or make a certain change.
The petition is typically signed by a number of people who agree with the subject or problem being raised.
Awareness-building: Petitions can help people learn more about the problems that young people in their communities are facing.Feedback: Petitions can give local authorities insightful information about the wants and concerns of young people in their communities.Petitions can be used as an advocacy tool to help young people express their needs and desires to people in positions of authority.know more about the petitions
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Write a congruence statement for the triangles
Answers
Triangle ABC is congruent to triangle PQR, where A corresponds to P, B corresponds to Q, and C corresponds to R.
What is congruency?
In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.
To write a congruence statement for two triangles, we need to identify their corresponding parts and ensure that they are congruent in both triangles.
The congruence statement can be written in the following form:
Triangle ABC is congruent to triangle PQR, where A corresponds to P, B corresponds to Q, and C corresponds to R.
For example, if we have two triangles with vertices A, B, and C and P, Q, and R respectively, and we know that the following pairs of corresponding parts are congruent:
AB ≅ PQ
BC ≅ QR
AC ≅ PR
Then, we can write the congruence statement as:
Hence, Triangle ABC is congruent to triangle PQR, where A corresponds to P, B corresponds to Q, and C corresponds to R.
The symbol ≅ means "is congruent to."
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According to the drag equation the velocity of an object moving through a fluid can be modeled by the equation 2 dvkvdt=− where k is a constant. a) Find the general solution to this equation. b) An object moving through the water has an initial velocity of 40 m/s. Two seconds later, the velocity has decreased to 30 m/s. What will the velocity be after ten seconds?
Answers
Answer:
The given differential equation is:
2dv/dt = -kv
where k is a constant.
a) To find the general solution to this differential equation, we can use separation of variables.
2dv/v = -k dt
Integrating both sides, we get:
2ln|v| = -kt + C1
where C1 is the constant of integration.
Taking exponential of both sides, we get:
|v|^2 = e^(C1) e^(-kt)
Since v can be either positive or negative, we can simplify the above equation as:
v = ±√(Ae^(-kt))
where A = e^(C1).
Therefore, the general solution to the given differential equation is:
v(t) = ±√(Ae^(-kt))
b) We are given that the initial velocity is 40 m/s and after 2 seconds, the velocity has decreased to 30 m/s. Let's use this information to find the value of A.
When t = 0, v = 40. Therefore,
40 = ±√(Ae^0)
40^2 = A
A = 1600
Now we can use the value of A to find the velocity after 10 seconds.
v(10) = ±√(1600e^(-10k))
We can use the information that the velocity has decreased to 30 m/s after 2 seconds to find the value of k.
30 = ±√(1600e^(-2k))
900 = 1600e^(-2k)
e^(-2k) = 0.5625
-2k = ln(0.5625)
k = -0.5 ln(0.5625)
Now we can substitute this value of k into the expression for v(10) to get:
v(10) = ±√(1600e^(5ln(0.5625)))
v(10) = ±√(1600(0.5625)^5)
v(10) = ±20.11 m/s
Therefore, the velocity of the object after ten seconds will be approximately 20.11 m/s.
Please solve for slope and y intercept with steps
Answers
The values are given as;
1. The intercept, c = -1.5, Slope, m = -1/4
2. Intercept = 0. 5, Slope, m = -3/5
How to determine the slopeThe general formula for the equation of a line is represented thus;
y = mx + c
Given that the parameter are;
y is a point on the y -axism is the slope or gradient of the linex is a point on the x - axisc is the intercept of the line on the y - axisFor the first graph
The intercept, c = -1.5
The slope, m = y₂ - y₁/x₂ - x₁
Substitute the values as shown in the graph
Slope, m = -2-(-3)/- 3- 1
Slope, m = -2 + 3/-4
Slope, m = -1/4
For the second graph, we have
Intercept = 0. 5
The slope, m = y₂ - y₁/x₂ - x₁
Substitute the values
Slope, m = 2 - (-1)/3 - (-2)
Slope, m = -3/5
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From Rs. 62,500 if we can by 500 US dollar. How much US dollar can be bought if Nepali currency is devaluated by 5%.
Answers
Devaluation of 5%:
125+(125x5%)=187,5
Many US dollars can be bought if Nepali currency is devaluated by 5%=
62.500:187,5=333 dollar
U6d11 area using trig
Answers
Area of a triangle is 88.37cm²
What is the area of triangle?The area of a triangle is given by the formula:
[tex]Area = \frac{(base * height)}{2}[/tex]
where "base" refers to the length of the base of the triangle, and "height" refers to the height of the triangle measured perpendicularly from the base to the highest point of the triangle.
As per figure we have to find out the area of triangle, first we have to find out the height of triangle,
by using sine rule:
Sin 67° = [tex]\frac{height}{12 cm}[/tex]
height = 12 Sin 67° (Here, Sin 67° = 0.92)
height = 12 × 0.92
height = 11.046cm
[tex]Area = \frac{(base * height)}{2}[/tex]
[tex]Area = \frac{(16 X 11.046)}{2}[/tex]
[tex]Area = 88.37cm^{2}[/tex]
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Suppose you have $8,295 in savings when the price level index is at 100. If inflation pushes the price level up by 9 percent, calculate the real value of your savings.
Answers
Answer: To calculate the real value of your savings after inflation, you need to adjust your savings for the increase in the price level. Here's how to do it:
Calculate the amount of inflation:
Inflation rate = (New price level index - Old price level index) / Old price level index
In this case, the old price level index is 100, and the new price level index is 109 (100 + 9%). So:
Inflation rate = (109 - 100) / 100 = 0.09
Adjust your savings for inflation:
Real value of savings = Nominal value of savings / (1 + inflation rate)
In this case:
Real value of savings = 8,295 / (1 + 0.09) = 7,614.22
Therefore, the real value of your savings after inflation is $7,614.22.
Step-by-step explanation:
what is the quotient of 14 divided by 5?
A. 4/5
B. 2 2/3
C.3 3/4
D. 4 4/15
Answers
Answer:
5divided by 14 is 2 with a remainder of 3
Jeff has 10 CDs and Ben has 15 CDs. In the inequalities below, k represents the number
of CDs Kevin has.
k> 10
k<15
How many CDs does Kevin have?
Group of answer choices
13
10
9
16
Answers
Answer:
Kevin has 11 CDs.
Consider the quadratic function f(x) = x2 – 5x + 12. Which statements are true about the function and its graph? Select three options.
A. The value of f(–10) = 82
B The graph of the function is a parabola.
C. The graph of the function opens down.
D. The graph contains the point (20, –8).
E. The graph contains the point (0, 0).
Answers
Answer:
B
Step-by-step explanation:
the graph of the function is a parabola
A straight line is drawn on the graph.
Enter an equation in slope-intercept form. Use y = mx + b, where m is the slope and b is the y-intercept.
Answers
An equation of the line in slope-intercept form drawn on the graph is y = 3x/2 + 8.
How to determine an equation of this line?In Mathematics and Geometry, the point-slope form of any straight line can be determined by using this mathematical expression:
y - y₁ = m(x - x₁)
Where:
x and y represent the data points.m represent the slope.Next, we would determine the slope;
Slope (m) = (y₂ - y₁)/(x₂ - x₁)
Slope (m) = (20 - 8)/(8 - 0)
Slope (m) = 12/8
Slope (m) = 3/2
At data point (0, 8) and a slope of 3/2, a linear equation for this line can be determined by using the point-slope form as follows:
y - y₁ = m(x - x₁)
y - 8 = 3/2(x - 0)
y = 3x/2 + 8
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-8 times f(0)+4 times g(-8)
Answers
The value of the given expression is -60. The solution has been obtained by using arithmetic operations.
What are arithmetic operations?
The four basic operations, also referred to as "arithmetic operations," are said to be able to describe all real numbers. The four mathematical operations following division, multiplication, addition, and subtraction are quotient, product, sum, and difference.
We are given two functions as f(x) = [tex]\frac{2x}{3}[/tex] + 7 and g(x) = 2x + 15
So,
⇒f (0) = [tex]\frac{0}{3}[/tex] + 7
⇒f (0) = 7
Similarly,
⇒g (-8) = 2 (-8) + 15
⇒g (-8) = -16 + 15
⇒g (-8) = -1
Now, on substituting these in the expression, we get
⇒-8 * f (0) + 4 * g (-8)
⇒-8 * 7 + 4 * (-1)
Now using the arithmetic operations, we get
⇒-56 -4
⇒-60
Hence, the value of the given expression is -60.
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Question:
If the following equations represent the functions f(x) and g(x).
f(x) = 2/3x + 7 and g(x) = 2x + 15
Calculate the value of -8 * f(0) + 4 * g(-8)
Prove that 7^7-7^6 is divisible by 6
Answers
Answer:
To prove that 7^7 - 7^6 is divisible by 6, we can factor out 7^6 from the expression:
7^7 - 7^6 = 7^6 (7 - 1)
Simplifying the expression, we get:
7^7 - 7^6 = 7^6 (6)
We can see that 7^6 is a common factor in the expression, and 6 is also a factor. Therefore, we can conclude that 7^7 - 7^6 is divisible by 6.
Step-by-step explanation:
We can factor out 7^6 from the expression to get:
7^7-7^6 = 7^6(7-1)
Simplifying this expression, we get:
7^7-7^6 = 7^6(6)
We can see that the expression is a multiple of 6, which means that it is divisible by 6. Therefore, we have proved that 7^7-7^6 is divisible by 6.
Wave heights at Folly Beach are normally distributed with a mean of 4 feet and a standard deviation of 1.5 feet.
What is the probability that a wave has a height of exactly 3 feet?
Answers
Answer:
Since wave heights at Folly Beach are normally distributed with a mean of 4 feet and a standard deviation of 1.5 feet, we can use the normal distribution formula to find the probability of a wave having a height of exactly 3 feet.
The formula for the normal distribution is:
P(x) = (1 / (σ * sqrt(2π))) * e^(-(x - μ)^2 / (2 * σ^2))
where:
P(x) is the probability density function
σ is the standard deviation
μ is the mean
e is the base of the natural logarithm
x is the value of the variable
Substituting the values given in the problem, we get:
P(3) = (1 / (1.5 * sqrt(2π))) * e^(-(3 - 4)^2 / (2 * 1.5^2))
P(3) = 0.176
Therefore, the probability that a wave has a height of exactly 3 feet is 0.176 or approximately 17.6%.
Find the value of the unknown base.
Log n 1/4=-1/2
Answers
Answer:
Therefore, the value of the unknown base is n = 16.
Step-by-step explanation:
Using the definition of logarithms, we have:
log_n(1/4) = -1/2
This equation can be rewritten as:
n^(-1/2) = 1/4
To solve for n, we can raise both sides to the -2 power:
n^(-1/2)^(-2) = (1/4)^(-2)
n = 16
Therefore, the value of the unknown base is n = 16.
Jose's family is renting a car for their family vacation. The car rental agency rents cars for $64 plus an additional charge of $0.20 per mile. The equation below represents the total cost of renting the car where x represents the number of miles driven.
Answers
No matter how many miles are travelled, only the standard rental fee of equation $64 is paid. The total cost of the rental is determined by multiplying the miles driven by the additional $0.20 per mile price.
What is equation?An equation is a mathematical statement that proves the equality of two expressions connected by an equal sign '='. For instance, 2x – 5 = 13. Expressions include 2x-5 and 13. '=' is the character that links the two expressions. A mathematical formula that has two algebraic expressions on either side of an equal sign (=) is known as an equation. It depicts the equivalency relationship between the left and right formulas. L.H.S. = R.H.S. (left side = right side) in any formula.
The following equation is used to calculate the overall cost of renting a car:
C(x) = 0.20x + 64
where x is the amount of miles driven and C(x) is the rental car's total cost.
No matter how many miles are travelled, only the standard rental fee of $64 is paid. The total cost of the rental is determined by multiplying the miles driven by the additional $0.20 per mile price.
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I will mark you brainiest!
Knowing that ΔQPT ≅ ΔARZ, a congruent angle pair is:
A) ∠Q ≅ ∠R
B) ∠P ≅ ∠A
C) ∠T ≅ ∠Z
D) ∠P ≅ ∠Z
Answers
Answer:
Since ΔQPT ≅ ΔARZ, we know that their corresponding angles are congruent.
Step-by-step explanation:
The congruent angle pair is:
C) ∠T ≅ ∠Z.
Note that ∠Q and ∠R are not necessarily congruent to each other, ∠P and ∠A are not necessarily congruent to each other, and ∠P and ∠Z are not necessarily congruent to each other.
Determine whether the following study described is observational or an experiment. If the study is an experiment, identify the control and treatment groups, and discuss whether single- or double-blinding is necessary. If the study is observational, state whether it is a retrospective study, and if so, identify the cases and controls.
A study by university social scientists found that 19.1
%
of first-year women students surveyed at a regional college experienced sexual assault during that first year.
Answers
The study that we have here is an observational study. The study is not retrospective in nature.
What type of research study is thisThis study is observational because the researchers are observing and collecting data on a group without manipulating any variables.
It is not a retrospective study because the data is collected in real-time during the first year of college, rather than looking back on past experiences.
Therefore, there are no treatment or control groups, and there is no need for blinding. The study simply reports the percentage of first-year women students who experienced sexual assault during their first year at the regional college.
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Question
What is the volume of the prism?
Enter your answer, as a mixed number in simplest form, in the box.
cm³
The figure contains a right rectangular prism. The length of the prism is labeled 3 and one half centimeters, the width is labeled 6 centimeters, and the height is labeled 5 and one half centimeters.
Answers
The volume of the prism is 115.5 cubic centimeters.
What is a rectangular prism?A three-dimensional structure called a right rectangular prism has six rectangular faces, each of which has four right angles. The length, breadth, and height of a right rectangular prism are perpendicular to one another, and the opposing sides are congruent and parallel to one another. The prism's dimensions—also referred to as its length, breadth, and height—can be utilised to determine its volume. Multiplying the prism's length, width, and height yields the volume of a right rectangular prism.
The volume of the rectangular prism is given by the formula:
Volume = Length x Width x Height
Substituting the values we have:
Volume = 3.5 cm x 6 cm x 5.5 cm
Volume = 115.5 cubic centimeters
Hence, the volume of the prism is 115.5 cubic centimeters.
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area is 14 sq feet what is it after dilation with a scale factor of 2
Answers
The area after dilation with a scale factor of 2 will be 56 sq feet.
How to calculate The area after dilationWhen a figure is dilated with a scale factor of 2, the area of the figure is multiplied by the square of the scale factor.
So, if the original area is 14 sq feet, then the new area after dilation will be:
New area = Original area x (scale factor)^2
= 14 x 2^2
= 14 x 4
= 56 sq feet.
Therefore, the new area of the figure after dilation with a scale factor of 2 is 56 sq feet.
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Find the linear function with the following properties.
f(0)=8
Slope of f=−6
Answer
Answers
Therefore , the solution of the given problem of function comes out to be f(x) = -6x + 8 .
Describe function.Numerous subjects, including numbers, numbers, as well as their subsets, as well as building, construction, and both real and fictitious geographic locations, are covered in the mathematics programme. The relationships between variable elements that all cooperate to create the same outcome are covered in a work. A utility is composed of several unique parts that, when combined, give particular outcomes for each input.
Here,
The linear function with a slope of -6 can be written in slope-intercept form as:
=> f(x) = -6x + b
where "b" is the y-intercept. We are also given that f(0) = 8, so we can substitute those values in to solve for "b":
=> f(0) = -6(0) + b = 8
=> b = 8
Therefore, the linear function we are looking for is:
=> f(x) = -6x + 8
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The weight of Amara to the weight of toyin is 3:5, if Amara weighs 30 kg , how many more does Toyin weight?
Answers
Answer:
Toyin weighs 50 kg, which is 20 kg more than Amara.
Step-by-step explanation:
If Amara weighs 30 kg and the ratio of the weight of Amara to the weight of Toyin is 3:5, then we can set up a proportion:
3/5 = 30/x
where x is the weight of Toyin in kg.
To solve for x, we can cross-multiply:
3x = 150
x = 50
Therefore, Toyin weighs 50 kg, which is 20 kg more than Amara.
Suppose c = 9 and A = 50 degrees.
Find:
Answers
Therefore , the solution of the given problem of triangle comes out to be a length measures about 5.64 units in length.
What is a triangle exactly?If a polygon has more than one extra sections, it is a hexagon. It has a straightforward square form. Something like this configuration can only be distinguished from a conventional triangle by the sides A and B. The borders continue to be precisely collinear, but Euclidean geometry only yields a section rather than the entire cube. A triangular has three sides and three angles.
Here,
Let's determine the length of side b using the sine function:
opposite/hypotenuse of sin(A)
=> b/9 sin(50) = 9*sin(50) b = 7.021
Side B is therefore roughly 7.021 units long.
We can use the knowledge that the sum of the angles in a triangle is 180 degrees to determine the size of angle B:
=> angle B = 180 - angle A - 90
=> angle B = 180 - 50 - 90
=> angle B = 40 degrees
B's angle therefore has a 40 degree value.
The Pythagorean formula can also be used to determine the length of side a:
The equation is:
=> a² + b² = c²
=> a² + 7.021² = 9²
=> a² + 49.24 = 81
=> a² = 31.76
=> a ≈ 5.64
Side a therefore measures about 5.64 units in length.
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o.ooo27 in scientific notation ?
Answers
Answer:
2.700 × 10-7
Step-by-step explanation:
Answer: 2.7*10^-4
step by step explanation:
1. Make the number a new number between 1 and 10
More Icon
Move the decimal point to make 0.00027 a new number between 1 and 10. Because our original number is less than one, we move the decimal point to the right. Drop any zeroes in front of the number. Keep track of how many times we move the decimal point.
0.00027 -> 2.7
Our new number is 2.7. We moved the decimal point 4 times.
2. Define the power of 10
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Because our original number was less than one, the exponent defining the power of 10 is negative. Remember, we moved the decimal point 4 times, so the exponent is negative 4
10^(-4)
3. Final result
2.7*10^(-4)
Find the following a-e please ! College algebra
Answers
Given functions f(x) = 2x² + 1 and g(x) = 7x - 8, then [tex]f[g(2) = 73\end{array}$[/tex].
What is function?A technical definitiοn οf a functiοn is: a relatiοn frοm a set οf inputs tο a set οf pοssible οutputs where each input is related tο exactly οne οutput.
This means that if the οbject x is in the set οf inputs (called the dοmain) then a functiοn f will map the οbject x tο exactly οne οbject f(x) in the set οf pοssible οutputs (called the cοdοmain).
The nοtiοn οf a functiοn is easily understοοd using the metaphοr οf a functiοn machine that takes in an οbject fοr its input and, based οn that input, spits οut anοther οbject as its οutput.
Given f(x) = 2x² + 1 and g(x) = 7x - 8
Then
a)
[tex]$\begin{array}{l c l}{f[g(2)]=f[(7*2-8)]=f(6)}\\ {f[g(2)]=2(6)^{2}+1=72+1}\\f[g(2) = 73\end{array}$[/tex]
b)
[tex]$\begin{array}{l}{{f[g(x)]=f(7x-1)=2[7x-8]^{2}+1}}\\ {{f[g(x)]=2(49x^{2}+64-112x)+1}}\\ {{f[g(x)]=98x^{2}-224x+129}}\end{array}$[/tex]
c)
[tex]$\begin{array}}{g[f(x)]=g[2x^{2}+1]=7[2x^{2}+1]-8}\\ {g[f(x)]=14x^{2}-1}\end{array}$[/tex]
d) (g · g)(x) = g{g(x)} = g(7x - 8)
= 7(7x - 8) - 8
= 49x−64
e) (f · f)(-2) = f{f(-2)} = f(2(-2)²)
= 2(2(-2)²)²
= 128
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Increase the number 15/16 by 3/5 of it of it. Then increase the resulting number by 3/5 of it.
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Answer:
Step-by-step explanation: 0.3375 by simple algebra.
Challenge) Solve for a. Answer as a mixed number. A=______
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Answer:
Step-by-step explanation:
what are the possible numbers of positive, negative, and complex zeros of f(x)=3x^4-5x^3-x^2
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Answer:
Counting the number of sign changes in the function f(x)
The number of positive roots is equal to the number of sign changes in the function f(x) or less than that by an even integer.
Here, f(x)=3x^4-5x^3-x^2 has two sign changes (from + to -, and from - to +), so the number of positive roots is either 2 or 0.
Counting the number of sign changes in the function f(-x)
The number of negative roots is equal to the number of sign changes in the function f(-x) or less than that by an even integer.
Here, f(-x)=3x^4+5x^3-x^2 has one sign change (from - to +), so the number of negative roots is either 1 or 0.
Counting the number of non-real roots (complex roots)
The number of non-real roots (complex roots) is equal to the difference between the total number of roots (4 in this case) and the number of real roots (which we found above).
Therefore, the possible number of positive roots is 2 or 0, the possible number of negative roots is 1 or 0, and the possible number of complex roots is 2 or 4.
Step-by-step explanation:
a) Assume that the height h(t) of the mass is a sinusoidal function, where t is the time in seconds, sketch a graph of h from t = 0 to t = 0.8 seconds. t = 0 is the time at which the mass is released
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The graph of the sinusoidal function shows that the mass oscillates up and down around the equilibrium position with an amplitude of 0.1 meters.
What is the graph of the functionwe can use a general form of sinusoidal function
h(t) = A sin(ωt + φ) + C
where:
A = amplitude (maximum displacement from the mean position)
ω = angular frequency (radians per second)
φ = phase angle (initial position of the oscillation)
C = vertical displacement (position of the mean or equilibrium position)
Since the mass is released from rest, we can assume that the initial displacement is zero, i.e., C = 0. Also, we know that the period of the oscillation is 1/f, where f is the frequency. Therefore, we can find the frequency from the given time period of 0.8 seconds.
Let's assume that the height of the mass oscillates with an amplitude of 0.1 meters and a frequency of 5π radians per second (corresponding to a period of 0.4 seconds), and the phase angle is 0.
h(t) = 0.1 sin(5πt)
We can plot this function for t ranging from 0 to 0.8 seconds using a graphing tool or by hand. Here is the resulting graph
|
0.1 + . * .
| .* *.
| . .
| * *
| * *
| . .
| . .
| .* *.
| .* *.
-0.1 +----------------------------------------------
0 0.2 0.4 0.6 0.8
The x-axis represents time in seconds, and the y-axis represents the height of the mass in meters. The graph shows that the mass oscillates up and down around the equilibrium position with an amplitude of 0.1 meters. The maximum height occurs at t = 0.1 seconds and t = 0.5 seconds, while the minimum height occurs at t = 0.3 seconds and t = 0.7 seconds. At t = 0.4 seconds and t = 0.8 seconds, the mass returns to the equilibrium position.
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