Answer:
Step-by-step explanation: the answer is 20, 48, 52
4. Prove/disprove: (a) The functionT:Rn→Rdefined byT(v)=∥v∥is a linear transformation. (b) Letx∈Rnbe a fixed vector. The functionT:Rn→Rdefined byT(v)=v⋅xis a linear transformation. (c) LetA∈Mn×n(F)be an invertible matrix, and letTA:Fn→Fnbe the linear transformation determined byA. For ally∈Fn, there exists a uniquex∈Fnsuch thatTA(x)=y(d) LetA,B∈Mn×n(F). Suppose thatAB=ATand thatAis invertible. ThenBmust be invertible.
a)T(v)=∥v∥ is not a linear transformation.
b)T(v)=v⋅x is a linear transformation.
c)TA(x)=y.
d)B must also be invertible.
a) The function T:Rn→R defined by T(v)=∥v∥ is not a linear transformation.
b) The function T:Rn→R defined by T(v)=v⋅x is a linear transformation.
c) Let A∈Mnxn(F) be an invertible matrix, and let TA:Fn→Fn be the linear transformation determined by A. For all y∈Fn, there exists a unique x∈Fn such that TA(x)=y.
d) Let A,B∈Mnxn(F). Suppose that AB=AT and that A is invertible. Then B must also be invertible.
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5/6 divided by 1 1/2
Answer:
5/33
Step-by-step explanation:
5/6 divided by 11/2
Invert the divider
5/6 x 2/ 11
10/66
Reduce to smallest fraction
5/33
Answer:
5/33
Step-by-step explanation:
Apply the fraction rule: a/b ÷ c/d = (a × d) ÷ (b × c) for 5/6 ÷ 11/2
a = 5
b = 6
c = 11
d = 2
(5 × 2) ÷ (6 × 11)
For "(6 × 11)", break 6 down into "2 × 3", so that you can cancel out the common factor, because there is also a 2 in "5 × 2".
= (5 × 2) ÷ (2 × 3 × 11) -- cancel out the 2's
= 5 ÷ (3 × 11) *****3 × 11 = 33*****
= 5/33
Which one of the following equations could describe the graph above?
A. V=1.5x+2) - 3
B. V=3-1
= (-3)* +
О с.
C.
-5-4-3-2-1-1 2 3 4 5 6 7 8
V=
+6
An equation that could describe the graph below include the following: A. y = (1/2)^x + 6.
What is an exponential function?In Mathematics, an exponential function can be modeled by using this mathematical expression:
f(x) = a(b)^x
Where:
a represents the base value or y-intercept.x represents time.b represents the slope or rate of change.Generally speaking, When the base value or y-intercept (a) is less than one (1), the graph of an exponential function increases exponentially to the left. Additionally, the smaller the value of y-intercept (a), the steeper the slope (b) of the line.
By critically observing the graph of this exponential function, we can logically deduce that it was vertically shifted up by 6 units.
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Which point has coordinates (4.9, 3.9)?
Answer:
C
Step-by-step explanation:
Answer: c.
Step-by-step explanation:
A data firm records large amount of data. Historically, .9% of the pages of data recorded by the firm contain errors If 200 pages of data are randomly selected, What is the probability that six or more pages contain errors? b What is the probability that more than 10 pages contain errors? What is the probability that none ofthe pages contain errors? d. What is the probability that fewer than five pages contain errors?
There is a 0.0876 percent probability that six or more pages will include mistakes. There is a 1.64 x 10⁻⁶ chance that more than 10 pages may include mistakes.
What purposes does probability serve?Information about possibility of occurrence is provided by probability. For instance, weather patterns are used by meteorologists to predict the possibility of rain. In order to understand the relationship between exposures and the risk of health outcomes, epidemiology uses probability theory.
a) Chance that six or more pages may have mistakes:
To get this probability, we can utilise the binomial probability formula:
P(X>=6) = 1-P(X<6)
where X is the number of pages with errors.
P(X<6) = P(X=0)+P(X=1)+P(X=2) +P(X=3)+P(X=4)+P(X=5)
For each term, using the binomial probability formula, we obtain:
P(X < 6) = 0.9124
Therefore, P(X >= 6) = 1 - P(X < 6) = 1 - 0.9124 = 0.0876
Hence, there is a 0.0876 percent chance that six or more pages will include errors.
b) Chance that there are errors on more than 10 pages:
Using the same method, we can discover:
P(X > 10) = P(X = 11) + P(X = 12) + ... + P(X = 200)
To find the likelihood, we can use a calculator or software because performing the math by hand can be very time-consuming. Using a calculator for the binomial distribution, we obtain:
P(X > 10) = 1.64 x 10⁻⁶ (rounded to 3 decimal places)
Hence, there is a roughly 10% chance that there will be errors on more than 10 pages. 1.64 x 10⁻⁶.
c) Probability that none of the pages contain errors:
P(X = 0) can be calculated using the binomial probability formula:
P(X = 0) = (200 choose 0) * (0.009)⁰ * (1 - 0.009)²⁰⁰
= 0.8196
As a result, 0.8196 percent of the pages are likely to be error-free.
d) Probability that fewer than five pages contain errors:
P(X<5)=P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)
Again, using the binomial probability formula for each term, we get:
P(X < 5) = 0.8151
Hence, the likelihood that fewer than five pages are mistake-free is 0.8151.
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The following list shows the number of new memberships that a gym has sold each day over the past week: 4, 2, 9, 2, 0, 3, 8. Which of the following is not a true statement?
Show quoted text
The median of the data is 2 is not a true statement. (second option).
What is the median?Mean is the average of a set of numbers.
Mean = sum of numbers / total numbers
(0 + 2 + 2 + 3 + 4 + 8 + 9) / 7 = 28 / 7 = 4
Median can be described as the number that occurs in the middle of a set of numbers that are arranged either in ascending or descending order.
The numbers in ascending order is: 0, 2, 2, 3, 4, 8, 9
Median = 3
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The able shows Claude's assets and liabilities. What is the total value of his liabilities? Responses $24,950 $24,950 -$24,520 -$24,520 $50,370 $50,370 $25,420
The total value of Claude's liabilities is $24,950.(option a).
In Claude's balance sheet, the liabilities are shown separately from the assets. The liabilities are the debts or obligations that Claude owes to others. To find the total value of Claude's liabilities, we need to add up all the amounts listed under liabilities.
However, before we do that, let's quickly review what assets are. Assets are things that have value and can be owned or controlled by a person or a company.
Next, we can eliminate the two larger options, $50,370 and $25,420, because they are higher than Claude's total assets, which are not given in the question. If the liabilities were greater than the assets, this would mean that Claude owes more than he owns, which would not be a good financial situation.
This leaves us with the remaining option, $24,950. This is the total value of Claude's liabilities based on the information given in the question.
Hence the option (a) is correct.
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Find the surface area from the box shown
Response to the given question would be that Hence, the surface area rectangular box has a surface area of 592 square cm.
what is surface area ?Surface area is a measure of how much space an object's surface takes up overall. The total area of a three-dimensional shape's surroundings is its surface area. Surface area refers to the total surface area of a three-dimensional form. You may compute the surface area of a cuboid with six rectangular sides by adding together their individual areas. Instead, you may use the following formula to name the box's dimensions: Surface (SA) = 2lh plus 2lw plus 2hw. A three-dimensional shape's surface area is calculated as the total amount of space it occupies (a three-dimensional shape is a shape that has height, width, and depth).
Size (l) equals 10 cm
Height (h) = 8 cm
Size (h) equals 12 cm
We may use the following formula to get the box's surface area:
Surface Area = 2(lh, lw, and wh)
Inputting the values provided yields:
Surface Area equals 2 (10 x 8, 10 x 12, and 8 x 12) square centimetres.
Surface Area is equal to 2 (80, 120, and 96 square cm).
Surface Area: 2(296) square centimetres
592 square cm is the surface area.
Hence, the rectangular box has a surface area of 592 square cm.
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5. Verify that the equation is an identity. a) \( \frac{\cos x}{\tan x+\sec x}=1-\sin x \) b) \( \frac{\sec x+1}{\sec x-1}=(\csc x+\cot x)^{2} \)
Use the identity \(\sec x=\frac{1}{\cos x}\) to simplify the expression:
\( (\csc x+\cot x)^{2}=\frac{\sec x+1}{\sec x-1} \)
To verify that the given equations are identities, we need to simplify the expressions on each side of the equation and show that they are equal. We can do this by using the trigonometric identities and algebraic manipulation.
a) \( \frac{\cos x}{\tan x+\sec x}=1-\sin x \)
Start by simplifying the left side of the equation:
\( \frac{\cos x}{\tan x+\sec x}=\frac{\cos x}{\frac{\sin x}{\cos x}+\frac{1}{\cos x}} \)
Multiply the numerator and denominator by \(\cos x\) to get rid of the fractions:
\( \frac{\cos x}{\tan x+\sec x}=\frac{\cos x \cdot \cos x}{\sin x+1} \)
Now, use the identity \(1-\sin^2 x=\cos^2 x\) to simplify the numerator:
\( \frac{\cos x}{\tan x+\sec x}=\frac{1-\sin^2 x}{\sin x+1} \)
Factor the numerator:
\( \frac{\cos x}{\tan x+\sec x}=\frac{(1-\sin x)(1+\sin x)}{\sin x+1} \)
Cancel out the common factor:
\( \frac{\cos x}{\tan x+\sec x}=1-\sin x \)
We have shown that the left side of the equation is equal to the right side, so the equation is an identity.
b) \( \frac{\sec x+1}{\sec x-1}=(\csc x+\cot x)^{2} \)
Start by simplifying the right side of the equation:
\( (\csc x+\cot x)^{2}=(\frac{1}{\sin x}+\frac{\cos x}{\sin x})^{2} \)
Expand the square:
\( (\csc x+\cot x)^{2}=\frac{1+2\cos x+\cos^2 x}{\sin^2 x} \)
Use the identity \(1-\cos^2 x=\sin^2 x\) to simplify the denominator:
\( (\csc x+\cot x)^{2}=\frac{1+2\cos x+\cos^2 x}{1-\cos^2 x} \)
Factor the denominator:
\( (\csc x+\cot x)^{2}=\frac{1+2\cos x+\cos^2 x}{(1-\cos x)(1+\cos x)} \)
Now, simplify the numerator by factoring:
\( (\csc x+\cot x)^{2}=\frac{(1+\cos x)^2}{(1-\cos x)(1+\cos x)} \)
Cancel out the common factor:
\( (\csc x+\cot x)^{2}=\frac{1+\cos x}{1-\cos x} \)
Finally, use the identity \(\sec x=\frac{1}{\cos x}\) to simplify the expression:
\( (\csc x+\cot x)^{2}=\frac{\sec x+1}{\sec x-1} \)
We have shown that the right side of the equation is equal to the left side, so the equation is an identity.
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Find the fourth term of an arithmetic sequence with a_(1)=3 and a recursive formula of a_(n)=a_(n-1)-9.
The fourth term of the arithmetic sequence is -24.
What is arithmetic sequence?An arithmetic sequence is a sequence of numbers in which each term after the first is formed by adding a constant, called the common difference, to the preceding term.
The fourth term of an arithmetic sequence with a1=3 and a recursive formula of an=an-1-9 can be found by using the recursive formula repeatedly until the fourth term is reached.
First, find the second term by plugging in n=2 into the recursive formula:
a2=a2-1-9=a1-9=3-9=-6
Next, find the third term by plugging in n=3:
a3=a3-1-9=a2-9=-6-9=-15
Finally, find the fourth term by plugging in n=4:
a4=a4-1-9=a3-9=-15-9=-24
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Find the logistic function f with the given properties.
f(0) = 1, f has limiting value 11, and for small values of x, f is approximately exponential and grows by 75% with every increase of 1 in x.
f(x) = ____
The logistic function is then given by f(x) = 11 / (1 + e^(-0.75x))
The logistic function with the given properties is given by: f(x) = 11 / (1 + e^(-0.75x))
To understand this, we can start by noticing that for small values of x, f is approximately exponential and grows by 75% with every increase of 1 in x. This implies that the function can be written as f(x) = e^(0.75x). Since the limiting value of f is 11, the function can be written as f(x) = 11e^(0.75x). The logistic function is then given by f(x) = 11 / (1 + e^(-0.75x)), which satisfies the given conditions.
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Draw the image of the given rotation of the preimage
the image of the point (x,y) after the rotation is (y,-x).
what is a graph?
A graph is a visual representation of data, relationships or information that is often used in mathematics, science, engineering, economics, and other fields.
A graph consists of a set of points, called vertices or nodes, that are connected by lines or curves, called edges or arcs. The vertices represent objects or events, while the edges represent the relationships or connections between them.
To apply the rotation of 90 degrees clockwise about the origin to the point (x,y), we use the following formula:
r(90,0)(x,y) = (y, -x)
To visualize this transformation, we can plot the preimage point (x,y) and the image point (y,-x) on a coordinate plane, and draw an arrow to represent the rotation.
Therefore, the image of the point (x,y) after the rotation is (y,-x).
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WHICH ONEEE III NEED HELPP
The table with order pairs (-3,2), (0, 2), (3, 7),(5, 8) (6, 8) represents the function.
What is a function?A relation is a function if it has only One y-value for each x-value.
In the given tables the third table represents the function.
The left bottom table is the third table.
As we observe the other tables there are repeating values of x.
In a function, for every y value there should be one unique x value.
(-3,2), (0, 2), (3, 7),(5, 8) (6, 8) represents the function.
Hence, the table with order pairs (-3,2), (0, 2), (3, 7),(5, 8) (6, 8) represents the function.
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HELP BRO MY BRAIN CELLS ARE DISSOLVING
Answer:
Step-by-step explanation:
(2x + 3) + (6x + 25) = 180 they are supplementary angles
8x + 28 = 180
8x = 180 - 28 = 252
x = 152/8 = 19
m∠EFG = 6x + 25 = 6(19) + 25 = 139
m∠IFH = 90 - (2x + 3) = 90 - 2(19) - 3 = 49
∠EFD≅∠GFH because they are vertical angles
Determine the domain of the quadratic function. f (x) = 7x2 - 10x + 10 Get Hint Enter Your Stop Here 16 8704 МАА 3
The domain of the quadratic function f(x) = 7x2 - 10x + 10 is all real numbers.
To determine the domain, set the equation equal to 0 and solve for x.
7x² - 10x + 10 = 0
7x2 - 10x = -10
7x(x - 10) = -10
x(x - 10) = -10/7
x = 10 or x = -10/7
The domain of a function is the set of all possible input values (x-values) for which the function is defined. In the case of the quadratic function f(x) = 7x^2 - 10x + 10, there are no restrictions on the values that x can take.
That is, any real number can be plugged in for x, and the function will give a real output. Therefore, the domain of this quadratic function is all real numbers, or (-∞, ∞).
Since there are no restrictions on x, the domain of the function is all real numbers.
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x^2+14x-51=0 I have solve by completing the square
Answer:
x=3 or x=−17
Step-by-step explanation:
Let's solve your equation step-by-step.
x2+14x−51=0
Step 1: Add 51 to both sides.
x2+14x−51+51=0+51
x2+14x=51
Step 2: The coefficient of 14x is 14. Let b=14.
Then we need to add (b/2)^2=49 to both sides to complete the square.
Add 49 to both sides.
x2+14x+49=51+49
x2+14x+49=100
Step 3: Factor left side.
(x+7)2=100
Step 4: Take square root.
x+7=±√100
Step 5: Add -7 to both sides.
x+7+−7=−7±√100
x=−7±√100
x=−7+10 or x=−7−10
x=3 or x=−17
a 5.23. Exercise. Make a ruler-and-compass construction of the center of given circle.
The center of the circle can be found using a ruler-and-compass construction by drawing two arcs that intersect at the center of the circle. The radius of the circle can also be found using a ruler by measuring the distance from the center of the circle to any point on the circumference of the circle.
To make a ruler-and-compass construction of the center of a given circle, follow these steps:
Draw a line segment through the center of the circle using a ruler. This line segment should intersect the circle at two points.
Using a compass, draw an arc with the center at one of the intersection points and the radius equal to the length of the line segment.
Draw another arc with the center at the other intersection point and the same radius.
The point where the two arcs intersect is the center of the circle.
Use a ruler to draw a line from the center of the circle to any point on the circumference of the circle. This line is the radius of the circle.
The center of the circle can be found using a ruler-and-compass construction by drawing two arcs that intersect at the center of the circle. The radius of the circle can also be found using a ruler by measuring the distance from the center of the circle to any point on the circumference of the circle.
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4) Let U = {x:6≤ x ≤40, x is a positive whole number} ,A = {1, 2, 3, 4, 5} , B = {2, 3, 4, 5} C = {x|x is an integer and 2x + 1 = 5 }and D = {x|x is a positive integer and x²/2 = even numbers less than 10}. Provide answer to the following: a) AUA =
The set that is resulting of the union operation between set A and it's complementary A' is given as follows:
U = {x:6≤ x ≤40, x is a positive whole number}.
How to obtain the union between two sets?The union operation of two sets is a mathematical operation that combines all the elements from two sets into a single set, without any duplicates. The union of two sets A and B is denoted as A ∪ B, and is defined as the set that contains all elements that are in at least one of the sets A and B.
The set A is given as follows:
A = {1, 2, 3, 4, 5}.
The complement of it's set is the set containing all the elements that are in the universe set and are not in A.
The union operation between a set and it's complement is always the universe set, hence it is given as follows:
U = {x:6≤ x ≤40, x is a positive whole number}.
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What is the relationship between a millimeter and a meter?
1 meter = 1000 millimeter
Answer: :)
Step-by-step explanation:
The metric system is based on the International System of Units (SI), which is a standardized system of measurement used worldwide. The SI unit for length is the meter, and all other units of length in the metric system are derived from it.
The relationship between millimeters and meters can be expressed mathematically as follows:
1 mm = 0.001 m
1 m = 1000 mm
This means that if you have a length measurement in millimeters, you can convert it to meters by dividing by 1000. Conversely, if you have a length measurement in meters, you can convert it to millimeters by multiplying by 1000.
For example, if you have a piece of wire that measures 500 mm long, you can convert this to meters as follows:
500 mm ÷ 1000 = 0.5 m
Similarly, if you have a piece of fabric that measures 2.5 m long, you can convert this to millimeters as follows:
2.5 m x 1000 = 2500 mm
In summary, the relationship between millimeters and meters is that they are both units of length in the metric system, with one meter being equal to 1000 millimeters.
To operate safely the total weight on an elevator must be less than 400 pounds
The inequality of the statement is x < 400
How to determine the inequality of the statementFrom the question, we have the following parameters that can be used in our computation:
To operate safely the total weight on an elevator must be less than 400 pounds
Represent the variable with x
So, we have
x must be less than 400 pounds
less than can be represented as <
Using the above as a guide, we have the following:
x < 400
Hence, the expression is x < 400
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A microwave was originally sold for $148 and has been marked up to $222. What is the percentage increase for the microwave? Round to the nearest one percent.
Answer:
I think the answer is 33%
Answer:
33%
Step-by-step explanation:
the percent difference of 222 to 148 is 66.66. meaning 100%-66.66% is 33.34%. rounding this the the nearest one percent is 33%. the microwave price has increased by 33% respectively .
I NEED HELP GUYS PLS NEED THIS BY FRIDAY
Answer:
∠M = ∠P = 112
∠N = ∠Q = 68
Step-by-step explanation:
The interior angles of a quadrilateral always sum up to 360°.
since NP//MP and NM//PQ => ∠N = ∠Q and ∠M = ∠P
and ∠N + ∠Q + ∠M + ∠P = 360
∠M = ∠P
6x - 2 = 4x + 36
6x - 4x = 36 + 2
2x = 38
x = 38/2 = 19
=> ∠M = ∠P = 4(19) + 36 = 112
∠N + ∠Q + ∠M + ∠P = 360
2∠N + 2∠M = 360
∠N + ∠M = 180
∠N = 180 - ∠M = 180 - 112 = 68
∠N = ∠Q = 68
Write a trinomial with the leading coefficient -10 in terms of x. A trinomial in simplest form with leading coefficient -10 in terms of x is
A trinomial with the leading coefficient -10 in terms of x in simplest form is -10x^2.
A trinomial with a leading coefficient of -10 in terms of x can be written as:
-10x^2 + bx + c
Where b and c are any real numbers. The simplest form of this trinomial would be:
-10x^2 + 0x + 0
Which can be simplified to:
-10x^2
A trinomial is an algebraic expression that has three non-zero terms and has more than one variable in the expression. This expression has three terms. Therefore, this expression is called a trinomial.
A trinomial is a type of polynomial but with three terms.
The examples of trinomials are: x + y + 7. ab + a +b,
3x+5y+8z with x, y, z variables
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I need help writing equations of a parallel and perpendicular equations
The equation of the line that is parallel to y = 2/5x + 2 and passes through (-5, 6) is y = (2/5)x + 8.
What connection exists between the slopes of parallel and perpendicular lines?Perpendicular line slopes are the negative reciprocals of one another. In other words, any line that is perpendicular to a line with a slope of m will have a slope of -1/m. If lines are parallel if they have the same slope.
The slope of the given line is 2/5.
We know that, any line perpendicular to it will have a slope that is the negative reciprocal of 2/5, which is -5/2.
Using the point-slope form of a line:
y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.
Substituting the values we know:
y - 6 = (-5/2)(x - (-5))
y - 6 = (-5/2)x - (25/2)
y = (-5/2)x + 17.5
Therefore, the equation of the line perpendicular to y = 2/5x + 2 and passes through (-5, 6) is y = (-5/2)x + 17.5.
The slope of parallel lines are same. Thus, using point slope form:
y - 6 = (2/5)(x - (-5))
y - 6 = (2/5)x + 2
y = (2/5)x + 8
Therefore, the equation of the line that is parallel to y = 2/5x + 2 and passes through (-5, 6) is y = (2/5)x + 8.
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Show that circle D with center open parentheses 8 comma negative 2 close parentheses and radius 8 is similar to circle E with center (5, 1) and radius 2.
The ratio of their radii is 8/2 = 4.
What is ratio?Ratio is a comparison of two or more quantities expressed in terms of their relative sizes. It is a way to express one number as a fraction of another number. For example, if a person has three apples and two oranges, the ratio of apples to oranges can be expressed as 3:2. Ratios can also be expressed as fractions or percentages.
To show that Circle D and Circle E are similar, we need to determine if the ratio of their radii is equal to the ratio of their distances from the origin. The ratio of their radii is 8/2 = 4. The distance from the origin for Circle D is sqrt((8)²+ (-2)²) = sqrt(64 + 4) = sqrt(68) = 8.24. For Circle E, the distance from the origin is sqrt((5)² + (1)²) = sqrt(26) = 5.1. The ratio of their distances from the origin is 8.24/5.1 = 1.61. Since the ratio of their radii (4) is equal to the ratio of their distances from the origin (1.61), we can conclude that Circle D and Circle E are similar.
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What is the experiment in relation to probability
Answer:
Step-by-step explanation:
The experiment is repeating and recording the result of an event in order to determine the events probability. For example, tossing a coin is an experiment.
Given f (x) = |x| - 3; x ≥
0, write an equation for f -1(x).
(Hint: Sketch f (x) and note the domain
and range.)
Select one:
a. f -1(x) = |x + 3|;
x ≥ 0
b. f -1(x) = |x | + 3;
x ≥ -3
c. f -1(x)
The correct answer is option b. f -1(x) = |x| + 3; x ≥ -3.
To find the inverse of a function, we can switch the x and y values and solve for y. In this case, we can start with the original equation:
f(x) = |x| - 3
Switch the x and y values:
x = |y| - 3
Solve for y:
|x| = y + 3
|y| = x + 3
Since the original function has a domain of x ≥ 0, the inverse function will have a range of y ≥ 0. This means that the absolute value of y will always be positive, so we can drop the absolute value bars:
y = x + 3
So the inverse function is:
f -1(x) = x + 3
And since the original function has a domain of x ≥ 0, the inverse function will have a domain of x ≥ -3. So the final equation for the inverse function is: f -1(x) = x + 3; x ≥ -3
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g or synthetic division to divide. Make sure to write your ans (2x^(3)-10x^(2)+14x-24)-:(x-4)
The Synthetic divison is (2x^(3)-10x^(2)+14x-24) ÷ (x-4) = 2x^(2) - 2x + 6.
To divide (2x^3 - 10x^2 + 14x - 24) by (x - 4) using synthetic division, the following steps should be taken:
1. Write the coefficients of the dividend in a row, placing the divisor to the extreme left of the row.
2. Bring the first coefficient of the divisor down.
3. Multiply the divisor by the number directly below it in the row and write the answer to the right of the number.
4. Add the two numbers to the right of the divisor and write the answer below.
5. Repeat steps 3 and 4 until the last number in the row is reached.
6. The last number in the row is the remainder, while the numbers to its left form the quotient.
For the example given, the process is as follows:
Therefore, the quotient is
.
To divide (2x^(3)-10x^(2)+14x-24) by (x-4) using synthetic division, we can follow the steps below:
Step 1: Write down the coefficients of the dividend polynomial in descending order of the exponents. In this case, the coefficients are 2, -10, 14, and -24.
Step 2: Write down the constant term of the divisor polynomial with the opposite sign. In this case, the constant term is -4, so we write down 4.
Step 3: Bring down the first coefficient of the dividend polynomial, which is 2.
Step 4: Multiply the number brought down by the constant term of the divisor polynomial, and write the result under the next coefficient of the dividend polynomial. In this case, 2 * 4 = 8, so we write 8 under -10.
Step 5: Add the numbers in the column, and write the result below. In this case, -10 + 8 = -2.
Step 6: Repeat steps 4 and 5 until all the coefficients of the dividend polynomial have been used. In this case, we get:
-2 * 4 = -8, 14 + (-8) = 6
6 * 4 = 24, -24 + 24 = 0
Step 7: The numbers in the last row are the coefficients of the quotient polynomial, and the last number is the remainder. In this case, the quotient polynomial is 2x^(2) - 2x + 6, and the remainder is 0.
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slope=-1 y-intercept=8
Answer: y = x + 8
Step-by-step explanation:
30% of the applications received for a position in a graduate school are rejected.
a)Determine the expected number of rejected applications among the next 10 applications.
b)What is the probability that among the next 15 applications, none will be rejected?
c)What is the probability that among the next ten applications, seven will be rejected?
d)What is the probability that among the next ten applications, between 1 and 8 applications (inclusively) will be rejected?
a. The expected number of applications rejected is 3
b. The probability that among the next 15 applications, none will be rejected is 0.042
c. The probability that among the next ten applications, seven will be rejected is 0.48
d. The probability that among the next ten applications, between 1 and 8 applications is 0.95
Binomial distribution formula:The binomial distribution formula is used to calculate the probability of getting a certain number of successes (x) in a fixed number of independent trials (n) with a known probability of success (p) for each trial. The formula is:
P(x) = (n choose x) × pˣ × (1 - p)⁽ⁿ⁻ˣ⁾Here we have
30% of the applications received for a position in a graduate school are rejected.
a) The number of rejected applications among the next 10 applications follows a binomial distribution with parameters n = 10 and p = 0.3.
The expected number of rejected applications is:
E(X) = np = 10 * 0.3 = 3
Hence, the expected number of applications rejected is 3
b) The probability of being rejected is 0.3
The probability that none of the next 15 applications will be rejected is:
P(X = 0) = (1 - p)ⁿ = (1 - 0.3)¹⁵= 0.042
Therefore, the probability that none of the next 15 applications will be rejected is 0.042 or approximately 4.2%.
c) The probability that 7 of the next ten applications will be rejected is:
By using the binomial distribution formula
P(X = 7) = (10, 7) × 0.3⁷ × 0.7³ =
= 6435 × 0.0002187 × 0.343 = 0.48
Therefore, the probability that 7 of the next 10 applications will be rejected is 0.48 or approximately 48%.
d) The probability that between 1 and 8 (inclusive) of the next ten applications will be rejected is:
P(1 ≤ X ≤ 8) = P(X ≤ 8) - P(X ≤ 0) = Σ P(X = i) for i = 1 to 8
We can use the complement rule and calculate the probability of having 0 or 9 rejected applications, and subtract that from 1:
=> P(1 ≤ X ≤ 8) = 1 - [P(X = 0) + P(X = 9) + P(X = 10)]
= 1 - [(1 - p)ⁿ + (n, 1) × p¹ × (1 - p)⁽ⁿ⁻¹⁾ + (n, 0) × p⁰ × (1 - p)ⁿ]
= 1 - [(0.7)¹⁰ + ((10,1) × 0.3 × 0.7⁹) + (10, 0) (0.3)¹⁰]
= 1 - [ 0.02824 + 0.01210 + 0.000006]
= 1 - [ 0.040346]
= 0.95
Hence, The probability that between 1 and 8 (inclusive) of the next ten applications will be rejected is 0.95 or approximately 95%
Therefore,
a. The expected number of applications rejected is 3
b. The probability that among the next 15 applications, none will be rejected is 0.042
c. The probability that among the next ten applications, seven will be rejected is 0.48
d. The probability that among the next ten applications, between 1 and 8 applications is 0.95
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