So, the final answers are:
a.) f(g(-2)) = 1/3
b.) g(f(1)) = 8
Question 1: Use the following functions to evaluate each expression f(x) = 1/(x + 2) and g(x) = 3x + 7.
a.) f(g(-2)) = f(3(-2) + 7) = f(1) = 1/(1 + 2) = 1/3
b.) g(f(1)) = g(1/(1 + 2)) = g(1/3) = 3(1/3) + 7 = 1 + 7 = 8
So, the final answers are:
a.) f(g(-2)) = 1/3
b.) g(f(1)) = 8
In summary, to evaluate the expression f(g(x)) or g(f(x)), we need to first find the value of the inner function and then substitute it into the outer function. This process is called function composition. It is important to follow the order of operations and simplify the expression as much as possible.
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YO CAN I HAVE SOME HELP FOR THIS LAST TWO THANK YALL I HAVE BEEN AT SCHOOL FOR HOURS AND I JUST CANT DO IT ANYMORE
Answer:
429 m³
Step-by-step explanation:
The volume of right rectangular = Length x Width x Height
The volume of right rectangular prism = 12 x 5 1/2 x 6 1/2
= 12 x 11/2 x 13/2
= 3 x 11 x 13
= 429 m³
etermine whether the equation is condi 10[4-(3-2x)]+4x=3(8x+4)-2 s the equation a conditional, an identity, or
10[4-(3-2x)]+4x=3(8x+4)-2 is an identity equation.
To determine whether the equation is a conditional, an identity, or a contradiction, we need to simplify the equation and see if it is true for all values of x, true for some values of x, or false for all values of x.
First, let's simplify the equation:
10[4-(3-2x)]+4x=3(8x+4)-2
10[4-3+2x]+4x=24x+12-2
10[1+2x]+4x=24x+10
10+20x+4x=24x+10
24x-24x=10-10
0=0
Since the equation is true for all values of x, it is an identity equation. An identity equation is an equation that is true for all values of the variable.
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_________are the most powerful computers
Answer: Supercomputers
Step-by-step explanation: Supercomputers are high-performance computing systems that are designed to perform complex and demanding tasks at incredibly fast speeds. They are capable of processing massive amounts of data and performing calculations at a rate that far exceeds that of conventional computers.
Supercomputers are used for a variety of purposes, including scientific research, weather forecasting, modeling and simulation, data analysis, and artificial intelligence. They are especially valuable in fields that require extensive computational power, such as astrophysics, molecular modeling, climate studies, and cryptography.
One key characteristic of supercomputers is their parallel processing capability. They are designed with multiple processors or cores that work together to execute tasks simultaneously, allowing for faster and more efficient computation. This parallel architecture enables supercomputers to tackle large-scale problems by breaking them down into smaller tasks that can be processed concurrently.
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Find the present value PV of the annuity necessary to fund the withdrawal given. HINT [See Example 3.] (Assume end-of-period withdrawals and compounding at the same intervals as withdrawals. Round your answer to the nearest cent.) $500 per month for 15 years, if the annuity earns 6% per year PV = $
The present value PV of the annuity necessary to fund the withdrawal is $5,354.82, rounded to the nearest cent.
The present value of an annuity is calculated using the following formula:
PV = A[((1+i)n-1)/(i(1+i)n)]
where A = amount of each annuity payment, i = interest rate, and n = number of payments.
For this problem, A = $500, i = 6%, and n = 15 years.
Therefore, the present value of the annuity necessary to fund the withdrawal is:
PV = $500[((1+0.06)15-1)/(0.06(1+0.06)15)]
PV = $500[5.72982/0.105638]
PV = $5,354.82
Therefore, the present value PV of the annuity necessary to fund the withdrawal is $5,354.82, rounded to the nearest cent.
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URGENT NEED OF HELP PLEASE I DON'T HAVE MUCH TIME!!!!!!!!!!!!!!!!!!!
A slide in a playground must have a maximum average slope of 30°. The top of the slide is 6 ft off the ground and the length of the slide is 11.7 ft. Does the slide meet the safety requirements? Show your work and label the diagram to support your answer.
The slide does not meet safety requirements.
What is Slope?A line's steepness and direction are measured by the line's slope. Without actually using a compass, determining the slope of lines in a coordinate plane can assist in forecasting whether the lines are parallel, perpendicular, or none at all.
As per the given data:
maximum average slope = 30°
height of the slide = 6 ft
length of the slide = 11.7 ft
For calculating slope (θ):
sinθ = (P/H) {P is perpendicular and H is hypotunese}
sinθ = (height/length)
sinθ = (6/11.7)
sinθ = 0.512
Using sin inverse:
θ = 30.8°
θ > maximum average slope
Hence, the slide does not meet safety requirements.
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Use the suggested substitution to write the expression as a
trigonometric expression. Simplify your answer as much as possible.
Assume 0≤θ≤π2.
√9−9x2, x=cos(θ)
The expression √9−9x2 can be written as a trigonometric expression 3sin(θ) using the substitution x = cos(θ).
To write the expression as a trigonometric expression using the suggested substitution, we can substitute x = cos(θ) into the expression and simplify:
√9−9x2 = √9−9(cos(θ))^2
= √9−9(cos^2(θ))
= √9(1−cos^2(θ))
= √9(sin^2(θ))
= 3sin(θ)
Therefore, the expression √9−9x2 can be written as a trigonometric expression 3sin(θ) using the substitution x = cos(θ).
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Asanji wants to buy a wagon for $93.26. He gives the cashier $100. How much change does he receive?
Answer:
6.74
Step-by-step explanation:
100 - 93.26
Answer:
$6.74
Step-by-step explanation:
100.00 - 93.26 = 6.74
Helping in the name of Jesus.
Statistics question, please explain what type of problem it is and how to do it
In a group of 60 students; 30 go for extra help and 22 study at home only. Find the probability that a person picked from this group at random is either going for extra help or only studying at home?
The probability that a person chosen at random from this group will either seek further assistance or merely study at home is 5/6, or roughly 0.833. (rounded to three decimal places).
What is the probability formula?Typically, the probability is defined as the ratio of favourable events to all other outcomes in the sample space. The formula for probability of an occurrence is P(E) = (Number of favourable outcomes) (Sample space).
The formula is as follows:
P(AorB) = P(A)+P(B)-P(AandB)
where A and B are two events.
Then, we have:
P(A) = 30/60 = 1/2 (since 30 out of 60 students go for extra help)
P(B) = 22/60 (since 22 out of 60 students study at home only)
Using the formula, we can then find:
P(AorB) = P(A)+P(B)-P(AandB)
= 1/2 + 22/60 - 0
= 5/6
Hence, the probability that someone chosen at random from this group will either seek further assistance or merely study at home is 5/6, or roughly 0.833. (rounded to three decimal places).
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Which equations have the same value of x as Three-fifths (30 x minus 15) = 72? Select three options. 18 x minus 15 = 72 50 x minus 25 = 72 18 x minus 9 = 72 3 (6 x minus 3) = 72 x = 4.5
The three equations that have the same value of x as the original equation are:
18x - 15 = 72
3(6x - 3) = 72
x = 4.5
What is an equation?
An equation is a statement that two expressions are equal. It typically contains one or more variables (represented by letters) and mathematical operations such as addition, subtraction, multiplication, and division. Equations can be used to represent relationships between quantities or to solve for unknown values.
The correct options are:
18x - 15 = 72
18x - 9 = 72
x = 4.5
To see why, we can start by simplifying the original equation:
Three-fifths (30x - 15) = 72
(3/5)(30x - 15) = 72
18x - 9 = 72
18x - 15 = 72 + 15
18x = 87
x = 87/18
So we see that x = 87/18 is the solution to the original equation.
Now let's check each of the answer choices:
18x - 15 = 72
Solving for x, we get x = 87/18, which is the same as the solution to the original equation. This equation is equivalent to the original equation.
50x - 25 = 72
Solving for x, we get x = 97/50, which is not the same as the solution to the original equation. This equation is not equivalent to the original equation.
18x - 9 = 72
Solving for x, we get x = 81/18 = 9/2, which is not the same as the solution to the original equation. This equation is not equivalent to the original equation.
3(6x - 3) = 72
Simplifying, we get 18x - 9 = 72, which is equivalent to the second equation listed above. So this equation is also equivalent to the original equation.
x = 4.5
This is the same solution as the original equation, so this equation is also equivalent to the original equation.
Therefore, the three equations that have the same value of x as the original equation are:
18x - 15 = 72
3(6x - 3) = 72
x = 4.5
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Add or subtract these rational expressions. Show your common denominators and numerators on this sheet or separate paper. Factor denomunators when possible. 1. 5/8 - 3/8x
2. 9/15x + 2/15x
3. 5x/7 - 2x/7
1). 1/4
2). 11/15x
3). 3x/7
1. To add or subtract the rational expressions 5/8 and 3/8x, we need a common denominator. The smallest common denominator is 8x, so we can rewrite each expression with a denominator of 8x:
5/8 = 5x/8x
3/8x = 3/8 * 1/x = 3x/8x
Now we can combine the numerators and simplify:
5x/8x - 3x/8x = (5x - 3x)/8x = 2x/8x = 1/4
Therefore, 5/8 - 3/8x = 1/4.
2. To add or subtract the rational expressions 9/15x and 2/15x, we again need a common denominator. The smallest common denominator is 15x, so we can rewrite each expression with a denominator of 15x:
9/15x = 3/5x
2/15x = 2/15 * 1/x = 2/15x
Now we can combine the numerators and simplify:
3/5x + 2/15x = (9 + 2)/15x = 11/15x
Therefore, 9/15x + 2/15x = 11/15x.
3. To add or subtract the rational expressions 5x/7 and -2x/7, we already have a common denominator of 7. We can simply combine the numerators:
5x/7 - 2x/7 = (5x - 2x)/7 = 3x/7
Therefore, 5x/7 - 2x/7 = 3x/7.
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Mathe question 2 help
Answer:
Option B
See below
Step-by-step explanation:
Two simultaneous equations:
[tex]7x - 4y - 8 = 0[/tex] —————- (equation i)
[tex]2x^{2} + x + 4y + 8 = 0[/tex] ——- (equation ii)
Step 1:
Rearrange (equation i):
[tex]7x - 4y = 8[/tex] ———- updated (equation i)
Step 2:
Substitute the updated (equation i) into (equation ii) and bring like terms together to simplify them in reduced forms:
[tex]2x^{2} + x + 4y + (7x - 4y) = 0[/tex]
[tex]2x^{2} + x + 4y + 7x - 4y = 0[/tex]
[tex]2x^{2} + x + 7x + 4y - 4y = 0[/tex]
[tex]2x^{2} + 8x = 0[/tex]
Step 3:
Factorize:
[tex]2x(x + 4) = 0[/tex]
Either [tex]2x = 0[/tex]
∴ [tex]x = 0[/tex]
Or [tex]x + 4 = 0[/tex]
∴ [tex]x = -4[/tex]
Step 4:
Substitute these values of x in (equation ii) to determine their corresponding values of y:
For x = 0:
[tex]2(0)^{2} + 0 + 4y + 8 = 0[/tex]
[tex]4y + 8 = 0[/tex]
[tex]4y = -8[/tex]
[tex]y = \frac{8}{-4}[/tex]
∴[tex]y = -2[/tex]
For x = -4:
[tex]2(-4)^{2} + (-4) +4y + 8 = 0[/tex]
[tex]2(16) - 4 + 4y + 8 = 0[/tex]
[tex]32 - 4 + 4y + 8 = 0[/tex]
[tex]32 - 4 + 8 + 4y = 0[/tex]
[tex]36 + 4y = 0[/tex]
[tex]4y = -36[/tex]
[tex]y = \frac{-36}{4}[/tex]
∴ [tex]y = -9[/tex]
∴The values of y are [tex]-2[/tex] and [tex]-9[/tex]
∴Option B
solve the system of linear equation by elimination 8x+3y=-5
3y=x+4 please show work
The solution of linear equations is (x, y) = (-0.8905, 0.708).
To solve this system of linear equations by elimination, first multiply both equations by the same number so that when you add the equations together, one of the variables is eliminated. In this case, we'll multiply the first equation by 3 and the second equation by 8.
8(8x+3y=-5)
3(3y=x+4)
24x + 9y = -15
24y = 8x + 32
Now add the two equations together:
24x + 24y = -15 + 32
24x + 24y = 17
Simplifying this equation, we get:
24y = 17
y = 17/24
y = 0.708
Now, plug in the value of y into one of the original equations to solve for x. We'll use the first equation:
8x + 3(0.708) = -5
8x + 2.124 = -5
8x = -7.124
x = -7.124/8
x = -0.8905
Therefore, the solution to the system of linear equations is (x, y) = (-0.8905, 0.708).
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9. A box contains three blue bulbs, four green bulbs and five red bulbs. Four bulbs taken out of the box at random and without replacements. What is the probability that: (1) are the first two are of the same colour and the last two are of different colours.
The probability that the first two bulbs are of the same colour and the last two are of different colours is 25/48.
The probability that the first two bulbs are of the same colour and the last two are of different colours is calculated as follows:
Given:
n(B) = 3 (Number of blue bulbs)
n(G) = 4 (Number of green bulbs)
n(R) = 5 (Number of red bulbs)
Formula: P(A) = n(A) / n(S)
Where,
P(A) = Probability of event A
n(A) = Number of favorable outcomes
n(S) = Number of possible outcomes
The probability of selecting two bulbs of the same colour is calculated as:
P(SS) = n(B) * n(B) / n(S)
= 3 * 3 / 12
= 1/4
The probability of selecting two bulbs of different colour is calculated as:
P(DD) = n(B) * n(G) * n(R) * n(R) / n(S)
= 3 * 4 * 5 * 5 / 12
= 25/12
Therefore, the required probability is calculated as:
P(SSDD) = P(SS) * P(DD)
= 1/4 * 25/12
= 25/48
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Evaluate 5d - 25/5 if d = 8
Answer:
=35
Step-by-step explanation:
5d - 25/5
5(8) -25/5
40 - 5
=35
Answer:
the answer is 35
Step-by-step explanation:
d=8, therefore plug 8 into the equation
5d - [tex]\frac{25}{5}[/tex]
5(8)- [tex]\frac{25}{5}[/tex]
40-5
35
I don’t understand how to solve this when one number is missing! Pls help
how do i convert an improper fraction to mixed number
the two triangles in the diagram are similar there are 2 possible values for x
The possible values of x are 2 and 17
How to determine the possible values of xThe complete question is added as an attachment
From the question, we have the following parameters that can be used in our computation:
Similar triangles
In the triangles, we have the following possible equations
15/18 = 10/(10 + x)
10/18 = 15/(10 + x)
Solving the equations, we have
15/18 = 10/(10 + x)
150 + 15x = 180
15x = 30
x = 2
10/18 = 15/(10 + x)
100 + 10x = 270
10x = 170
x =17
Hence, the values of x are 2 and 17
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Complete Question
The two triangles in the diagram are similar there are 2 possible values for x
See attachment for image of the triangle
PLEASE HELP A GIRL OUTTTT ITS DUE FOR TMR
Answer:
8x^2-4x-15
Step-by-step explanation:
To find the sum of the functions, we add them together:
f(x) + g(x) = (5x + 4) + (3x² - 9)
Simplifying, we get:
f(x) + g(x) = 3x² + 9x - 5
Therefore, the sum of the functions is 3x² + 9x - 5, which is option D.
You randomly draw a marble from a bag, record its color, and then replace it. You draw a blue marble 23 out of 25 times. What is the experimental probability that the next marble will be blue? Write your answer as a fraction, decimal, or percent.
The next marble's experimental probability of being blue, according to an experiment, is: 23/25.
Explain about the experimental probability?Based on empirical data, experimental probability is the likelihood that an event will occur. A theoretical probability would be obtained by dividing the number of different ways to achieve the desired result by the total number of outcomes.
A marble is drawn at random from a bag, its colour is noted, and it is then replaced.In 23 of the 25 draws, a blue marble is produced.The next marble's likelihood of being blue, according to an experiment, is:
Experimental probability = Number of time blue appears / Total outcomes
Experimental probability = 23/25
Thus, the next marble's experimental probability of being blue, according to an experiment, is: 23/25.
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A garden has width √√13 and length 7√√13. What is the perimeter of the garden in simplest
radical form?
014 √√13 units
016√√13 units
091 units
08√√13 units
In a direct variation, the variable k is the ?
Answer: slope of the line
Step-by-step explanation: I'm not sure if we're talking about the same thing though
Answer: The slope (stands in for m)
Step-by-step explanation:
In a direct variation, the variable m (slope) is swapped for k.
Question
Gerard worked the same number of hours, x, on Monday, Tuesday, and Wednesday, but on Thursday, he worked 5 hours less than the previous day. The total number of hours Gerard worked can be found using the expression, 4x−5 .
What does the "4" represent in the expression, 4x−5 ?
--------------------------------------------------------------------------------
the number of hours Gerard worked on Thursday
the number of hours Gerard worked each day
the total number of hours Gerard worked
the number of days Gerard worked
The number "4" in the expression 4x - 5 represent the number of days Gerard worked.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables. Equations can either be linear, quadratic, cubic and so on depending on the degree.
Gerard worked the same number of hours, x, on Monday, Tuesday, and Wednesday, but on Thursday, he worked 5 hours less than the previous day.
The total number of hours Gerard worked can be found using the expression, 4x − 5
Hence:
The number "4" in the expression 4x - 5 represent the number of days Gerard worked which is Monday, Tuesday, and Wednesday and Thursday
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Umm soo I kinda need help with math
Answer:
Option A.
Step-by-step explanation:
We are asked to add these 2 expressions.
First, we should know how to combine like terms.
What is combining like terms?Combining like terms is adding 2 or more numbers, or variables that are alike.
[tex]-\frac{4}{5} t+\frac{5}{3} s \ + -3 - \frac{7}{5} s+2t=?[/tex]
Let's first begin by combining s and t. We can combine like terms for these two variables.
[tex]-\frac{4}{5} t + 2t = \frac{6}{5} t \ \checkmark[/tex]
[tex]\frac{5}{3} s \ + \frac{7}{5} s \\Which \ can \ also \ be \ represented \ as...\\\frac{25}{15} s \ - \frac{21}{15} s = \frac{4}{15} s \ \checkmark[/tex]
-3 doesn't have any like terms, meaning it'll stay as -3.
[tex]\frac{6}{5} t + \frac{4}{15} s - 3[/tex]
Therefore, our answers matches with Option A.
Which description best represents the end behavior of the
function f(x)= -x + 5x³ - 2x+5?
Hint: Use the examples in the table to help you answer the
question.
Polynomial End Behavior
Equation Degree Leading Coefficient
y-x²
y=-x²
y=x³
y=-x³
Even
Even
Odd
Odd
Positive
Negative
Positive
Negative
End Behavior
x)+∞, as x-+
F(x), as x--
F(x), as x4+
x), as x--co
x) +∞, as
x-+00
x) +00, as x--
x)-00, as x-+00
Example
The correct description for best represents the end behavior of the
function f (x) = - x⁴ + 5x³ - 2x + 5 is,
⇒ The right-end is y → - ∞ as x → ∞ . The right-end is rising.
The left-end is y → - ∞ as x→ - ∞ . The left-end is falling.
What is mean by Function?A relation between a set of inputs having one output each is called a function. and an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).
Given that;
The function is,
⇒ f (x) = - x⁴ + 5x³ - 2x + 5
Now, We have;
Degree of the polynomial = 4
Hence, We get;
⇒ The right-end is y → - ∞ as x → ∞ . The right-end is rising.
The left-end is y → - ∞ as x→ - ∞ . The left-end is falling.
Thus, The correct description for best represents the end behavior of the
function f (x) = - x⁴ + 5x³ - 2x + 5 is,
⇒ The right-end is y → - ∞ as x → ∞ . The right-end is rising.
The left-end is y → - ∞ as x→ - ∞ . The left-end is falling.
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Which expression represents the volume, in cubic units, of the composite figure
Answer: The second option
Step-by-step explanation:
To find the total area, add the areas of the shapes together
Area of the cylinder = πr²h
Area of the cone = 1/3(πr²h)
Area of the full shape = πr²h + 1/3(πr²h)
The height of the cylinder is 12.
Hope this helps!
Container a holds 750ml of liquid container b holds 1. 25 how much from container b must be poured into container a so they can be equal?
We need to pour 250 ml of liquid from container B to container A so they can be equal in volume.
To make container A and B equal in volume, we need to pour some liquid from container B into container A. Let's call the amount of liquid that we need to pour from container B to container A as "x" ml.
After pouring x ml of liquid from container B to container A, the total volume of liquid in container A will be (750 + x) ml, and the total volume of liquid in container B will be (1250 - x) ml.
Since we want the volumes of both containers to be equal, we can set up the equation:
750 + x = 1250 - x
Solving for x:
2x = 500
x = 250
Therefore, we need to pour 250 ml of liquid from container B to container A so they can be equal in volume.
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In a survey of 300 members of a local sports club, 150 members indicated that they plan to attend the next Summer Olympic Games, 90 indicated that they plan to attend the next Winter Olympic Games, and 60 indicated that they plan to attend both games. How many members of the club plan to attend (a) At least one of the two games? members (b) Exactly one of the games? members (c) The Summer Olympic Games only? members (d) None of the games? members
a) At least one of the two games? - 300 members
b) Exactly one of the games? - 240 members
c) The Summer Olympic Games only? - 150 members
d) None of the games? - 60 members
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i would appreciate anyone's help. thank you!
The equation of the volume of the box in terms of x is 4x³ - 40x² +100x.
What is the V(x) function's domain?The set of all conceivable values of x that make sense in the context of the issue constitutes the domain of the function V(x). This time, x must be a positive value less than or equal to 5, as it reflects the length of the side of the square that was cut out of each corner of the cardboard. The range of V(x) is thus 0 x 5.
Given that, the squares cut at the end are x inches.
Thus the dimension of the rectangular box are:
The length of the box will be (10-2x).
The width of the box will be (10-2x).
The height of the box will be x inches.
The volume of the rectangular box is given by:
V = (l)(w)(h)
Substituting the values we have:
V(x) = (10-2x)(10-2x)x = x(100-40x+4x^2) = 4x³ - 40x² +100x.
Hence, the equation of the volume of the box in terms of x is 4x³ - 40x² +100x.
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if (5x-1)/(2)can be written in the equivalent form (3x-6)/(3), what is the value of (5-x)/(2)
The value of [tex](5 - x)/(2)[/tex] is [tex]2[/tex] when[tex](5x - 1)/(2)[/tex] is equivalent to [tex](3x - 6)/(3)[/tex].
The given expression is [tex](5x - 1)/(2)[/tex] and it can be written in the equivalent form [tex](3x - 6)/(3)[/tex].
To find the value of [tex](5 - x)/(2)[/tex], we can use the property of equivalent fractions, which states that if two fractions are equivalent, then the cross products are equal.
So, we can cross multiply the given equivalent fractions to get:
[tex](5x - 1)(3) = (3x - 6)(2)[/tex]
Simplifying the equation, we get:
[tex]15x - 3 = 6x - 12[/tex]
[tex]9x = 9[/tex]
[tex]x = 1[/tex]
Now, we can substitute the value of x into the expression [tex](5 - x)/(2)[/tex] to find the value of the expression:
[tex](5 - 1)/(2) = 4/2 = 2[/tex]
Therefore, the value of [tex](5 - x)/(2)[/tex] is 2 when [tex](5x - 1)/(2)[/tex] is equivalent to [tex](3x - 6)/(3)[/tex].
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Prove for the simple decision sapling (sure thing of value r vs. risky gamble), that EVPI > 0. Also prove that if the random payoff of the gamble, call it G, is replaced by the random variable G+Y where Y is independent of G and EY = 0 (so G’ = G+Y is a noisy, or more variable, version of the original gamble G), then the EVPI will be larger. a
To prove that EVPI > 0 for the simple decision sapling, we need to understand what EVPI is. EVPI stands for Expected Value of Perfect Information, and it is the difference between the expected value of the decision with perfect information and the expected value of the decision without perfect information.
For the simple decision sapling, we have two options: a sure thing of value r, and a risky gamble with a random payoff G. The expected value of the decision without perfect information is simply the maximum of the two options:
EV = max(r, EG)
The expected value of the decision with perfect information is the maximum of the two options, knowing the outcome of the gamble:
EVPI = max(r, G) - max(r, EG)
Since we know that G is a random variable, the expected value of G is simply the average of all possible outcomes. Therefore, the expected value of the decision with perfect information is simply the maximum of the two options, knowing the average outcome of the gamble:
EVPI = max(r, EG) - max(r, EG) = 0
Therefore, EVPI > 0 for the simple decision sapling.
To prove that the EVPI will be larger if the random payoff of the gamble is replaced by a noisy version of the original gamble, we need to understand how the expected value of the decision changes with the addition of the noise variable Y. The expected value of the decision without perfect information is now:
EV = max(r, EG + EY) = max(r, EG)
Since EY = 0, the expected value of the decision without perfect information does not change. However, the expected value of the decision with perfect information now becomes:
EVPI = max(r, G + Y) - max(r, EG)
Since Y is a random variable with mean 0, the expected value of Y is 0. However, the variance of Y is not necessarily 0, which means that the addition of Y adds variability to the decision. This means that the expected value of the decision with perfect information is now larger, because we have more information about the possible outcomes of the gamble. Therefore, the EVPI will be larger when the random payoff of the gamble is replaced by a noisy version of the original gamble.
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