We proved the expression rank(AB)=rank(B) and rank(BA)=rank(B),
where A and B are the two matrices of same order.
Given, A is a square matrix.
If A is invertible and a n×n square matrix of full rank.
Since both AB and BA exist, where B is also a n×n square matrix.
rank(B)=rank(BT)
On using the rank–nullity theorem, we get
nullity of a matrix be n minus the rank of the matrix.
So let U be a matrix whose columns are a basis for the nullspace of AB, so that ABU = 0.
Pre-multiplying both sides by A−1 , we obtain BU=0,
As B and AB have the same rank.
Similarly, let V be a matrix whose columns are a basis for the null space of ATBT, so that ATBTV=0 .
Pre-multiplying both sides by (A−1)T , we get
BTV=0.
Since rank(BT)=rank(B) and rank(ATBT)=rank((BA)T)=rank(BA)
we get rank(AB)=rank(B) and rank(BA)=rank(B).
So, rank(AB)=rank(B) and rank(BA)=rank(B).
Hence, rank(AB)=rank(B) and rank(BA)=rank(B).
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compared to smaller firms that export, larger companies tend to multiple choice systematically scan foreign markets to see where export opportunities lie. delay exporting until after their domestic market is fully saturated. be intimidated by the complexities and mechanics of exporting to foreign countries. be reactive about seeking opportunities for profitable exporting. hesitate to seek export opportunities because they do not know how big the opportunities actually are.
According to the profit point of view, compared to smaller firms that export larger companies tend to systematically scan foreign markets to see where export opportunities lie.
What does profit mean?
In math, profit means the excess of returns over expenditure in a transaction or series of transactions.
Here we have given that, compared to smaller firms that export, larger companies tend to to be what.
And for the reason more firms are not proactive is that they are unfamiliar with foreign market opportunities; they simply do not know how big the opportunities actually are or where they might lie. Simple ignorance of the potential opportunities is a huge barrier to exporting
Here we know that, smaller firms, are often intimidated by the complexities and mechanics of exporting to countries where business practices, language, culture, legal systems, and currency are very different from the home market
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The polynomial function p(x)=x³-3 x²+x+205 has x=-5 as a root.
What are the remaining roots of p(x) ? Separate multiple answers with a comma.
Enter the factored form of p(x).
The remaining roots of p(x) is x = -5 , 4 + 5i , 4 - 5i and the factored form of p(x) = ( x + 5 ) ( x - ( 4 + 5i ) ( x - ( 4 - 5i )
Given :
The polynomial function p(x)=x³-3 x²+x+205 has x=-5 as a root.
using synthetic division :
x = -5
5 1 -3 1 205
0 5 10 55
1 2 11 260
on solving using the quadratic formula or calculator the roots are :
x = -5 , 4 + 5i , 4 - 5i
so quadratic form of p ( x ) = ( x + 5 ) ( x - ( 4 + 5i ) ( x - ( 4 - 5i )
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write an expression for the sequence of operations described below. triple s, subtract u from the result, then subtract t from what you have do not simplify any part of the expression.
The expression for the sequence of operations is 3s - u - t .
In the question ,
it is given that ,
the sequence of operation is given as " triple s, subtract u from the result, then subtract t from what you have" .
we have to write an expression for the above operation ,
So ,
first step is : triple s ,
that means the expression is 3s .
next subtract u from the result , that means ⇒ 3s - u .
next we have to subtract t from what we have ,
that means ⇒ 3s - u - t .
Therefore , the final expression is 3s - u - t .
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What is the answer?
For the given circle the value of cotθ is -√30/√19.
What is Trigonometry?Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles.
The terminal side of an angle θ.
The standard position intersects the unit circle at (√30/7, -√19/7).
We need to find the value of cotθ.
Suppose that θ is an angle in standard position whose terminal side intersects the unit circle at (√30/7, -√19/7)
cotθ=x/y
=-√30/7/√19/7
=-√30/7×7/√19
=-√30/√19
Hence, the value of cotθ is -√30/√19 for the given circle.
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Y^5y^5-(y^-2)^4 complete the equation
Answer: x =2, -1
Step-by-step explanation:
Solve the system using elimination.
3x + 10y = -4
2x + 2y = 2
([?], [ ])
Answer:
Step-by-step explanation:
that normally means we want to add our subtract a certain of the 2 equations, so that we are led with one summary equation with only one variable.
the easiest way here is possibly to multiply the second equation by 5 (to get also 10y) and then subtract it from the first equation :
3x + 10y = -4
- 10x + 10y = 10
-----------------------
-7x + 0 = -14
7x = 14
x = 2
2x + 2y = 2
gives us then
2×2 + 2y = 2
4 + 2y = 2
2y = -2
y = -1
x = 2
y = -1
Answer:
x = 2
y = -1
Step-by-step explanation:
3x + 10y = -4
- 10x + 10y = 10
7x + 0 = -14
7x = 14
x = 2
2x + 2y = 2
2×2 + 2y = 2
4 + 2y = 2
2y = -2
y = -1
Shaquanna kicks a football. It’s height in feet is given by h= -16t^2 + 16t where t represents the time in seconds after kick. After how many seconds does the football hit the ground?
Answer:
Step-by-step explanation:
the question is asking, or implying, when is [tex]t^{2}[/tex] = t :?
if you are thinking at 1, you're correct. so the ball will hit the ground at 1 second
taking . definitions: a nullcline of this system is a line on which either or . more precisely, a vertical nullcline of this system is a line on which , and likewise, a horizontal nullcline of this system is a line on which . (a) write an equation for a vertical nullcline that is not a coordinate axis:
The a) equation for the vertical nullcline is y = -4x + 4 and horizontal nullcline is x = -3y + 1. The b) equilibrium points are (0,0) , (0 , 0.8) , (0.75 , 0), and (0.375,0.5)
We have
dx/dt = x(1 - x/3 - y)
dy/dt = y(1 - y/4 - x)
a) To get the equation for vertical nullcline, not on the coordinate axis we take
dy/dt = 0
or, y(1 - y/4 - x) = 0
or, 1 - y/4 - x = 0
or, y/4 = 1 - x
or, y = -4x + 4
Similar;y for horizontal nullcline not on coordinate axis we take
dx/dt = 0
or, (1 - x/3 - y) = 0
or, x/3 = 1 - y
or, x = -3y + 1
b)
To get the equilibrium points of the nullcline system we simply take
Here the x nullcline will intersect y nullcline at
x = 0 and x = 1 - x/3 - y
y = 0 and y = 1 - y/4 - x
therefore we have equilibrium points
for x = 0 and y = 0 at(0,0)
x = 0 and y = 1 - y/4 - x at (0, 4/5)
x = 1 - x/3 - y and y = 0 at (3/4 , 0)
x = 1 - x/3 - y and y = 1 - y/4 - x at (3/8 , 1/2)
Hence the equilibrium points are (0,0) , (0 , 0.8) , (0.75 , 0), and (0.375,0.5)
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Complete Question
(Image Attached)
How many miles is it from Austin Texas to Seattle Washington by car?
Answer:
The total driving distance from Austin, TX to Seattle, WA is 2,128 miles
Step-by-step explanation:
Answer:
i believe it is a 2,116.4 mile drive.
Step-by-step explanation:
Alice and Bob are playing a game where in each round, each of them rolls a fair die independently. If both roll the same number, then the game is repeated. Otherwise the player with the larger number wins. Let X be the number of rounds until the game is decided.
(a) Determine the probability mass function of X.
(b) Compute E[X].
(c) Compute Probability[ Alice win].
(d) Assume that you get paid 10USD for winning in the first round, 1USD for winning in any other round, and nothing otherwise. Compute your expected winnings.
(a) The probability mass function of X will be P(X=n) = 5/(6n)
(b) E[X] =1.2
(c) Probability [ Alice win] will be =0.5
(d) The value of expected winnings will be 4.25
(a) Let 'p' represent the probability of getting the same number on both the dices on a roll of pair of die
And (1-p) = probability of not getting the same number on both the dices on a roll of pair of die
Sample space for the same number appearing on both the dices on a roll of pair of die = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}
As there are a total of 36 outcomes on a roll of pair of dice,
p = 6/36 = 1/6
Let the events be defined as:
Dn : Both Alice and Bob roll the same number on the nth roll
Wn : The game is finished in the nth roll (i.e. Alice and Bob roll different numbers (as the numbers are different, one will be larger))
Then,P(W) = P(W) , n = 1,2,3,4....infinity
Wn takes place when event D takes place in the first (n-1) rolls and Event W takes place in the nth roll
[tex]$\mathrm{P}(\mathrm{W})=\sum P\left(W_n\right), \mathrm{n}=1,2,3,4 \ldots .$[/tex]. infinity
And, W_n takes place when
Event D takes place in the first (n-1) rolls and event W takes place in the [tex]$n^{\text {th }}$[/tex] roll
[tex]P\left(W_n\right)=P\left(D_1\right)^* P\left(D_2\right)^* P\left(D_3\right) \ldots . .{ }^* P\left(D_{n-1}\right) P\left(W_n\right)P\left(D_n\right)\\=(p)P\left(W_n\right)\\=(1-p)[/tex]
Implies
[tex]$$\begin{aligned}& \left.P\left(W_n\right)=(p)^{n-1}\right)^*(1-p) \\& =\left((1 / 6)^{n-1}\right)^*(1-(1 / 6)) \\& =\left((1 / 6)^{n-1}\right)^*(5 / 6) \\& =5 /\left(6^n\right)\end{aligned}$$[/tex]
(b)
[tex]$$\begin{aligned}& \mathrm{E}[\mathrm{X}]=\sum n * P(X=n), \mathrm{n}=1,2,3,4 \ldots . . \text { infinity } \\& =\sum n * \frac{5}{6^n} \\& =5 * \sum n * \frac{1}{6^n}\end{aligned}$$[/tex]
If [tex]$\mathrm{S}=\sum n * \frac{1}{6^n}$[/tex]
Then S is a sum of infinite aritho-geometric series of the form
[tex]$$\sum_{k=1}^{\mathrm{inf}} k * r^k \text {, }$$[/tex]
The sum of which is given as [tex]$S=\frac{r}{(1-r)^2}$[/tex] for 0 < r < 1
Here, r=(1 / 6)
[tex]\Rightarrow & S=\frac{\frac{1}{6}}{\left(\frac{5}{6}\right)^2} \\& =\frac{6}{25}[/tex]
E[X] = (5) x (6/25) = (6/5)= 1.2
(c) Since the dies are fair, and the outcomes are independent, the probability of each player winning is equal
Therefore,
P(Alice WIn) = P(Bob WIn) = 0.5
(d) If X represent the round in which the game ends, then the probability of us winning in that round
P(W|X=n) = 0.5*P(X=n)
= (0.5)*(5/(6n))
P(W|X=1) = (0.5)*(5/(61)) = (5/12)
P(W|X=2,3,4...) = 0.5 - P(W|X=1) = (1/2) - (5/12) = (1/12)
Leth the variable W represent the winnings
(W|X=1) = 10
(W|X=2,3,4...) = 1
E[W] = P(W|X=1)*(W|X=1) + P(W|X=2,3,4...)*(W|X=2,3,4...)
= (10 x (5/12)) + (1 x (1/12))
= (50/12) + (1/12)= (51/12)= 4.25
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Raúl is pouring a concrete walkway in his backyard. He decides to put a turn of 30° from the horizontal in the walkway to go around a tree and flowerbed. Once the walkway is past the tree, Raúl wants to continue the walkway in the same direction as before. At what angle should he put the second turn so that the walkway will continue in the same direction as before it reached the flowerbed? Why?
Answer:
Step-by-step explanation: Raúl should put a turn of -30° in the walkway after the tree in order to continue in the same direction as before. This is because the negative angle will cancel out the positive angle and the walkway will continue in the same direction as before.
Question 2
What is the length of DE? (number only)
The length of the side DE, that is parallel to the side AC is 6.3 units
Here the line DE is parallel to the side AC of the triangle
In the given triangle
The length of the side BD = The length of the side AD
The length of the side BE = The length of the side EC
That mean the parallel line AC is equally dividing the two side of the triangle
Therefore, the length of the line DE will be be exactly half of the line AC
The length of the side AC = 12.6 units
The length of the side DE = The length of the side AC / 2
Substitute the values in the equation
= 12.6 / 2
= 6.3 unit
Therefore, the length of DE is 6.3 units
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The original price of a scarf was $16, During a store closing sale, a shopper saved $12 on the scarf, What percentage discount did she receive? Explain or show your reasoning.
Answer:
The shopper received a 75% discount
Step-by-step explanation:
ok so first we turn it into an equation
let x=percentage discount
16(x)=12
x=.75
discount=75%
Write the equation in slope-intercept form.
–2x + 4y = 12
Answer:
y = [tex]\frac{1}{2}[/tex] x + 3
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
given
- 2x + 4y = 12 ( add 2x to both sides )
4y = 2x + 12 ( divide through by 4 )
y = [tex]\frac{2}{4}[/tex] x + [tex]\frac{12}{4}[/tex] , that is
y = [tex]\frac{1}{2}[/tex] x + 3
find the value h(-6) where h(t) = 6t-2
Answer:
Step-by-step explanation:
Quite literally you plug in -6 into t. So therefore:
h(t)=6t-2
h(-6)=6(-6)-2
h(-6)=-36-2
h(-6)=-38
-38 is your answer
h can be replaced with f and t can be replaced with x to get your standard f(x) equation if that helps you understand it better. Remember f(x) really just means the y value of the function with an input of value x. So f(x) = x is really just y=x or a linear line.
A soccer coach surveyed the players to determine the number who preferred selling coupon books, magazine subscriptions, or both for their fundraiser. The results are given in the Venn diagram. To the nearest whole percent, what is the value of a in the relative frequency table for the survey results? a = 27% a = 42% a = 81% a = 88%
The value of a in the relative frequency table for the survey results a = 27%
From the given Venn diagram, the total number of players = 11 + 3 + 7 + 5 = 26
The number of players preferred magazines books which are not coupon books = 7
The relative frequency for the players who preferred magazines books which are not coupon books (a)=
magazine books but not coupon books/ total player = 7 / 26 = 0.269
In percent,
7/26 x 100 = 26.9 = 27%
Hence, a = 27%
Therefore, the value of a in the relative frequency table for the survey results a = 27%
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f(x)=x²-8x³ +26x² - 88x - 111; x = −11
Answer:
14772
Step-by-step explanation:
7x-8y=10 and x-8y=-26 using elimination
x= 6;
y= 4.
Step-by-step explanation:1. Write the equations.[tex](1)7x-8y=10\\\\ (2)x-8y=-26\\[/tex]
2. Subtract equation 2 from equation 1.[tex](1)7x-8y=10\\-\\ (2)x-8y=-26\\--------\\\\6x+0y=36\\ \\6x=36[/tex]
3. Solve for x.[tex]\frac{6x}{6} =\frac{36}{6} \\ \\x=6[/tex]
4. Use the calculated value of "x" in any of the equation to the value of "y".For this solution, we are going to use equation 2 to get the value of "y".
[tex](6)-8y=-26\\\\-8y=-26-6\\ \\-8y=-32\\ \\\frac{-8y}{-8} =\frac{-32}{-8} \\ \\y=4[/tex]
5. Verify.If the answers are correct, both equations should return the same value in both sides of the equal (=) sign.
[tex]7(6)-8(4)=10\\ \\42-32=10\\ \\10=10[/tex]
That's correct.
[tex](6)-8(4)=-26\\ \\6-32=-26\\ \\-26=-26[/tex]
That's correct.
Hence, the answers are correct.
3. Net income = gross profits - (cost to bring goods into store + ___)
A. long-term liabilities
B. total assets
C. operating expenses
D. cost of merchandise sold
Answer: D & C
Step-by-step explanation:
The net Income of the business is represented as
Net profits or income = Net sales revenue (gross income) – Cost of goods and services sold – Administration, operating, marketing, and advertising costs, taxes, interests, and other expenses.
Here C is an option because we need to take operating expenses into consideration but if you consider cost of goods brought into store as operating then option changes to D.
D represents the sales which helps us determine the net income of the business.
4) How long will Lourdes have to wait for her plant to grow a total 14 1/4 inches
Answer:
10 [tex]\frac{5}{28}[/tex] days
Step-by-step explanation:
There are a lot of places to make mistakes, but I think that I am good.
Make a proportion:
[tex]\frac{Days}{Inches}[/tex] = [tex]\frac{Days}{Inches}[/tex] Put in the three number that you know and solve for the number of days.
[tex]\frac{7\frac{13}{21} }{10\frac{2}{3} }[/tex] = [tex]\frac{x}{14\frac{1}{4} }[/tex] Cross multiply and solve for x
10 [tex]\frac{2}{3}[/tex] x = [tex]7\frac{13}{21}[/tex]([tex]14\frac{1}{4}[/tex]) Change everything into improper fractions
[tex]\frac{32}{3}[/tex]x = ([tex]\frac{160}{21}[/tex])[tex]\frac{57}{4}[/tex] I am going to cross cancel the 160 and 4 They both can be divided by 4 and I am going to cross cancel the 21 and 57. They are both divisible by 3
[tex]\frac{32}{3}[/tex] x =( [tex]\frac{40}{7}[/tex])[tex]\frac{19}{1}[/tex]
[tex]\frac{32}{3}[/tex] x = [tex]\frac{760}{7}[/tex] Multiply both sides by [tex]\frac{3}{32}[/tex]
([tex]\frac{3}{32}[/tex])[tex]\frac{32}{3}[/tex]x =([tex]\frac{3}{32}[/tex])[tex]\frac{760}{7}[/tex] I can cross cancel 32 and 760. They both can be divided by 8
x = ([tex]\frac{3}{4}[/tex])[tex]\frac{95}{7}[/tex]
x = [tex]\frac{285}{28}[/tex]
x = 10 [tex]\frac{5}{28}[/tex] Days
(please explain and show work/drawing. thank you so much!) In a certain Algebra 2 class of 22 students, 5 of them
play basketball and 11 of them play baseball. There are 3
students who play both sports. What is the probability that
a student chosen randomly from the class plays basketball
or baseball?
Answer:
59%
Step-by-step explanation:
First, let's find the number of students who play basketball but not baseball: 5 students - 3 students = <<5-3=2>>2 students
Next, let's find the number of students who play baseball but not basketball: 11 students - 3 students = <<11-3=8>>8 students
Now, let's add up the number of students who play basketball but not baseball, the number of students who play baseball but not basketball, and the number of students who play both sports to find the total number of students who play basketball or baseball: 2 students + 8 students + 3 students = <<2+8+3=13>>13 students
Finally, we can divide the total number of students who play basketball or baseball by the total number of students in the class to find the probability that a student chosen at random plays basketball or baseball: 13 students / 22 students = 0.59
Therefore, the probability that a student chosen at random from the class plays basketball or baseball is 0.59.
Answer:
13/22
Step-by-step explanation:
The class has 22 students.
5 play basketball.
11 play baseball.
3 play both.
Break down the 5 who play basketball into:
2 play only basketball
3 play both basketball and baseball
Break down the 11 who play baseball into:
8 play only baseball
3 play both basketball and baseball (These 3 are the same 3 above)
Now we have:
2 play only basketball
8 play only baseball
3 play both
This is a total of 13.
The class has 22, so 9 don't play any sport.
That means out of 22 students, 13 play either sport, and 9 play nothing.
p(basketball or baseball) = 13/22
Right Answers Only! thank you
Answer: A
Step-by-step explanation:
D is wrong because its saying 3 times of the flat fee of 52.5 per month.
B is wrong because it is saying 52.5$ per gigabyte
C is wrong because the total amount is not 64.2
What is 1 2/3 - 5/6 ??
The Subtraction of the given fraction using the number line is 5/6.
The correct answer option is option A
How to subtract fraction?Fraction refers to a number which consists of a numerator and a denominator.
A numerator is the upper value of a fraction while a denominator is the lower value or bottom value of a fraction.
1 2/3 - 5/6
= 5/3 - 5/6
Find the lowest common multiple of the denominators and divide
= (10-5) / 6
= 5/6
In conclusion, 5/6 is the answer to the subtraction of fraction.
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find the integral please
Done :) Refer to the photo taken. Let me know if this is what you where looking for. Sorry for the sideways image.
True or False. If a cylinder and cone
have the same height and diameter, the
volume of the cylinder will always be
greater than that of the cones.
Answer:
This statement is false
Step-by-step explanation:
This statement is false. The volume of a cylinder with a given height and diameter will always be greater than that of a cone with the same height and diameter if the base of the cone is smaller than the base of the cylinder. However, if the base of the cone is the same size as the base of the cylinder, then the volume of the cone will be equal to that of the cylinder.
Suppose your waiting time for a bus in the morning is uniformly distributed on [0, 12], whereas waiting time in the evening is uniformly distributed on [0, 14] independent of morning waiting time.(a) If you take the bus each morning and evening for a week, what is your total expected waiting time? (Assume a week includes only Monday through Friday.) [Hint: Define rv'sX1, , X10and use a rule of expected value.]min(b) What is the variance of your total waiting time? (Round your answer to two decimal places.)min2(c) What are the expected value and variance of the difference between morning and evening waiting times on a given day? (Use morning time − evening time. Round the variance to two decimal places.)expected value minvariance min2(d) What are the expected value and variance of the difference between total morning waiting time and total evening waiting time for a particular week? (Use morning time − evening time. Assume a week includes only Monday through Friday.)expected value minvariance min2
The total expected waiting time including the morning and the evening time is 41/3
given that your waiting time for a bus in the morning is uniformly distributed on [0, 8], whereas waiting time in the evening is uniformly distributed on [0, 10] independent of morning waiting time.
Sum of both waiting times = X+Y
Where X = morning wait time is U(0.8) and
Y = evening wait time is U(0,10)
Since X and Y are independent
Var(x+y) = Var(x)+Var(y)
Var(x) = (8^2 - 0^2)/12 = 16/3
Var(Y) = (10^2 - 0^2)/12 = 25/3
Var(x+y) = 16/3 + 25/3
= (16 + 25)/3
= 41/3
Therefore, the total expected value of waiting time including the morning and the evening time is 41/3.
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8.11 a total of 725 auto-mode social/recreational trips are made from an origin (residential area) during the peak hour. a logit model estimation is made, and three factors were found to influence the destination choice: (1) population at the destination, in thousands (coefficient
Based on this information, we can estimate that a 1,000-person increase in population at the destination, an increase of 441 auto-mode social/recreational trips from the origin during the peak hour time
a 10-mile increase in distance to the destination, and the presence of recreational facilities at the destination are associated with an increase of 725 x (0.43 + (-0.07) + 0.62) = 441 auto-mode social/recreational trips from the origin during the peak hour.
1. Calculate the total increase in auto-mode social/recreational trips from the origin during the peak hour associated with a 1,000-person increase in population at the destination, a 10-mile increase in distance to the destination, and the presence of recreational facilities at the destination by multiplying the coefficients of the logit model by 725 (the total number of trips):
725 x (0.43 + (-0.07) + 0.62) = 441
2. Therefore, we can estimate that a 1,000-person increase in population at the destination, a 10-mile increase in distance to the destination, and the presence of recreational facilities at the destination are associated with an increase of 441 auto-mode social/recreational trips from the origin during the peak hour time.
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Let X and Y be IID Unif(0, 1). Find the expected value and the standard deviation of X-Y. i.e., the distance between X and Y.
Let X and Y be IID Unif(0, 1). the distance between X and Y.Expected value: 0 and Standard deviation: [tex]$\sqrt{\frac{1}{3}}$[/tex]
Expected value:
E(X-Y)
= E(X) - E(Y)
= 0 - 0
= 0
Standard deviation:
Var(X-Y) = Var(X) + Var(Y) = 1 + 1 = 2
SD(X-Y) = [tex]$\sqrt{2}$[/tex]
= [tex]$\sqrt{\frac{1}{3}}$[/tex]
The expected value of X-Y is the difference of the expected values of X and Y, which are both 0 since they are both IID Unif(0, 1). The variance of X-Y is the sum of the variances of X and Y, which are both 1. Since the standard deviation is the square root of the variance, the standard deviation of X-Y is the square root of 2.
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on my knees begging pls im tired and exhausted
A) If 3/5 of the 595 students are seventh graders, and 5/8 of them participated in the competition, about how many seventh graders participated in the competition? Describe the process you used to find your answer. (2 points)
b) If 2/5 of the 595 students are eighth graders, and 7/8 of them participated in the competition, about how many eighth graders participated in the competition? Describe the process you used to find your answer. (2 points).
c) About how many total students participated? Describe the process you used to find your answer. (2 points) Using the process shown above, I added 206 and 215 to get 421 total participants
a) The number of seventh graders that participated in the competition was of 223.
b) The number of eight graders that participated in the competition was of 208.
c) The total number of students that participated in the competition is of 431.
How to obtain the amounts?The amounts in this problem are obtained applying the proportions given by the fractions.
For item a, we have that the amount is 5/8 of 3/5 of 595, hence the total amount is obtained as follows:
5/8 x 3/5 x 595 = 223 students.(rounding to the nearest integer).
For eight graders in item b, we have that 7/8 of 2/5 of 595 students participated, hence the number of students that participated in the competition is given as follows:
7/8 x 2/5 x 595 = 208 students. (rounding to the nearest integer).
Then the total number of students that participated in the competition is given by the sum of these amounts as follows:
223 + 208 = 431 students.
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For the following exercises, express each function H as a composition of two functions F and G where H(x) = (fog)(x)
42) The functions F and G are G(x) = (2x - 1)/ (3x + 4) and F(x) = √x
43) The functions F and G are G(x) = 3x² - 4 and F(x) = 1/x⁻³.
What is a function?
A function in mathematics from a set X to a set Y allocates exactly one element of Y to each element of X. The sets X and Y are collectively referred to as the function's domain and codomain, respectively.
Given composite function is
H(x) = (fog)(x).
42)
H(x) =√[(2x - 1)/ (3x + 4)]
Rewrite the above equation
H(x) =√[G(x)] where G(x) = (2x - 1)/ (3x + 4)
H(x) = F(G(x)) where F(x) = √x
42)
H(x) =1/(3x² - 4)⁻³
Rewrite the above function
H(x) =1/[G(x)]⁻³ where G(x) = 1/(3x² - 4)
H(x) = F(G(x)) where F(x) = 1/x⁻³
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