23. determine whether each of these functions is a bijection from r to r. a) f (x) = 2x 1 b) f (x) = x2 1 c) f (x) = x3 d) f (x) = (x2 1)∕(x2 2)
Answers
a) This function is a bijection from R to R. b) This function is not a bijection from R to R. c) This function is a bijection from R to R. d) This function is a bijection from R to R.
To determine whether each of these functions is a bijection from R to R, we need to consider two conditions: injectivity (one-to-one) and surjectivity (onto).
a) f(x) = 2x + 1:
This function is a bijection from R to R.
Injectivity: If f(x₁) = f(x₂), then 2x₁ + 1 = 2x₂ + 1, which implies x₁ = x₂. Therefore, the function is one-to-one.
Surjectivity: For any y in R, we can solve 2x + 1 = y to find x = (y - 1)/2. Therefore, the function is onto.
b) f(x) = x² + 1:
This function is not a bijection from R to R.
Injectivity: If we consider x₁ = -1 and x₂ = 1, we have f(x₁) = f(x₂) = 2. Therefore, the function is not one-to-one.
Surjectivity: The function only maps to values greater than or equal to 1, so it does not cover the entire range of R. Therefore, the function is not onto.
c) f(x) = x³:
This function is a bijection from R to R.
Injectivity: If f(x₁) = f(x₂), then x₁³ = x₂³, which implies x₁ = x₂. Therefore, the function is one-to-one.
Surjectivity: For any y in R, we can solve x³ = y to find x = ∛(y). Therefore, the function is onto.
d) f(x) = (x² + 1)/(x² + 2):
This function is a bijection from R to R, excluding the value x = ±√2.
Injectivity: If f(x₁) = f(x₂), then (x₁² + 1)/(x₁² + 2) = (x₂² + 1)/(x₂² + 2), which implies x₁ = x₂. Therefore, the function is one-to-one.
Surjectivity: For any y in R, we can solve (x² + 1)/(x² + 2) = y to find x. The only exception is when y = 1, which corresponds to x = ±√2. Therefore, excluding these two values, the function is onto.
In summary:
a) f(x) = 2x + 1 is a bijection from R to R.
b) f(x) = x² + 1 is not a bijection from R to R.
c) f(x) = x³ is a bijection from R to R.
d) f(x) = (x² + 1)/(x² + 2) is a bijection from R to R, excluding x = ±√2.
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Related Questions
Last night, Troy had homework in math and science. He spent 112 minutes altogether on his homework. He spent 3 times as long on science homework as on math homework.
How many minutes did Troy spend on his math homework?
Answers
112=x+(x*3)
112=x+3x
112=4x
4x=112
x=28
there are 8 ounces in 1/2 pound . how many ounces are in 7 3/4 lbs ?
Answers
Answer:
There are 124 ounces in 7 3/4 Ibs
In 73/4 lbs there are 124 ounces
What is unit conversion?Unit conversion is a process with multiple steps that involves multiplication or division by a numerical factor or, particularly a conversion factor. The process may also require selection of the correct number of significant digits, and rounding. Different units of conversion are used to measure different parameters.
Different units are used to measure different quantities. Let us explore the units used to measure the following:
LengthTemperatureAreaVolumeWeightGiven:
in 1/2 pound= 8 ounces
As 1 lbs= 1 pound
1 pound = 16 ounces
In 7 3/4 lbs = 31/4 lbs or pounds
=31/4*16
=31*4
=124 ounces
Hence, in 7 3/4 lbs there are 124 ounces.
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[Please read first] I will NOT be taking links or download files as an answer you may NOT get credit for it thank you.
A rectangular prism has a length of 9 in., a width of 4 in., and a height of 1 1/2 in.
The prism is filled with cubes that have edge lengths of 1/2 in.
How many cubes are needed to fill the rectangular prism?
Enter your answer in the box.
To fill the rectangular prism, [ ] cubes are needed.
![[Please read first] I will NOT be taking links or download files as an answer you may NOT get credit](https://i5t5.c14.e2-1.dev/h-images-qa/contents/attachments/gGAW817TgkBMUu1UtjaiT7BBAhjB2Pb2.png)
Answers
Answer:
108
Step-by-step explanation:
To find the volume of a rectangular prisms just multiply its dimensions together then you should get a volume of 54 in² then divide by 1/2 to find out how many cubes will fit in the prism
Fill in the missing
reasons for the proof of Theorem 1-4.
Given: zf and zG are right angles.
Prove: zF £ zG
Statements Reasons
1) zF and zG are 1) Given
right angles
2) mzF = 90 and 2)
mzG = 90
3) mzF = mzG 3)
4) zF £ zG 4)
Answers
Answer:
1) Given
2) Defintion of Right Angle
3) Transitive Property of Equality
4) Definiton of Congruent Angles
Step-by-step explanation:
Could someone help me out please
Answers
Answer:
The answer is
GCF=8
56-48=8*(7-6)
simple exponential smoothing models are useful for data, which have group of answer choices a downward time trend an upward time trend neither an upward or downward time trend pronounced seasonality all the above
Answers
Simple exponential smoothing models are useful for data that do not have any significant upward or downward time trend. (Option 3)
The model works well for data that have a consistent level of variation over time and do not exhibit any pronounced seasonality. However, if the data exhibit a significant trend or seasonality, other modeling techniques may be more appropriate. Simple exponential smoothing models are often used for short-term forecasting or for smoothing out noise in the data to identify underlying patterns.
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Complete Question:
simple exponential smoothing models are useful for data, which have __. group of answer choices:
a downward time trend, an upward time trend, neither an upward or downward time trend, pronounced seasonality, all the above.on january 1, year 1, billy co signed a three-year lease for office space. the lease required monthly rent of $8,000 for the first year, $8,100 fo rhte second year
Answers
If on january 1, year 1, billy co signed a three-year lease for office space The amortization expense on this leasehold improvement that Billy should record for Year 2 is: $1,000.
How to find the amortization expense?Since we were told that the new overhead lighting fixtures is the amount of $2,000 now let determine the Amortization expense using this formula
Amortization expense = New overhead lighting fixtures / 2 years
Let plug in the formula
Amortization expense = $2,000 /2
Amortization expense = $1,000
Therefore the Amortization expense is the amount of $1,000.
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The complete question:
On January 1 , Year 1, Billy Co. signed a three year lease for office space. The lease required monthly rent of 8,000 for the first year. 8,100 for the second year and 8,200 for the 3rd year. Billy has the option to renew the lease for a fourth year at 8,300 per month. On January 1, Year 2, Billy permanently installed new overhead lighting fixtures at a cost of $2,000. How much amortization expense on this leasehold improvement should Billy record for Year 2?
What is 56,291.8 rounded to the nearest thousand
Answers
Answer:
56,000
Step-by-step explanation:
Answer:
56000
Step-by-step explanation:
6,2 is closest to 6
question content area the function f (x, y) = x 2 y 2 has a single global minimum and is relatively easy to minimize. (True or False)
Answers
The given function is f(x, y) = x²y². We are to determine if this function has a single global minimum and is relatively easy to minimize or not. To check if a function has a global minimum, we need to take the partial derivative of the function with respect to x and y.
Let's evaluate it.∂f/∂x = 2xy² ∂f/∂y = 2x²y Equating both the partial derivatives to zero, we get;2xy² = 0 => xy² = 0 => x = 0 or y = 0 2x²y = 0 => x²y = 0 => x = 0 or y = 0
Hence, the stationary points are (0,0), (0, y), and (x,0). Let's use the second derivative test to determine if the stationary point (0,0) is a global minimum, maximum, or saddle point.∂²f/∂x² = 2y² ∂²f/∂y² = 2x² ∂²f/∂x∂y = 4xy
The determinant is;∆ = ∂²f/∂x² * ∂²f/∂y² - (∂²f/∂x∂y)² = 4x²y²Since (0,0) is a stationary point, we have ∂f/∂x = 0 and ∂f/∂y = 0 which implies that x = 0 and y = 0, respectively. Thus, ∆ = 0 which means that the second derivative test is inconclusive and we cannot determine if (0,0) is a global minimum, maximum, or saddle point. The given function does not have a single global minimum and is not relatively easy to minimize.
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which expression is equivalent to 7(6m - 7) + 9?
- 42 + 2
- -7(6m + 7) + 9
- 7 (-7 + 6m) + 9
- -40m + 42
Answers
Answer:
\(7( - 7 + 6m) + 9\)
A tusmith wants tomake a smali windowizi planter from a 42 cm×18 crm sheet of copper. Shell form it by culting equally sized squares from each of the four corners of the sheet, foldingup the resulang flaps to form the sides of the planter, and then soidering the four vertical edges, If she wants the ratio of the planter's width to height to be 4 , what will be the ratio of its lenghh to width? Write the exact answer. Do not round.
Answers
Thhe ratio of the planter's length to width is 4. Length : Width = 4 : 1
How to solve the ratioWe are given that the ratio of the planter's width to height is 4. Mathematically, this can be expressed as:
w/h = 4
Substituting the expressions for w and h, we have:
(42 - 2x)/(18 - 2x) = 4
Let's solve the equation to find the exact value.
First, cross-multiply:
(42 - 2x) * 1 = (18 - 2x) * 4
42 - 2x = 72 - 8x
6x = 30
x = 5
Now we can substitute x = 5 into the expression (42 - 2x)/(18 - 2x):
(42 - 2(5))/(18 - 2(5)) = 32/8 = 4
Therefore, the ratio of the planter's length to width is 4. Length : Width = 4 : 1
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make x the subject of formula
\((xy - fx) = 2\)
Answers
Answer:
\(\huge\boxed{\sf x=\frac{2}{y-f}}\)
Step-by-step explanation:
Given equation:xy - fx = 2
Take x common
x(y - f) = 2
Divide y-f to both sides
\(\displaystyle x=\frac{2}{y-f} \\\\\rule[225]{225}{2}\)
Two sides of a triangle have lengths 1010 and 1515. The length of the altitude to the third side is the average of the lengths of the altitudes to the two given sides. How long is the third side
Answers
The length of the third side is BC ≈ 1315.5.
Let the two sides of the triangle with given altitudes be AB=1010 and AC=1515. Let h be the length of the altitude from A to BC.
Let D and E be the feet of the perpendiculars from A to BC and from B to AC, respectively. Then we have:
BD = AB - AD = 1010 - h
CE = AC - AE = 1515 - h
Since the length of the altitude from A to BC is the average of the lengths of the altitudes to the two given sides, we have:
h = (BD + CE) / 2
h = [(1010 - h) + (1515 - h)] / 2
2h = 2525 - h
3h = 2525
h = 2525 / 3
Now we use the Pythagorean theorem on the right triangle ABD:
\(AD^2 + BD^2 = AB^2\)
\(h^2 + (1010 - h)^2 = 1010^2\)
Expanding and simplifying, we get:
\(2h^2 - 2020h + 515 = 0\)
Solving for h using the quadratic formula, we get:
h = \((2020 ± sqrt(2020^2 - 4(2)(515))) / (2(2))\)
h ≈ 808.6 or h ≈ 1511.4
We take the smaller value of h since the altitude is shorter than the length of the side opposite to it. Therefore, h ≈ 808.6.
Now we can use the Pythagorean theorem on the right triangle ABC:
\(BC^2 = AC^2 - h^2\)
\(BC^2 = 1515^2 - (808.6)^2\)
BC ≈ 1315.5
Therefore, the length of the third side is BC ≈ 1315.5.
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The mean mass of five men is 76 kg. The masses of four of the men are 72 kg, 74 kg, 75 kg and 81 kg. What is the mass of the fifth man
Answers
Answer:
78kg
Step-by-step explanation:
76×5= 380kg
380-72-74-75-81= 78kg
x=(76.5)-302
x=380-302=78
There are two numbers between 30 and 40 that have just two factors.
What are they?
Answers
Answer:
31 and 37
Step-by-step explanation:
Those are the only two numbers
Answer:
The two numbers between 30 and 40 which have only 2 factors are -
31 and 37
Hope it helps you.
hi can someone help me with this proof?
Answers
I really need help hurry
Answers
Answer:
the answer is A. {-1, 1, 3}
Step-by-step explanation:
Which expression is equivalent to
…..
Answers
According to the following expression, what is \( z \) if \( x \) is 32 and \( y \) is 25 ? \[ z=(x
Answers
When x = 32 and y = 25, the value of z is calculated as 3200 using the given expression.
According to the following expression, the value of z when x = 32 and y = 25 is:
[z = (x+y)² - (x-y)²]
Substitute the given values of x and y:
\(\[\begin{aligned}z &= (32+25)^2 - (32-25)^2 \\ &= 57^2 - 7^2 \\ &= 3249 - 49 \\ &= \boxed{3200}\end{aligned}\]\)
Therefore, the value of z when x = 32 and y = 25 is \(\(\boxed{3200}\)\).
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Complete Question:
Identify the correct statement for the following system x + y = k ; 3 x - k y = 2.
1.Solution of the system is x= (3k-2) /(k+3) ; y = k- ((3k-2)/ (k+3));
2.System is inconsistent, if k = -3
3.Solution of the system is x= k- ((3k-2)/ (k+3)); y = (3k-2) /(k+3)
4.System is consistent , if k = -3
Answers
The correct statement for the given system x + y = k; 3x - ky = 2 is: Solution of the system is x = k - ((3k - 2) / (k + 3)); y = (3k - 2) / (k + 3).
To determine the correct statement, we need to solve the system of equations and find the values of x and y in terms of k.
First, let's solve the system by using the method of substitution. We have:
Equation 1: x + y = k
Equation 2: 3x - ky = 2
From Equation 1, we can express y in terms of x: y = k - x. Substituting this into Equation 2, we get:
3x - k(k - x) = 2
\(3x - k^2 + kx = 2\)
\((3 + k)x - k^2 = 2\)
Now, we solve for x:
\((3 + k)x = k^2 + 2\)
\(x = (k^2 + 2) / (3 + k)\)
Substituting this value of x back into Equation 1, we can solve for y:
y = k - x
\(y = k - (k^2 + 2) / (3 + k)\)
Simplifying the expression for y, we have:
y = (3k - 2) / (k + 3)
Therefore, the correct statement is: Solution of the system is x = k - ((3k - 2) / (k + 3)); y = (3k - 2) / (k + 3). Option 3 is the correct choice.
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Matt earns $6 per hour spreading mulch and $4 per hour pulling weeds last week mat spent 3 hours spreading mulch and 2 hours pulling weed how many money did Matt earn?
Answers
From the word problem, we know that:
• Matt earns $6 per hour spreading mulch
,• Matt earns $4 per hour pulling weeds
Since Matt spent 3 hours spreading mulch, we can find his earnings multiplying 3 by $6.
\(3\cdot\text{\$}6=\text{\$}18\)Since Matt spent 2 hours pulling a weed, we can find his earnings multiplying 2 by $4.
\(2\cdot\text{\$}4=\text{\$}8\)Finally, we add up Matt's earnings from doing both activities.
\(\text{\$}18+\text{\$}8=\text{\$}26\)Therefore, last week Matt earned $26.
If there are 3 fiction books sold for every 4 nonfiction books sold, how many nonfiction books are sold when 10 fiction books are sold?
Answers
By evaluating a proportional relation, we can see that if 10 fiction books are sold, then 13 non-fiction books are sold.
How many nonfiction books are sold when 10 fiction books are sold?Here we know that 3 fiction books are sold for every 4 non-fiction books sold.
Then if x fiction books are sold, the number of non-fiction books sold are given by the proportional relation:
y = (4/3)*x
Here we know that 10 fiction books are sold, so x = 10, replacing that we will get:
y = (4/3)*10 = 40/3 = 13.33
So 13 non-fiction books are sold.
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19. Two angles are supplementary. One angle is three
times the size of the other angle. Give the measures
of the two angles.
Answers
Answer: 135 and 45
Step-by-step explanation:
Solve for b.49 = b 2b = = 1or
Answers
√49 = +/- 7
√16 = +/- 4 as you calculated
Congrats, you did it!
for A=s^2 find A when s=25 ft
Answers
The value of A = 625 ft.
What is substitution?
Substitution is the algebraic method to solve simultaneous linear equations. As the word says, in this method, the value of one variable from one equation is substituted in the other equation.
Given,
A = \(s^{2}\)
when s = 25, A = 25(25)
A= 625 ft.
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Apex Financial Literacy: Comparing Credit and APR
Jesse has a balance of $1200 on a credit card with an APR of 18. 7%, compounded monthly. About how much will he save in interest over the course of a year if he transfers his balance to a credit card with an APR of 12. 5%, compounded monthly? (Assume that Jesse will make no payments or new purchases during the year and ignore any possible late payment fees. )
A. $87. 33
B. $85. 77
C. $181. 46
D. $117. 85
Answers
To calculate the interest savings, we need to find the difference in the amount of interest paid between the two credit cards.
For the first credit card with an APR of 18.7% compounded monthly, the annual interest can be calculated as follows:
Annual interest = Balance * (APR/100)
= $1200 * (18.7/100)
= $224.40
For the second credit card with an APR of 12.5% compounded monthly, the annual interest can be calculated as follows:
Annual interest = Balance * (APR/100)
= $1200 * (12.5/100)
= $150.00
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What is 1836837292 times 816282?
Answers
Answer:
The product is 1.4993772e+15.
Step-by-step explanation:
Thanks for the points! Brainliest? Thanks!
Answer:
1499377218388344
Step-by-step explanation:
What are the x-intercepts of this function? f(x)=-9x^2+3x1
Answers
\( - 9 {x}^{2} + 3x = 0 \)
\(3x( - 3x + 1) = 0\)
\(3 x = 0 \\ x = 0\)
And
\( - 3x + 1 = 0 \\ - 3x = - 1 \\ x = \frac{ - 1}{ - 3} \\ x = \frac{1}{3} \)
So the x-intercepts are :
\(x = 0 \: \: \: \: \: and \: \: \: \: \: x = \frac{1}{3} \\ \)
how many distinct ordered pairs of positive integers $(m,n)$ are there so that the sum of the reciprocals of $m$ and $n$ is $\frac14$?
Answers
There are no such pairs. The sum of the reciprocals of any two positive integers is always greater than\($\frac14$\).
The sum of the reciprocals of any two positive integers is \($\frac{1}{m} + \frac{1}{n}$\), which is always greater than \($\frac14$\). Therefore, there are no distinct ordered pairs of positive integers \($(m,n)$\) so that the sum of the reciprocals of m and n is \($\frac14$\)
For any two positive integers m and n, the sum of their reciprocals is \($\frac{1}{m} + \frac{1}{n}$\) which is always greater than \($\frac14$\) This is because \(\frac{1}{m}$ ,$\frac{1}{n}\) are both positive numbers, and so their sum must be greater than \($\frac14$\)Therefore, there are no distinct ordered pairs of positive integers \($(m,n)$\) so that the sum of the reciprocals of m and n is \($\frac14$\).
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Find the area of a regular hexagon with a perimeter of 12cm.
Answers
Answer:
\(6\sqrt{3}\) \(cm^{2}\)
Step-by-step explanation:
Here is a fact to remember. A hexagon is always made up of 6 equilateral triangles.
So if the perimeter of the Hexagon is 12 cm, then each side is 2 cm.
The area of an equilateral triangle is \(\frac{s^{2}\sqrt{3} }{4}\).
So plugging in 2 gets us the area of one triangle.
s = 2
\(2^{2}\sqrt{3}/4 = \sqrt3\)
The area of 6 triangles will be \(6\sqrt{3}\) \(cm^{2}\).
