Find the inverse of the function. y = 2x2 –4
Answers
x=2y²-4
x+4=2y²
(x+4)/2=y²
√((x+4)/2)=y
Answer:
The inverse function is [tex]y=\sqrt{\frac{x+4}{2}}[/tex]
Step-by-step explanation:
The given function is [tex]y=2x^2-4[/tex].
This function is only invertible on the interval, [tex]x\ge 0[/tex].
To find the inverse on this interval, we interchange [tex]x[/tex] and [tex]y[/tex].
[tex]x=2y^2-4[/tex]
We now make [tex]y[/tex] the subject to get,
[tex]x+4=2y^2[/tex]
[tex]\Rightarrow \frac{x+4}{2}=y^2[/tex]
[tex]\Rightarrow \pm \sqrt{\frac{x+4}{2}}=y[/tex]
But the given interval is [tex]x\geq 0[/tex], This implies that, [tex]y\geq 0[/tex].
[tex]y=\sqrt{\frac{x+4}{2}}[/tex]
Related Questions
delia spent 45 minutes working on her book report she finished at 6:10 p.m. at what time did delia start working on her paper
Answers
6:10 - 10 = 6:00
6:00 - 35 =5:25
When she finished the work, the time was 6:10 p.m. In other words, 6 hours and 10 minutes.
It says that she took 45 minutes to complete the work.
We need to find the time when she started working on it.
Start Time = Finish Time - time taken to complete work.
Start Time = (6 hours and 10 minutes) - 45 minutes
(Hint:- 1 hour = 60 minutes)
Start Time = (5 hours and 70 minutes) - 45 minutes
Start Time = 5 hours and 25 minutes i.e. 5:25 p.m.
It means that she started the work at 5:25 p.m.
What is the approximate volume of the come 8cm and 12cm use 3.14 for pie
Answers
For ΔABC, the measure in degrees of angles A, B, and C are 60, 55, and x + 20 respectively. What is the value of x?
Answers
135 + x = 180
x = 180 - 135 = 45 degrees Answer.
Answer:
45
Step-by-step explanation:
60 + 55 + x + 20 = 180
135 + x = 180
x = 180 - 135 = 45
Answer: 45
is 7/8×6/6 greater then, equal to, or less than 7/8
Answers
6/6 simplified equals 1.
7/8 times 1 would equal 7/8.
Therefore, 7/8 times 6/6 is equal to 7/8.
Which classification best describes the following system of equations?
12x+5y-3z=36
x-2y+4z=3
9x-10y+5z=27
inconsistent and dependent
consistent and dependent
consistent and independent
inconsistent and independent
Answers
Answer: These three planes are consistent and independent.
Explanation:
Since, if the system of planes has a solution then it is called Consistent, While, if it does not have any solution then it is called inconsistent.
Further, If the consistent system has infinite solution then it is dependent but if it has only a unique solution then it is called independent.
Here, given equations of planes are
12x+5y-3z=36 -------(1 )
x-2y+4z=3 -------(2)
9x-10y+5z=27 -------(3)
From equation (1), 3z=12x+5y-36⇒z=4x+5y/3-12 ------(4)
after putting this value in equation (2) and (3), we will get two equation in variables x and y
So, x-2y+4(4x+5y/3-12)=3 ⇒x-2y+16x+20y/3-48=3⇒3x-6y+48x+20y-144=9⇒51x+14y=153 --------(5)
And, 9x-10y+5(4x+5y/3-12)=27⇒ 9x-10y+20x+25y/3-60=27⇒ 27x-30y+60x+25y-180=81⇒87x-5y=261 --------(6)
after solving equation equation (5) and (6) we will get x=3 and y=0
substituting these values in equation (4) we will get z=0
Thus the solution of these three plane (1), (2) and (3) is x=3, y=0 and z=o
Which is the unique solution, Thus the given planes are consistent and independent.
The given system of equations is consistent and independent so, [tex]\fbox{\begin\\\ \bf option (3)\\\end{minispace}}[/tex] is correct.
Further explanation:
A system of equations is said to be a consistent system if the solution exists and if the solution does not exist then it is an inconsistent system.
If a consistent system has a unique solution then it is an independent system of equations but if the number of solutions is infinite then it is a dependent system of equations.
Label the given equations as shown below:
[tex]\boxed{x-2y+4z=3}[/tex] …… (1)
[tex]\boxed{12x+5y-3z=36}[/tex] …… (2)
[tex]\boxed{9x-10y+5z=27}[/tex] …… (3)
The augmented matrix for the above equations is, as follows:
[tex]\left[\begin{array}{ccc}1&-2&4\\12&5&-3\\9&-10&5\end{array}\Biggm\vert\begin{array}{c}3\\26\\27\end{array}\right][/tex]
Apply row transformation [tex]R_{2}\rightarrow R_{2}-12R_{1}[/tex] and [tex]R_{3}\rightarrow R_{3}-9R_{1}[/tex] as,
[tex]\left[\begin{array}{ccc}1&-2&4\\0&29&-51\\0&8&-31\end{array}\Biggm\vert\begin{array}{c}3\\0\\0\end{array}\right][/tex]
Now, apply row transformation [tex]R_{2}\rightarrow R_{2}-4R_{3}[/tex] and [tex]R_{3}\rightarrow R_{3}-3R_{2}[/tex] as,
[tex]\left[\begin{array}{ccc}1&-2&4\\0&-3&73\\0&-1&188\end{array}\Biggm\vert\begin{array}{c}3\\0\\0\end{array}\right][/tex]
Now, apply row transformations [tex]R_{2}\rightarrow -R_{2}+2R_{3}[/tex] and [tex]R_{3}\rightarrow R_{3}+R_{2}[/tex] as,
[tex]\left[\begin{array}{ccc}1&-2&4\\0&1&303\\0&0&491\end{array}\Biggm\vert\begin{array}{c}3\\0\\0\end{array}\right][/tex]
The equations obtained from the above augmented matrix are,
[tex]\begin{aligned}x-2y+4z&=3\\y+303z&=0\\491z&=0\end{aligned}[/tex]
The first equation is simplified to obtain the value of z as,
[tex]\begin{aligned}491z&=0\\z&=0\end{aligned}[/tex]
Substitute [tex]0[/tex] for [tex]z[/tex] in the equation [tex]y+303z=0[/tex] to obtain the value of [tex]y[/tex] as,
[tex]\begin{aligned}y+(303\cdot0)&=0\\y&=0\end{aligned}[/tex]
Now, substitute [tex]0[/tex] for [tex]z[/tex] and [tex]0[/tex] for [tex]y[/tex] in the equation [tex]x-2y+4z=3[/tex] to obtain the value of [tex]x[/tex] as,
[tex]\begin{aligned}x-(2\cdot0)+(4\cdot0)&=3\\x&=3\end{aligned}[/tex]
Therefore, the value of [tex]x[/tex] is [tex]3[/tex], the value of [tex]y[/tex] is [tex]0[/tex] and the value of [tex]z[/tex] is [tex]0[/tex] and the system has a unique solution.
Thus, the given system of equations is consistent and independent.
Option (1)
Here, the first option is inconsistent and dependent.
There exists a solution of the given system of equations so it is consistent.
Therefore, option (1) is incorrect.
Option (2)
Here, the second option is consistent and dependent.
There exists a solution of the given system of equations so it is consistent.
Also, the solution is unique therefore the system of equations is independent.
Therefore, option (2) is incorrect.
Option (3)
Here, the third option is consistent and independent.
There exists a solution of the given system of equations so it is consistent.
Also, the solution is unique therefore the system of equations is independent.
Therefore, option (3) is correct.
Option (4)
Here, the fourth option is inconsistent and independent.
There exists a solution of the given system of equations so it is consistent.
Therefore, option (4) is incorrect.
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Answer details
Grade: High school
Subject: Mathematics
Chapter: System of linear equations
Keywords: Equations, unique solution, independent, dependent, consistent, inconsistent, infinite solutions, homogeneous equation, non- homogeneous equation, determinant.
Milo wants to make a mixture that is 50% lemon juice and 50% lime juice.
How much 100% lemon juice should he add to a juice mixture that is 20% lemon juice and 80% lime juice to make 4 gallons of the 50% lemon/50% lime juice mixture?
A. 0.5 gallon
B. 1.5 gallons
C. 2 gallons
D. 2.5 gallons
Answers
its actually 1.5 but ok
Answer:
The correct option is B.
Step-by-step explanation:
It is given that Milo wants to make a mixture that is 50% lemon juice and 50% lime juice.
Milo has a juice mixture that is 20% lemon juice and 80% lime juice. Milo add a 100% lemon juice.
Let the 100% lemon juice added by milo be x.
In 4 gallon of mixture contains 50% lemon juice and 50% lime juice. It means 2 gallon lemon juice and 2 gallon lime juice.
If we add 100% lemon juice, then the quantity of lime juice will remains same.
80% of the fist mixture is 2 gallon.
20% of the first mixture is 0.5 gallon.
It means the first mature contains 2 gallon lime juice and 0.5 gallon lemon juice.
Milo will add 100% lemon juice
[tex]2-0.5=1.5[/tex]
Therefore option B is correct.
Find a linear differential operator that annihilates the given function. (use d for the differential operator.) e−x + 8xex − x2ex
Answers
Explanation:
Let
[tex]f(x) = e^{-x} + 8xe^x - x^2 e^x[/tex].
[tex]d^n (f(x))[/tex] = nth derivative of f
Note that
[tex](d + 1) (e^{-x}) = d (e^{-x}) + e^{-x} = -e^{-x} + e^{-x} = 0 \\ \boxed{\Rightarrow (d + 1)e^{-x} = 0} [/tex]
So, (d + 1) annihilates [tex]e^{-x}[/tex].
Note that
[tex](d + 1) (8x e^x - x^2 e^x) = d(8x e^x - x^2 e^x) + (8x e^x - x^2 e^x) \\ = (8e^x + 8x e^x - 2x e^x - x^2 e^x) + (8x e^x - x^2 e^x) \\ \boxed{(d + 1) (8x e^x - x^2 e^x) = 8e^x + 14x e^x - x^2 e^x}[/tex]
Let [tex]g(x) = 8e^x + 14x e^x - x^2 e^x[/tex]
For any real number [tex]\alpha[/tex] and positive integer n,
[tex](d - \alpha)^n (c_1 e^{\alpha x} + c_2 xe^{\alpha x} + c_3 x^2 e^{\alpha x} + ... + c_n x^{n - 1} e^{\alpha x}) = 0 [/tex] (1)
So, for g(x), [tex]\alpha = 1, n = 3[/tex]. Thus, using equation (1),
[tex](d - 1)^3 (8e^x + 14x e^x - x^2 e^x) = 0[/tex] (2)
Since, [tex]8e^x + 14x e^x - x^2 e^x = (d + 1) (8x e^x - x^2 e^x)[/tex] ,
[tex](d - 1)^3 ((d + 1) (8x e^x - x^2 e^x)) = (d - 1)^3 (8e^x + 14x e^x - x^2 e^x) = 0[/tex]
Moreover, since [tex](d + 1) (e^{-x}) = 0[/tex]
[tex](d - 1)^3(d + 1) (e^{-x}) = (d - 1)^3 (0) [/tex]
[tex]\\ \boxed{(d - 1)^3(d + 1) (e^{-x}) = 0 }[/tex] (3)
Hence based on equations (2) and (3),
[tex](d - 1)^3(d + 1) (e^{-x} + 8xe^x - x^2e^x) \\ = (d - 1)^3(d + 1)(e^{-x}) + (d - 1)^3(d + 1)(8xe^x - x^2e^x) \\ = 0 + 0 \\ \boxed{(d - 1)^3(d + 1) (e^{-x} + 8xe^x - x^2e^x) = 0}[/tex]
Therefore, the linear operator [tex](d - 1)^3(d + 1)[/tex] annihilates [tex]e^{-x} + 8xe^x - x^2e^x[/tex].
To find the differential operator that annihilates the given function e^(-x) + 8xe^x - x^2e^x, we individually find differential operators for each term. The operators d^2, d^3, and d^4 annihilate e^(-x), 8xe^x, and x^2e^x respectively. The least common multiple of these operators is d^4, which is the common operator that annihilates the entire function.
Explanation:To find a linear differential operator that annihilates the function e−x + 8xex − x2ex, we must apply an operator that, when used on these functions, yields zero. For simplicity, we can use d to represent the differential operator d/dx. We annihilate each term independently and find a differential operator common to all terms.
Starting with the first term e−x, we note that d(e−x) = −e−x. Applying d again gives us d(d(e−x)) which is e−x. Thus, applying d2 annihilates e−x.
Annihilating 8xex and x2exNext, consider the second term 8xex. Applying the operator d3 to 8(xex) would be: d3(8xex) = d3(8(ex + xex))= d3(8ex) + d3(8xex) which results in zero.
For the third term x2ex, applying d4 annihilates the term, following the pattern: d4(x2ex) = d(d(d(d(x2ex)))) = 0.
The common differential operator that annihilates all three terms is the least common multiple of the individual operators for each term, in this case, d4.
Rosa made 12 gallon of lemonade to sell at a lemonade stand.How many pints of leamonade did she make
Answers
12 gallon equals 12x8 pints
12 x 8 = 96 pints
I need someone’s help!
Answers
Check out the attached image to see the steps on how to get that answer
Notice how I'm marking the rows and columns, and then multiplying the corresponding values together. Also make note of the color-coding.
There are a total of 26 students in Tom's class. There are 4 more boys than girls.
Which system of equations represents this situation? Let g be the number of girls, and let b be the number of boys.
A
g+b=26
g=4+b
B
g+b=26
b=4+g
C
g+b=26
g=4b
D
g+b=26
b=4g
Answers
4 more boys than girls so:
b = g + 4
And the total number of student is 26 so:
g + b = 26
The graph below shows the solution set to which system of inequalities?
Answers
using a graph tool
I proceed to verify each case to get the answer
see the attached figure
case A)
y> 2x-4 y>=-x y<=3
case B)
y> 2x-4 y>=-x y<3
case C)
y< 2x-4 y>=-x y<=3
the answer is the option A)
y> 2x-4 y>=-x y<=3
write a whole number as a fraction the number 5
Answers
5/1, as 5 divided by 1 = 5
Hope this helps
Hope this helped!
Lydia drove 302 miles in 10 hours. on average, how fast did she drive in miles per hour? express your answer in simplest form.
Answers
Answer:
30.2 mph
Step-by-step explanation:
Hello, I think I can help you with this
the speed average is the is the quotient of the distance traveled and the time used to do it, mathematical speaking
[tex]Average\ speed =\frac{distance}{time}\\[/tex]
Step 1
Put the values into the equation
Let
Distance:302 miles
Time:10 hours
[tex]Average\ speed =\frac{distance}{time}\\\\Average\ speed =\frac{302\ miles}{10\ hours}\\Average\ speed =30.2 \frac{miles}{hour} \\[/tex]
Average speed=30.2 mph
Have a great day.
If the rules for the lottery game powerball required participants to choose 5 unique numbers (ranging from 1 to 59) in any order along with one "powerball" (ranging from 1 to 35), what is the probability of winning the jackpot under these game rules?
Answers
5,006,386 x 35 = 175,223,510
probability 1: 175,223,510
To calculate the probability of winning the Powerball, you need to identify all possible combinations for drawing 5 unique numbers from a set of 59, then multiply that with the separate 35 options for the 'powerball'. Your chance of winning equals one (the exact match to your ticket) divided by the total possible combinations.
Explanation:The probability of winning the Powerball jackpot can be determined using the concept of combination in probability theory. Probabilities are calculated by taking the total number of successful outcomes or combinations and dividing it by the total number of possible outcomes or combinations.
In the case of Powerball, there are 5 unique numbers to be drawn from a pool of 59 (ignoring order), and 1 'powerball' from a pool of 35. For the 5 unique numbers, there are C(59,5) combinations where C(n,r) = n! / [r!(n-r)!], n being the total number of options available and r being the number of options chosen at a time.
For the 'powerball', since it is a separate pool of 35 numbers and only 1 is chosen, the number of combinations is simply 35. As such, you multiply the two outcomes for total possible combinations: C(59,5)*35.
For the probability of winning, you want to know combinations that match exactly your selection, of which there is only one. Therefore the probability is 1 / [C(59,5)*35], resulting in extremely low chances of winning.
Learn more about Powerball Probability here:
https://brainly.com/question/33176496
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what is equivalent to (81m^6)^1/2
Answers
= 9 * m^(6*1/2)
= 9m^3
Answer: 9m^3
Step-by-step explanation:= 81^(1/2) * (m^6)^(1/2)
= 9 * m^(6*1/2)
= 9m^3
Lalasa and yasmin are designing a triangular banner to hang in the school gymnasiums. They first draw the design on paper. The triangle has a base of 5 inches and a height of 7 inches. If 1 inch on the drawing is equivalent to 1.5 feet on the actual banner, what will the area of the actual banner be?
A. 17.5 ft2(squared)
B. 52.5 ft2
C. 39.375 ft2
D. 78.75 ft2
Answers
The correct answer is C. 39.375 square feet.
The student's question pertains to finding the area of the actual banner when a scale transformation is applied to a triangular design. To calculate the area of the actual banner, we need to scale the base and the height of the triangle from the drawing size to the actual size and then use the formula for the area of a triangle.
First, we convert the measurements of the base and the height from inches to feet using the given scale of 1 inch = 1.5 feet:
Base in feet: 5 inches × 1.5 feet/inch = 7.5 feetHeight in feet: 7 inches × 1.5 feet/inch = 10.5 feetWe then apply the area formula for a triangle which is Area = (base × height) / 2:
Area = (7.5 feet × 10.5 feet) / 2 = 39.375 square feet
Therefore, the correct answer is C. 39.375 square feet.
What is the area of this triangle?
Picture below, will give brainliest!!!! please help
can you give it to me in a decimal? Thank you
Answers
[tex]\text {Area = } \dfrac{1}{2} (3.2)(4.7)sin (62 \textdegree)[/tex]
[tex]\text {Area = } 6.64 \ yd^2[/tex]
what is the correct answer to this question?
a
b
c
d
Answers
.. from 275/6.58 = $41.79
.. to 275/6.25 = $44.00
which is a change of $2.21.
Evaluate. 134 × -7
A)-2,158
B)-828
C)-938
D)938
Answers
Because you are multiplying a positive and a negative your answer should be negative
if f(x) varies directly with x2, and f(x) = 96 when x = 4, find the value of f(6).
Answers
96=k(4)
k=24
f(x)=kx
f(6)=24(6)
f(6)=144
Please help with the question. For some reason, I find it confusing and I need some assisstance. Thank you!
Answers
C=4x5+2.
C=20+2
C=22
You need 22 chairs.
Answer: i think it is 22
Step-by-step explanation:
a taxi ride costs $3 plus $2.50 per mile. write and graph an equation in two variables that represents the total cost of a taxi ride.
Answers
the 2 variables in the equation are as follows;
y = total cost of taxi ride
x = number of miles travelled per taxi ride
we need to write the equation in slope- intercept form
y = mx + c
m = gradient /slope
mx = cost per mile travelled x number of miles travelled
c= intercept. a fixed cost for every taxi ride
m = $2.50 per mile
c = $3
the graph equation is as follows;
y = 2.50x + 3
Nacir is buying school supplies. He purchases 8 notebooks for $12.48. How much does Nacir pay for 1 notebook?
Answers
Simplify (SecX-TanX)(1+sinX) X stands for theta
Answers
[tex]\bf [sec(\theta )-tan(\theta )][1+sin(\theta )]\implies \left[ \frac{1}{cos(\theta )}-\frac{sin(\theta )}{cos(\theta )} \right][1+sin(\theta )] \\\\\\\ \left[\frac{1-sin(\theta )}{cos(\theta )} \right][1+sin(\theta )]\implies \cfrac{\stackrel{\textit{difference of squares}}{[1-sin(\theta )][1+sin(\theta )]}}{cos(\theta )} \\\\\\ \cfrac{1^2-sin^2(\theta )}{cos(\theta )}\implies \cfrac{cos^2(\theta )}{cos(\theta )}\implies cos(\theta )[/tex]
If a canoe travels with a speed of A mph for 3 hours, and then with a speed of B mph for the rest of the journey, how long does it take to travel C miles
Answers
.. speed = distance/time
in any of its various forms.
The distance traveled at A mph is
.. distance = speed * time
.. distance = A*3
The time required for the rest of the journey is
.. time = distance / speed
.. time = (C -3A)/B
The the total time required to travel C miles is the 3 hours for the first part plus the time for the rest of the journey.
.. total journey time = 3 + (C -3A)B . . . . . hours
t = (C - 3*A)/B + 3
i dont know if its correct but hope it helps!
PLZ HELP ME FAST! Given that ∠MQL = 180° and ∠XQR = 180°, which equation could be used to solve problems involving the relationships between ∠MQR and ∠XQL?
A) (−5b + 115) = (125 − 10b)
B) (−5b + 115) + (125 − 10b) = 180
C) (−5b + 115) − (125 − 10b) = 180
D) (−5b + 115) − 180 = (125 − 10b)
Answers
because XQL = MQR
Given
∠MQL=180°
∠XQR=180°
Hence ∠MQL=∠XQR
(We know that ∠MQL is the sum of ∠MQR and ∠RQL
and ∠XQR is the sum of ∠XQM and ∠MQR)
lets plug in these in our equation
∠MQL=∠XQR
∠MQR + ∠RQL = ∠XQM + ∠MQR
We can cancel out ∠MQR
hence ∠RQL = ∠XQM
hence we can infer that the opposite angles of intersecting lines are equal
similarly
∠MQR = ∠XQL
which is
125-10b = -5b+115
Hence the right option is A)
The graph shows a system consisting of a linear equation and a quadratic equation.
What is the solution(s) to the system?
Question 10 options:
(3, 4) only
This system has no solution.
(3, 4) and (5, 12)
(0, 7) and (2, 0)
Answers
Answer:
(3, 4) and (5, 12)
Step-by-step explanation:
The points where the two curves intersect are the solutions to the system of equations:
(3, 4) and (5, 12)
Answer:
Step-by-step explanation:
A pond is freshly stocked with 6 brown trout and 18 lake trout. The first fisherman in the area catches and releases a trout. He catches another trout a while later. What is the probability that the fisherman caught brown trout each time?
Answers
24÷6= 4 so probability of 1 out of 4. times 2 because he caught 2 fish. so then that would be 1 out of 8. :)
Jean adds 35 and 9.how can she solve using only equations to model her thinking?
Answers
Hope this helps! ;)
Answer:
Step-by-step explanation: 35+(5+4)=35+5+4=40+4=44
A fleet of vehicles is comprised of 60 vans, 20 limos, and X sedans. If 10% of all vehicles are limos, how many sedans are in the fleet?
Answers
Answer:
120 sedans are in the fleet.
Step-by-step explanation:
A fleet of vehicles is comprised of 60 vans, 20 limos and X sedans.
It is given that 10% of all vehicles are limos.
Let all vehicles be n
10% × n = 20
0.10n = 20
n = [tex]\frac{20}{0.10}[/tex]
n = 200
Now we have to calculate the number of sedans.
So 60 vans + 20 limos + x sedans = 200
80 + x = 200
x = 200 - 80
x = 120
Therefore, 120 sedans are in the fleet.
Final answer:
By knowing that 10% of the fleet are limos and that there are 20 limos, we determine that the fleet has 200 vehicles. Subtracting the known number of vans and limos from the total, we find there are 120 sedans in the fleet.
Explanation:
To solve the problem, we need to determine the total number of vehicles in the fleet and use the information that 10% of all vehicles are limos. We know there are 60 vans and 20 limos already, and we want to find the number of sedans, denoted as X.
Since 10% of the vehicles are limos and there are 20 limos, that means there must be 200 vehicles in total (because 20 is 10% of 200).
So the equation to find the total number of vehicles is: 60 vans + 20 limos + X sedans = 200 vehicles. We already know that there are 60 vans and 20 limos, so we can say: 60 + 20 + X = 200. Solving for X, we subtract 60 and 20 from 200: X = 200 - 80 = 120.
Therefore, there are 120 sedans in the fleet.