Let f:X→Ybe a function. Then the function f is a bijection if the relation {(y ,x)∈Y×X∣ (x ,y)∈f} is a function.
Assume that X→Y is a bijective function.
i.e. f:X→Y is one-one and onto function.
so different elements in X have different images in Y
and b ∈ Y there exists a ∈ X such that f(a)=b.
R= {(y ,x)∈Y×X∣(x ,y)∈f} is a relation
we wish to prove R is a function.
R is a relation from Y to X
let Y∈Y by assumption f is bijection function from X to Y
for all y ∈ Y there exists X∈ X such that f(x)=y.
so for every element in Y there is mapping element in X.
if (x ,y) ∈ f then (y ,x) ∈ R
therefore R is a function.
let X1, X2, ∈ X then there exists some y1, y2,∈ y such that
take f ( x1 )= y1 and f( x2)=y2
by assumption R is a function from Y to X that is inverse of f is existing means f is bijection.
for bijective functions only inverse exists.
If f:X→Y is a function. Then the function f is a bijection if the relation
{(y ,x)∈Y×X∣(x, y)∈f} is a function.
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Dhara will be 26 years old when she graduates. If her mean bi-weekly wage over her career is $1,875.00, what will her lifetime earnings be when she retires at the age of 65?
Dhara's Lifetime earnings when she retires at the age of 65 is; $1,901,250
How to find the lifetime earnings?The first thing we will do here is to find the difference between Dhara's current age and the age he will be when he retires. Thus;
Difference in age = 65 - 26
Difference in age = 39 years old
Bi- weekly wage = 52 weeks/2
Bi- weekly wage = 26 paychecks per annum
Thus;
Lifetime earnings = $1875.00 × 26 paychecks per annum × 39
Lifetime earnings = $1,901,250
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Graph the line with y-intercept.....
Can you also mark the two points on the graph.
Given the equation v over 9.4 equals 13.7, solve for v.
4.3
23.1
128.87
128.78
Therefore the solution to the given question is option d)128.78 for this particular equation.
What is Equation?Many different definitions exist for a mathematical equation. In algebra, a statement that mathematically establishes the equality of two mathematical expressions is known as an equation.. Consider the equation 3x + 5 = 14, where 3x + 5 and 14 are two expressions that are separated by the symbol "equal." Mathematical variables are used in even the most fundamental and straightforward algebraic equations.
Here,
=> v/9.4 = 13.7
=> v = 13.7*9.4
=> v= 128.78
Therefore the solution to the given question is option d)128.78 for this particular equation
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Suppose your gross monthly income is $6,500 and your current monthly payments are $525. If the
bank will allow you to pay up to 38% of your gross monthly income (less current monthly payments)
for a monthly house payment, what is the maximum loan you can obtain if the rate for a 25-year
mortgage is 8.65%?
The maximum loan you can obtain if the rate for a 25-year mortgage is 8.65% is $298,853.76
How to find monthly payment?A monthly payment can be calculated as;
Monthly payment = (Monthly income - payment)(percentage payment)
38% of your gross monthly income is
0.38 × $6,500 = $2470
If you're already spending $525 on monthly payments, the amount the bank says you can afford for a mortgage loan;
Then
$2470-525= $1945
The amortization formula can be used to find the corresponding loan value.
A = P(r/25)/(1 -(1 +r/25)^(-25t))
where A is the monthly payment on a loan of P at interest rate of r for time t years.
Filling in the known values, we have ;
1945= P(.0525/25)/(1 -(1 +.0525/25)^(-12·30))
≈ 0.00515636808·P
Then the maximum loan can obtain is
P = $1945/0.00515636
P ≈ $298,853.76
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Departments D1 and D2. Each unit of product1 requires processing for 1 minute at D1 and 3 minutes at D2; each unit of product 2 requires processing for 2 minutes at D1 and 2 minutes at D2. Machine time available per day is 860 minutes at D1 and 1200 minutes at D2. How much of product 1 and 2 should be produced every day so that total profit is maximum. Make the mathematical model for the given problem
Answer:
510
Step-by-step explanation:
1200-860x3 /2 = 510
In 2010, a total of 2187 of the employees at Zoe's company owned a petrol
car.
In 2013, there were 1029 employees with petrol cars.
Assuming this number decreases exponentially, work out how many
employees owned a petrol car in 2017.
Give your answer to the nearest integer.
The number of employees with petrol cars is 423
Exponential Decay:In exponential decay, a quantity initially drops gradually before decreasing quickly. Finding the exponential drop of a fast reduction over time is made easier by the exponential decay formula.
The exponentially decreasing formula is used to calculate the radioactivity decay, population decay, and half-life.
The formula for Exponential Decay is given by
f(x) = a (1 - r)ˣHere we have
In 2010, a total of 2187 of the employees at Zoe's company owned a petrol car.
In 2013, there were 1029 employees with petrol cars.
The number of cars decreases exponentially,
As we know the formula for Exponential Decay, A = P (1 - r)^t
From the given data, a = 2187 and t = 3
The number of cars decreased = 2187 (1 - r)³
=> 2187 (1 - r)³ = 1029
=> (1 - r)³ = 1029/2187
=> (1 - r)³ = 343/729
=> (1 - r)³ = 7³/ 9³
=> 1 - r = 7/9
=> r = 1 - 7/9
=> r = 2/9
=> r = 0.2222
The Number of cars in 2017 i.e after 4 years from 2013
=> (1029)(1 - 0.2222)⁴
=> 423.0721516 ≅ 423
Therefore,
The number of employees with petrol cars is 423
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DIRECTIONS: Use this information to answer Parts A and B.
A faucet supplies water at a constant rate. After 3 minutes, the faucet supplies 12 liters of water.
Part A
Use the Line Tool to graph the proportional relationship that gives the number of liters of water the facet supplies, y, after x minutes.
The required proportional relationship is given as y = 4x and shown in the graph.
What is proportionality?proportionality is defined as between two or more sets of values, and how these values are related to each other in the sense are they directly proportional or inversely proportional to each other.
here,
According to the question,
The proportional relationship gives the number of liters of water the facet supplies, y, after x minutes,
y = kx - - - - - (1)
Put x = 3 so y = 4
12 = k[3]
k = 4
From equation 1,
y = 4x
Thus, the required proportional relationship is given as y = 4x and shown in the graph.
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HELPPP PLEASE ASPPP!!!!!!
Examine the graph below. Calculate the slope.
Slope of line is 1/2.
What is slope?
In arithmetic, the slope or gradient of a line is a range that describes each the direction and therefore the gradient of the road.
Main body:
to find the slope from a graph , we need to fix two co-ordinates at slope for axis
let point at x be = (40,40)
let point y = (80,60)
Formula for slope ,
m = y₂-y₁/x₂-x₁
m = 60-40/80-40
m = 20/40
m = 1/2
hence , slope of line is 1/2.
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7.) Given the following triangle, what is the value of x and the value of y ?
Answer: x = 36; y = 36
Step-by-step explanation:
This is an isosceles triangle, so the 2 "bottom" angles are equal:
2x = 72
x = 72/2 = 36
sum of interior angles = 180
180 - 72 - 72 = 36
Melvin and Sylvia are married with no children, and they're filing their federal income tax return. Melvin had a gross income of $44,500 last year, while Sylvia had a gross income of $51,200, and they plan to use the standard deduction. They're trying to decide whether to file their return jointly or separately, so they want to calculate how much less they would pay in federal income taxes if they filed jointly rather than separately. part 2
Answer: 3702
Step-by-step explanation: from plato
a building has a height of 250 in length of 160 m on a scale drawing of the building the height is 50 cm what is the length of the building on the scale drawing a centimeters
The required length of the building is given as 32 cm.
What is the scale factor?The scale factor is defined as the ratio of the modified change in length to the original length.
Here,
The height of the building is 250 meters, the length of the building is 160m, and on the drawing, the height of the building is 50 cm,
Scale factor = height on drawing / original height
Scale factor = 50 / 250
Scale factor = 1/5
Now,
Length in drawing = scale factor × original length,
length in drawing = 1 / 5 × 160
= 31
Thus, the length of the building as of the scale factor 1/5 is given as 31 cm.
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The classroom outside bag had 5 frisbees in it. If 25% of the outside toys are frisbees, then how many total toys are in the bag?
Answer: 15, not including the frisbees. (The answer would be 20 if the frisbees were in there)
Step-by-step explanation: 25% is equivalent to 1/4, and 5 is equal to 1/4. The simplest way to solve this is to multiply 5 by 4, and you'll get 20. Then subtract 5 (the frisbees) and you'll get 15. (Sorry if that doesn't make much sense haha) Hope this helps!!
a circular plate has a radius of 12 inches complete the table included a number and a unit in the last column
The area and the circumference of the circular plate is 452.16 in² and 75.36 in
What is area and circumference of circular plate?
The measurement of the circle's perimeter, also known as its circumference, is called the circle's boundary. The region a circle occupies is determined by its area. The length of a straight line drawn from the centre of a circle equals the diameter of that circle. It is typically expressed in units like cm or unit m.a closed circle geometric shape in space. A circle is a locus of a point moving around a fixed point at a defined distance from the point in mathematical terms. A circle is essentially a closed curve with an outer line that is equally spaced from the center.Area of circular plate , A = πr²
=3.14 * 12 * 12
= 452.16 in²
Circumference of the circular plate, C = π * 2r
= 3.14 * 2 * 12
= 75.36 in
Hence, the area and the circumference of the circular plate is 452.16 in² and 75.36 inches.
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bus company uses a grid to locate bus stops. The bus station is at (3, 4). The point (7, 6) represents the first bus stop. The location of each stop after that is determined using the same relationships as those between the coordinates of the bus station and the first bus stop. A coordinate plane is titled Which point represents the fourth bus stop? A. (11, 8) B. (13, 12) C. (15, 10) D. (19, 12) 7 / 7 6 of 7 Answered
Answer:
I think that's 11,8
Step-by-step explanation:
Hope its right
6. Jesse roasted 12 marshmallows in 4 minutes. How many marshmallows could she roast in 7 minutes?
Answer: 21
Step-by-step explanation: 12 in 4 minutes means every minute 3 marshmallows are roasted, 4x3=12, so add three more minutes and you get 21. 7x3=21
Answer: 21 marshmallows
Step-by-step explanation:
4 minutes x 1.75 = 7 minutes
multiply marshmallows by 1.75 as well.
12 marshmallows x 1.75 = 21 marshmallows
Determine whether the equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement.
log(x+3)-log(8x)=log(x+3)/log(8x)
if false make necessary changes to produce a true statement
So the RHS of the given equation is incorrect.
[tex]\lim_{n \to \pi/2} (tanx)^{2cosx}[/tex]
John paid $280 for a new mountain bicycle to sell in his shop. He wants to price it so that he can offer a 30% discount but still make 10% of the price he paid for it. At what price should the bike be marked?
By using the concept of discount, it can be calculated that
The bike should be marked at $440
What is discount?
Sometimes during sale, the items are sold at a reduced price from the original price of the item. The difference between the reduced price and the original price of the item is called discount.
Price paid by John for a new mountain bike = $280
Markup price = [tex]\frac{10}{100} \times 280[/tex] = $28
Price after markup = $(280 + 28) - $308
Let the marked price be $x
Rate of discount = 30%
Discount = [tex]\frac{30}{100}\times x[/tex]
Price after discount = [tex]x - \frac{30}{100}\times x[/tex]
= [tex]\frac{100x - 30x}{100}[/tex]
= [tex]\frac{70}{100}x[/tex]
By the problem,
[tex]\frac{70}{100}x = 308\\x = \frac{308 \times 100}{70}\\[/tex]
x = $440
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what is the solution for
an= -3an-1-3an-2-an-3, a0=1, a1=0, a2=5
a) (-1-4n+3n^2)(-1)^n
b) (1-4n-3n^2)(-1)^n
c) (1+4n+3n^2)(-1)^n
Answer:
none of the above
(1 -4n +3n^2)(-1)^n
Step-by-step explanation:
You have the recursive relation {a[0]=1, a[1]=0, a[2]=5, a[n]=-3a[n-1] -3a[n-2] -a[n-3]} and you want to know the explicit relation.
Test a fewThe given initial value a[0] = 1 rules out choice A.
The value a[1] = 0 rules out choices B and C.
Actual solutionWe observe that changing the sign of the constant in the first choice will make it match the sequence:
(1 -4n +3n^2)(-1)^n . . . . . . the solution for the recursive equation
Write the systems of equations from the table below and find the solution?
The system of equations is y = 4x - 3 and y = -1/3x + 4 and the solution is (21/13, 45/13)
How to determine the system of equations?From the question, we have the table of values as the parameters that can be used in our computation:
From the table, we have
Table 1
(x, y) = (0, -3) and (4, 13)
A linear equation is represented as
y = mx + c
Where
Slope = m
c = y when x = 0
This means that
c = -3
So, we have
y = mx - 3
The point (4, 13) implies that
13 = m * 4 - 3
So, we have
4m = 16
m = 4
So, the equation is
y = 4x - 3
Table 2
(x, y) = (0, 4) and (4, 6)
A linear equation is represented as
y = mx + c
This means that
c = 4
So, we have
y = mx + 4
The point (4, 6) implies that
6 = m * 4 + 4
So, we have
4m = -2
m = -1/3
So, the equation is
y = -1/3x + 4
Substitute y = -1/3x + 4 in y = 4x - 3
-1/3x + 4 = 4x - 3
So, we have
-x + 12 = 12x - 9
Evaluate the like terms
13x = 21
Divide
x = 21/13
y = -1/3x + 4 implies that
y = -1/3 x 21/13 + 4
So, we have
y = -7/13 + 4
Evaluate
y = 45/13
So, the solution is (21/13, 45/13)
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Suppose you can afford to pay $ 325 a month for 9 years towards a new car with no down payment. If the current interest rates are 4.75%, how expensive a car can you afford?
Car sticker price =
Based on the given parameters, the car would be $497.9 expensive in 9 years
How to determine the amount in nine years?The given parameters about the compound interest are
Principal Amount, P = $325Interest Rate, R = 4.75%Time, t = 9 yearsCompound interests are different from simple interest, and they are calculated using the following compound interest formula
CI = P(1 + R)^t - P
To calculate the amount, we have:
A = P + CI
So, the equation becomes
A = P + P(1 + R)^t - P
Evaluate the like terms
A = P(1 + R)^t
In this case, we introduce the variable n
So, we have
A = P(1 + R/n)^nt
Where n is the number of times in a year
i.e. n = 12
Substitute the known values in the above equation
A = 325 * (1 + 4.75%/12)^(12 * 9)
Evaluate the percentages and the quotients
A = 325 * (1 + 0.0039583)^(108)
Evaluate the sum
A = 325 * (1 .0039583)^(108)
Evaluate the exponent
A = 325 * 1.532
Evaluate the product
A = 497.9
Hence, the value of the investment after 9 years is $497.9
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In a random sample of 125 people, 42 of them owned two vehicles. The sample proportion of people who own two vehicles is
In a random sample of 125 people, 42 of them owned two vehicles. The sample proportion of people who own two vehicles is 0.336
The percentage of people in a sample that exhibit a particular attribute or trait is expressed by the sample proportion (p).
Divide the total number of individuals (or things) in the sample by the number of individuals (or items) who share the characteristic of interest to obtain the sample proportion.
p = x/n, where x stands for the sample's success rate and n for sample size, gives the sample proportion p.
According to the question,
Sample size = 125
Number of people owned two vehicles = 42
Sample Proportion (p) = x/n
=> p = 42 / 125
=> p = 0.336
Therefore , The sample proportion of people who own two vehicles is 0.336
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rectangle abcd is similar to rectangle wxyz what is the value of ab if wx is 8 feet ad is 12 feet and wz is 16feet
The value of AB is 6 units.
What are Similar Rectangles ?
Similar rectangles states that the corresponding sides are in proportion.Two rectangles are similar when the corresponding adjacent sides have the same ratio. We do not need to check the angles as all angles in a rectangle are 90 degrees.
As per the statement:
Rectangle ABCD is similar to rectangle WXYZ.
WX = 8 feet, AD = 12 feet and WZ = 16 feet.
Since. Rectangle ABCD and rectangle WXYZ are similar
then;
their corresponding sides are in proportion:
[tex]\frac{AB}{WX} =\frac{BC}{XY} = \frac{CD}{YZ} =\frac{AD}{ZW}[/tex]
To find the value of AB:
[tex]\frac{AB}{WX} =\frac{AD}{ZW}[/tex]
Substitute the given values we have;
[tex]\frac{AB}{8} =\frac{12}{16}[/tex]
Multiply both sides by 8 we have;
[tex]AB = \frac{12}{16} * 8 = 6[/tex]
Therefore, the value of AB is 6 units.
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Solve x2 = 36 for x.
Answer:
x=18
36/2=18
all good.
Find the absolute extrema if they exist, as well as all values of x where they occur, for the function f(x)=(x²-64)^{1/11} on the domain [-8,9]
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
A. The absolute maximum is , which occurs at x= (Round the absolute maximum to two decimal places as needed. Type an exact answer for the value of x where the maximum occurs. Use a comma to separate answers as needed.)
B. There is no absolute maximum.
The absolute maximum is 1.29, which occurs at x= 9
What is absolute extrema?
An absolute extremum of a perform in an exceedingly given interval is that the purpose at that a most or minimum worth of the perform is obtained. Frequently, the interval given is that the function's domain, and therefore the absolute extremum is that the purpose admire the most or minimum worth of the complete perform.
Main body:
Given =
f(x) = [tex](x^{2} -64)^{1/11[/tex]
f' (x) = derivative of function f(x)
f'(x) = 1/11[tex](x^{2} -64)^{-10/11[/tex] . 2x
= [tex]2 \frac{x}{11}[/tex][tex](x^{2} -64)^{-10/11[/tex]
putting f'(x) = 0
2x/11 [tex](x^{2} -64)^{-10/11[/tex] = 0
[tex](x^{2} -64)^{-10/11[/tex] = 0
[tex]x^{2} -64[/tex] = 0
x² = 64
x = ±8
We are given interval [-8,9].
Hence, calculating f(x) at -8, 8, 9.
f(-8) = 0
f(8) = 0
f(9) = [tex]17^{1/11}[/tex]
= 1.29
Hence, absolute maximum value is 1.29 at x=9.
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Use the formula A=P(1+r/n)ⁿt to solve the compound interest problem
Find how long it takes a $ 1700 investment to earn $ 500 interest if it is invested at 2 % interest compounded quarterly.
$ 500 interest will be earned in approximately years. (Round to the nearest tenth.)
The compound interest is 61.342 years.
What is compound interest?
Compound interest is the interest calculated on the principal and the interest accumulated over the previous period. It is different from simple interest, where interest is not added to the principal while calculating the interest during the next period.
To find the time period the interest must be compounded quarterly.
Calculated twelve times a year is interest that is compounded monthly. Interest is calculated on a quarterly compound basis, four times a year.
Given that;
A = 1700 P = 500
r = 2% n = 4
= 2/100
r = 0.02 /year
Using the formula A=P(1+r/n)ⁿt find the time period (t)
t = ln(A/P) / n[ln(1 + r/n)]
t = ln(1,700.00/500.00) / ( 4 × [ln(1 + 0.02/4)] )
t = ln(1,700.00/500.00) / ( 4 × [ln(1 + 0.005)] )
t = 61.342 years
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A mural is feet long and is divided into 7 equal-length panels. How many feet long is each panel?
The required length of each panel would be 1 feet.
Given that,
A mural is 7 feet long and is divided into 7 equal-length panels. How many feet long is each panel to be determined.
The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.
here,
The required length of each panel = total length of the panel / 7
= 7 / 7 = 1 feet
Thus, the required length of each panel would be 1 feet.
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Write the expression x2 - 6x-2 in the form (x + a)² + b
Answer:
(x - 3)² - 11--------------------------
GivenExpression x² - 6x - 2.Convert this to vertex form by completing the square.
Recall identity (a ± b)² = a² ± 2ab + b² and apply as given below:
x² - 6x - 2 = x² - 2*3*x + 3² - 3² - 2 = (x - 3)² - 9 - 2 = (x - 3)² - 11Answer:
[tex](x-3)^2-11[/tex]
Step-by-step explanation:
Given quadratic expression:
[tex]x^2-6x-2[/tex]
To write the given expression in the form (x + a)² + b, complete the square.
Add and subtract the square of half the coefficient of the term in x:
[tex]\implies x^2-6x+\left(\dfrac{-6}{2}\right)^2-2-\left(\dfrac{-6}{2}\right)^2[/tex]
Simplify:
[tex]\implies x^2-6x+\left(-3\right)^2-2-\left(-3\right)^2[/tex]
[tex]\implies x^2-6x+9-2-9[/tex]
The first three terms x² - 6x + 9 form a perfect square trinomial.
A perfect square trinomial can be written as the square of a binomial.
Factor the perfect square trinomial:
[tex]\implies(x-3)^2-2-9[/tex]
Simplify:
[tex]\implies(x-3)^2-11[/tex]
Therefore, the given expression written in the form (x + a)² + b is:
[tex]\boxed{(x-3)^2-11}[/tex]
Hassan is cooking some rice the recipe says he needs 450 grams of rice for 6 people how many kilograms or rice would he need for 18 people?
Answer:
450 g = 6 people
? kg = 18 people
450 x 18 = 8100
8100 = 6x
8100/6 = 6x/6
x = 3150 kg of rice is needed for 18 people.
a sample of 79 eggs yields a sample mean weight of 1.00 ounces. assuming that the population standard deviation
The margin of error for 16 eggs with sample means 2 ounce and standard deviation as 42 ounces is 18.407 .
In the question ,
it is given that ,
the number of eggs = 16
the sample means weight is = 2 ounce .
the standard deviation is = 42 ounces .
we have to find the margin of error .
degree of freedom is 16 - 1 = 15 .
and the t score for the 90% confidence interval is
t₁₅,₀.₀₅ = 1.753
So , the margin of error is = t₁₅,₀.₀₅×(standard deviation/√n)
Substituting the values ,
we get ,
= 1.753×(42/√16)
On simplifying further ,
we get ,
= 18.407
Therefore , the margin of error is = 18.407 .
The given question is incomplete , the complete question is
A sample of 16 eggs yields a sample mean weight of 2.00 ounces and a sample standard deviation = 42.0 ounces, find the margin of error in estimating a confidence interval estimate at the 90% level of confidence. Round your answer to three decimal places. You may assume that egg weights are normally distributed .
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