The functions f, g and h on the set of real numbers are defined by f(x) = x² + 1, g(x) = 2x - 3 and h(x) = 4x + 5 respectively. Determine the formulae for the composite functions:
(a) f [g(x)]
(b) g [h(x)]
(c) h [g {f(x)} ]​

Answer:

We can start solving the problem by using algebra. Let x be the number of points scored by each girl, and y be the number of points scored by each boy. Then, based on the information given in the problem, we have the following equations:

x + y = 27 (the total number of points scored by the team)

3x - 2y = 15 (the difference between the number of points scored by the 3 girls and the 2 boys)

To solve for x and y, we can use the second equation to eliminate one of the variables. Multiplying both sides of the second equation by 2, we get:

6x - 4y = 30

Now we can substitute this into the first equation:

6x - 4y + y = 27

6x = 27 + 4y

x = (27 + 4y)/6

Now we can substitute this back into the second equation:

3((27 + 4y)/6) - 2y = 15

27 + 2y - 2y = 15

27 = 15

This is a contradiction, so the given information is insufficient to find the number of points scored by each girl and each boy.

Step-by-step explanation:

Answer:

1,3,5,15

Step-by-step explanation:

A factor cannot be greater then the number itself so we can scratch 30 and 45. 15 is divisible by itself and 1, as every other number. 2 is not a factor of 15 because 15 is not even. Also not 9. But it does have factors 5 and 3 because 5×3=15.

Hope that helps, let me know if you have any further questions!

Answer: 20/3 / 1/6 is a division problem that can be simplified as follows:

20/3 divided by 1/6 = (20/3) * (6/1) = (206) / (31) = 120/3 = 40

The value of the division is 40. However, without knowing the context, it is impossible to know what the meaning of this value is. In particular, without knowing the units of 20/3 and 1/6, it is not possible to interpret the division as miles per gallon or any other unit.

It is important to note that 20/3 and 1/6 should be represented as mixed numbers (6 2/3 and 1/6) or as decimal numbers (6.66 and 0.16666) in order to be able to proceed with the division.

Step-by-step explanation:

Answer: ASA

Step-by-step explanation: Hello! You can use ASA to prove that the triangles ABC and ABD are congruent.  This is because you are given two angles of each triangle.  You know that your answer should be ASA or AAA, which are the only two triangle congruence postulates that involve two or more congruent angles.  Lastly, you can use the reflexive property.  An example of reflexive property is 3 = 3, which means anything is congruent to itself.  You know that both triangles share line segment AB, meaning that they have one side and two angles that are congruent.  Since you know that the two angles and the included side are congruent, ASA proves that both triangles ABC and ABD are congruent.  I hope this helps you!

Answer: about 0.79

Step-by-step explanation:

First divide 8 by 9 --> (8/9) = about 0.8889

then square that value --> (0.8889)^2 = about 0.79.

PEMDAS is your friend!

Answer:

57 = 33+24 = 3a+3c

Step-by-step explanation:

Let a= adults, c = children

total costs = 11a+8c

57 = 33+24 = 3a+3c

Answer: We can use the variables x and y to represent the number of adult tickets and children's tickets, respectively. The cost of adult tickets is $11 each and the cost of children's tickets is $8 each. So, we can use the following equation to represent the total cost for the movie tickets:

C = 11x + 8y

Where C is the total cost of the tickets, x is the number of adult tickets and y is the number of children's tickets

One combination of tickets that would total $57 is:

C = 11x + 8y = $57

x + y = 7 (number of tickets)

One way to satisfy this equation is to have 5 adult tickets and 2 children's tickets

x = 5

y = 2

C = 115 + 82 = $57

So, 5 adult tickets and 2 children's tickets is one combination that would total $57.

Step-by-step explanation:

Answer:

see below

Step-by-step explanation:

11: write 3 equiv ratio of 1:3, so it' could be 3:9, 2:6, 4:12

12.  20:15 = 4:3, 12:9, 8:6

No, Andy do not have enough loan, he need $200 more dollars to purchase Michael Jackson white glove.

Percentage is defined as a given part or amount in every hundred. It is a fraction with 100 as the denominator and is represented by the symbol "%".

Given that, Adams awesome pawnshop will lend up to 42% of the value of a borrowers collateral.

Andy want wants to use his $10000 signed Michael Jordan Rookie card. his $3000 signed Joe Montana cleats and his $2000 signed Tiger Woods golf ball as collateral for a loan.

Now, 42% of (10000+3000+2000)

= 42/100 ×15000

= 0.42×15000

= $6300

Cost of Michael Jackson white glove costs $6500

No, Andy do not have enough loan, he need $200 more dollars to purchase Michael Jackson white glove.

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Applying the corresponding angles theorem, the values of x and y are:

x = 13

y = 24

The Corresponding Angles Theorem states that if two lines are cut by a transversal and the pairs of corresponding angles are congruent, then the two lines are parallel.

From the diagram given, we have:

63 + (5x - 8) + (7x - 31) = 180 [corresponding angles theorem]

63 + 5x - 8 + 7x - 31 = 180

12x + 24 = 180

12x = 180 - 24

12x = 156

x = 13

5x - 8 + 4y + 27 = 180

Plug in the value of x:

5(13) - 8 + 4y + 27 = 180

65 - 8 + 4y + 27 = 180

84 + 4y = 180

4y = 180 - 84

4y = 96

4y/4 = 96/4

y = 24

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Answer:

  9.  reduction, 1/2

  10.  enlargement, 3/2

  11.  enlargement, 3

  12.  reduction, 1/3

Step-by-step explanation:

You want to know whether the dilations shown are reduction or enlargement.

Dilation

If the image (figure B) is larger, the dilation is an enlargement. (10, 11)

If the image (figure B) is smaller, the dilation is a reduction. (9, 12)

The scale factor is the ratio of a length on the image to the corresponding length on the original figure. It will also be the ratio of distances from the center of dilation (the origin) of corresponding points.

The ratio of line lengths B/A is 3/6, a reduction by a factor of 1/2.

The ratio of distances from the origin on the x-axis is 3/2, an enlargement by a factor of 3/2.

The ratio of distances from the origin on the x-axis is 6/2, an enlargement by a factor of 3.

The ratio of distances from the origin on the x-axis is 2/6, a reduction by a factor of 1/3.

The height of the triangle with area of 60 square inches and the base of 12 inches is 10 in.

An equation is an expression that shows the relationship between numbers and variables.

The area of a triangle is the amount of space it occupies. The area is the product of the base and height divided by 2. It expressed as:

Area = (1/2) * base * height

Given that the area is 60 square inches and the length of the base must be 12 inches. Let h represent the height

Therefore:

Area = (1/2) * base * height

60 = (1/2) * 12 * h

6h = 60

h = 10 in

The height of the triangle is 10 in.

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Answer:

Step-by-step explanation:

The slope of the given line in the graph is "undefined".

Slope, usually referred to as rise over run, is the "steepness" of the line.

By dividing the difference in y values between two points by the difference in x values, we may determine slope.

Thus, as the line is passing through the y axis and parallel to the x axis, the slope of the given line is undefined.

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None of the options are correct. A variable can only hold one piece of data at a time.

A variable is a named storage location in a computer's memory that is used to store a single value. The value can be of any data type, such as a number, a string, or a boolean. A variable can only hold one piece of data at a time, meaning that it cannot store multiple pieces of data simultaneously. This is why it is important to use multiple variables if you need to store multiple pieces of data. Each variable should be given a unique name so that it can be easily referenced and identified when needed.

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Based on the information, it can be inferred that Nata has to work 7 days a week.

To calculate how many days Nata works we must perform the following mathematical operation.

We must find how many days in total Ana and Mary work together, that is, we must add the days that they work.

Based on the above, we can say that Ana and Mary work 7 days a week, so Ana could work Monday, Tuesday and Wednesday and Mary could work Thursday, Friday, Saturday and Sunday.

So, Nata will have to accompany both of them every day, so she will have to work 7 days.

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The questionnaire given to patients about their quality of life before and after the surgery included in the research design does add risks to the overall study design. Hence, the given statement is false.

Even though the questionnaire itself may not cause any additional physical risks, it can still have an impact on the overall study design. The questionnaire can add additional burden to the participants, who may already be dealing with physical and emotional stress from the surgery. Also, the questionnaire results may not be accurate if participants are not fully honest or unable to understand the questions being asked. Therefore, the inclusion of such a questionnaire in the research design should be carefully considered after evaluating its potential impact on the overall study.

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A system of equations that could be used to find the number of adults and the number of children in the theatre is:

x + y = 362

y = x + 140

In order to write a system of linear equations to describe this situation, we would assign variables to the number of adults and number of cats respectively, and then translate the word problem into algebraic equation as follows:

Since the total number of adults and children at this theatre is 362, a linear equation which represents this situation is given by;

x + y = 362

Additionally, there are 140 more adults than children;

y = x + 140

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If heights of a species of plant are approximately Normally distributed, have a mean of 9.31 cm, and standard deviation of 0.55 cm , If 20 of the plants are randomly selected , then the probability that the mean plant height is less than 9.5 cm is 0.9388  .

We can use the Z-score formula to find the probability that a randomly selected plant has a height less than 9.5 cm:

Z  = (x - mean)/(standard deviation/sqrt(sample size))  ;

Where x is the value we want to find the probability for, in this case x = 9.5

So , Z = (9.5 - 9.31)/(0.55/√20) = (0.19)/(0.123) = 1.54

Now we can use the standard normal table to find the probability that a Z-score is less than 1.54  .

From the z table ; we get , the p value is 0.9388  .

Therefore , The probability that a randomly selected plant from this species has a height less than 9.5 cm is about 0.9388  .

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On solving the equations x + y = 104 and 0.05x + 0.25y = 22, the number of nickels and quarters Lenny has is 20 and 84 respectively.

What is an equation?

A mathematical definition of an equation is a claim that two expressions are equal when they are joined by the equals sign ("=").

The value of 1 nickel is = $0.05

The value of 1 quarter is = $0.25

The total amount of coins is = $22.0

The total number of coins is = 104

Let the number of nickel be = x

Let the number of quarter be = y

The equation for total number of coins =

x + y = 104 ....(1)

The equation for total amount of coins =

0.05x + 0.25y = 22 ...(2)

Multiply equation (1) by 0.05 -

0.05(x + y) = 104 × 0.05

0.05x + 0.05y = 5.2 ... (3)

Subtracting equation (3) from (2) -

0.05x + 0.25y - (0.05x + 0.05y) = 22 - 5.2

0.05x + 0.25y - 0.05x - 0.05y = 22 - 5.2

0.20y = 16.8

y = 84

Substitute the value of y in equation (1) -

x + 84 = 104

x = 104 - 84

x = 20

Therefore, the value for x and y is obtained as x = 20 and y = 84.

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The distance that exists between star A and B is given as 29.87 light years

We can solve this by creating a triangle where Point a Is star a, point b is star b and the point c is earth

the angle is given as 60 degrees

AC = 22 light years

BC = 34 light years

ab² = ac² + bc² - 2 *ac * bc* cos60

we would have to put in the values

ab² = 22² + 34² - 2 * 34 * 22 * cos 60

ab² = 484 + 1156 - 748

ab² = 892

square both sides

ab = 29.87 light years

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Answer:4.5

Step-by-step explanation:

a) The probability that a wash-down motor will have to be replaced free of charge - 1,904  

b) A percentage of wash-down motors can be expected to operate at 2,28 %

c)  The average hours of operation before failure would be 13500 h

Normal Distribution

μ₀  = 14000    hours   and        σ  =  1050    hours

a) Probability of X  (replacement of motor free of charge)

Z = ( X - μ₀ ) /  σ

Z  =  ( 12000- 14000) / 1050

Z  =  - 2000/1050

Z = - 1,904    

From z Table we get

for  z = -1.904       P [ X ≤ 12000 ]  = 0.0262

P [ X ≤ 12000 ]  = 0.0262      or      2.62 %

b) What % of motors can be expected to operate more than 16100 hours

z = ( 16100 - 14000) / 1050

z = 2100/1050     ⇒   2

From the z table, we find with z = 2 total number of f motors operating up to 16100 hours

From the z table, we find 0,9772

Then the probability of motors operating for more than 16100 hours is:

P [ X > 16100]  =  1  -  0.9772    =  0,0228

Then

P [ X > 16100]  = 0,0228    or   2,28 %

c) The average hours of operation before failures if only 1% of motors would be replaced free of charge is:

1%  = 0,01    Probability then z score is  z = 0,4960

0,4960  =  ( 14000 - X ) / 1050

0,4960 * 1050 = ( 14000 - X )

520,8 - 14000 =  - X

X = 13479,2 hours  as is not common such offer company could set up a guarantee for  13500 h

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Answer:

672 in.³

Step-by-step explanation:

V = LWH

V = 8 in. × 12 in. × 7 in.

V = 672 in.³

Answer: D 672

Step-by-step explanation: The formula for volume is L x W x H = V so if you plug all those numbers in and multiply u get 672

The probability of getting orange 21% more in that large bag is 0.

Probability simply means the chance that a particular thing or event will happen. It is the occurence of likely events. It is simply the area of mathematics that deals with the numerical estimates of the chance that an event will occur or that a particular statement is true.

In this situation, the company claims that in the large bag of M&M's 20% of them were orange. Since the oranges are 20%, the probability of getting orange 21% more in that large bag is not possible. This will be zero.

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As per the compound interest, the lost annual income from savings on those refunds is $21,555,000

In math the term called compound interest is defined as a type of interest that not only accrues on the principal but also on the accrued interest and the interest rates on loans are quoted in terms of annual percentage rate, which does not account for the compounding of interest, and therefore understates the true cost of borrowing.

Here we know that the 479,000 people each receive an average refund of $4,500, based on an interest rate of 1 percent.

Then the lost annual income from savings on those refunds is calculated as,

As we all know that the lost annual income is the amount of interest these people could have earned on their income, in which  the extra tax had not been deducted.

Then With $4,500 each person,

Then the total amount of the deducted income is calculated as,

=> 479,000 x 4,500 = 2,155,500,000.

Here we have the  1% interest rate, then the net return on these fund if they were saved

=> 2,155,500,000 x 1% = 21,555,000.

Therefore the total lost annual income is 21,555,000

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Answer:

Step-by-step explanation:

Answer:

angle 1 = 118 deg

angle 2 = 62 deg

Step-by-step explanation:

To find angle one you must know that angles 1 and 118 deg are alternate angles. Therefore since these lines are parallel, angles 1 and 118 are equal.

To find angle 2 to you must know that angle 1 + angle 2 = 180.

angles on a straight line always equal 180 deg

therefore:

180

118 -

62 deg

hope this helps :)

Answer:

m<1 = 118°

m<2 = 62°

Step-by-step explanation:

The horizontal lines are parallel.

Angles 1 and 118° are alternate interior angles of parallel lines cut by a transversal, so they are congruent.

m<1 = m<2 = 118°

Angles 1 and 2 are a linear pair, so they are supplementary. That means that their measures add to 180°.

m<1 + m<2 = 180°

118° + m<2 = 180°

m<2 = 62°

i. ∃x ∈ Z, x² = 2x, ii. ∀y ∈ school buses, y is not purple and iii. ∀x ∈ R, if x is even, then x² is even.

i. The equivalent predicate statement for the first statement is:

∃x ∈ Z, x² = 2x

The symbol "∃" means "there exists", the symbol "∈" means "is an element of", and the symbol "Z" represents the set of integers.

ii. The equivalent predicate statement for the second statement is:

∀x, (x is a school bus) → (x is not purple)

The symbol "∀" means "for all", and the symbol "→" means "implies".

iii. The equivalent predicate statement for the third statement is:

∀x ∈ R, (x is even) → (x² is even)

The symbol "∈" means "is an element of", and the symbol "R" represents the set of real numbers.

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Answer:$742; $38,584

Step-by-step explanation: 14% of $5,300 is $742 (0.14 x 5300). There are 52 weeks in a year so $742 x 52 weeks = $38,584.