What it means when a function and its inverse is the same?

The first ferry's speed is 18.5 mph and the second ferry, which is slower, travels at 13.5 mph.

Let x represent the first ferry's speed.

The second, slower ferry's speed is then x-5 (since 5 miles per hour slower)

First ferry time equals distance/speed equals

50/x

Time spent on the second ferry:

50 / x - 5

The difference in times is 1 hour because the second ferry departs an hour earlier.

50 / x - 5 - 50/x = 1

50x - 50(x - 5) = x (x - 5 )

250 = x² - 5x

x² - 5x - 250 = 0

(x - 10 )(x + 5) = 0

x = 18.5 - 13.5

Speed can never be detrimental.

Therefore, the first ferry's speed is 18.5 mph.

and the second ferry, which is slower, travels at 13.5 mph.

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The Additive inverse of 3/2 is -2/3.

Additive Reciprocal

An additive reciprocal is a number that is added to a given number to bring the sum to zero. For example, adding -3 to the number 3 will result in zero. So the additive reciprocal of 3 is -3. In our daily life, we come across situations where we cancel the value of a quantity by forming its additive reciprocal.

Additive Reciprocal in Fraction:

The additive reciprocal of the fraction a/b is -a/b and vice versa. This is because a/b + (-a/b) = 0. The additive reciprocal of a positive fraction is the same fraction with a negative sign, but for negative fractions the additive reciprocal is the same fraction without a negative sign.

Given, the number is 2/3

We have to find the additive inverse of the given number.

In mathematics, the additive inverse of a number ‘a’ is the number that, when added to ‘a’, yields zero.

So, 2/3 + (-2/3)= 0

Therefore, the additive inverse of 2/3 is -2/3.

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Answer:

Step-by-step explanation

The triangle base would be 6

There are 5/12 strawberries were left. This can be solved using the concept of fraction addition.

A fraction is a piece of the entire. The number is stated in arithmetic as a quotient, which signifies the numerator divided by the denominator. Both are integers in a straightforward fraction. The numerator or denominator of a complex fraction is a fraction. A suitable fraction has a numerator that is smaller than its denominator.

Three main fractional kinds exist. They come in three varieties: appropriate fractions, incorrect fractions, and mixed fractions. The numerator and denominator words are known as fractions. We define its kinds in light of these two words.

Given that,

A farmer took 2/3 of the strawberry that he harvested to a market.

At the market, the farmer sold 1/4th of the strawberry.

Now, the farmer has left no of strawberries were:

= (2/3) - (1/4)

= (8 - 3) / 12

= 5 / 12

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The measure of  angle B is m ∠B = 107°.

What is similar triangles?

Triangles that are similar in shape but differ in size are known as similar triangles. Examples of related objects include all equilateral triangles and squares with any side length. To put it another way, if two triangles are similar, then their corresponding angles and sides are congruent and equal in size.

Given:   Δ NPZ ~ Δ HBZ

As the triangles NPZ and triangle HBZ are similar that means the corresponding angles are congruent.

⇒ ∠N = ∠H,  ∠P = ∠B,  ∠Z = ∠Z

Since,

∠P = (8x - 37)°,  ∠Z = 42°,  ∠H = (2x - 5)°

⇒ 2x - 5 = ∠H,  ∠B = 8x - 37,  Z = 42°

Since, the sum of three angles of triangle is 180°.

⇒              ∠H + ∠B + ∠Z = 180°

(2x - 5)° + (8x - 37)° + 42° = 180°

                10x - 42° + 42° = 180°

                                  10x = 180°

                                     x = 18°

So,

∠B = (8x - 37)° = 8(18) - 37 = 144 - 37 = 107°.

Hence, the measure of  angle B is m ∠B = 107°.

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According to the solving by  using substitution method the  x and y are x = 6/5 and y = -8/5 respectively.

The algebraic approach to solving simultaneous linear equations is known as substitution method. The value of one variable through one equation is substituted in the second equation in this procedure, as the name implies.

given equations are 2x+4y = 4 and -2x+ y = -4

Now, simplifying -2x+y = -4

-2x+y = -4

∴ y = -4 + 2x

i.e. y = 2x - 4

By substitution method,

y = 2x - 4 in other given equation 2x+4y = 4,

 2x+4y = 4

 2x+4(2x - 4) = 4

∴ 2x + 8x - 8 = 4

∴ 10x = 4 + 8

∴10x = 12

∴ x = 12/10

∴ x = 6/5

Now ,

substituting x = 6/5 in y = 2x - 4,

 y = 2(6/5) - 4

∴ y = - 8/5

∴ x = 6/5  &  y = - 8/5.

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An inequality to find how many days he will have to go kayaking. is; d ≥ 4

We are told that Jay wants to go paddle boarding at least 8 hours each week. This means greater than or equal to 8. That is the minimum number of hours each week is 8 hours.

Now, we are told that he averages 2 hours per day. If the number of days is given by d, then we have the inequality as;

2d ≥ 8

Divide both sides by 2 to get;

d ≥ 4

That is the domain of the number of days required to go kayaking

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The solution to the system of equations is x = 4 and y = 5.

3x + 2y = 7

y = 3x - 11

Substitute 3x - 11 for y in the first equation:

3x + 2(3x - 11) = 7

Simplify:

3x + 6x - 22 = 7

Combine like terms:

9x - 22 = 7

Add 22 to both sides:

9x = 29

Divide both sides by 9:

x = 29/9

x = 4

Change x in the second equation to 4:

y = 3(4) - 11

Simplify:

y = 12 - 11

y = 1

The solution to the system of equations is x = 4 and y = 1.

The solution to the system of equations 3x + 2y = 7 and y = 3x - 11 is x = 4 and y = 1. To solve this system, we first substituted 3x - 11 for y in the first equation. We then simplified the equation and combined like terms to get 9x - 22 = 7. We then added 22 to both sides, divided both sides by 9, and got x = 29/9 which is x = 4. We then substituted 4 for x in the second equation, simplified, and got y = 1. Thus, the solution to the system of equations is x = 4 and y = 1.

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The number of students who will not be able to ride on the bus for the school trip is given by the equation A = 15 students

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the equation be represented as A

Now , the value of A is

The total number of students for the trip = 235 students

The number of buses = 5 buses

The number of students in a bus be = B

Now , each seat on the bus can occupy = 2 students

So , 22 seats can occupy = 22 x 2 students

22 seats in the bus can occupy = 44 students

So , the number of students in a bus B = 44 students

And , the total number of students in 5 buses = 5 x number of students in a bus B

Substituting the values in the equation , we get

The total number of students in 5 buses = 5 x 44

The total number of students in 5 buses = 220 students

So , the remaining number of students A = total number of students for the trip - total number of students in 5 buses

Substituting the values in the equation , we get

The remaining number of students A = 235 - 220

The remaining number of students A = 15 students

Therefore , the value of A is 15 students

Hence , the number of students is 15 students

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Answer:   [tex]f^{-1}(\text{x}) = \frac{2-\text{x}}{3}[/tex]

Explanation:

Swap x and y, then solve for y to get the inverse.

[tex]f(\text{x}) = -3\text{x}+2\\\\\text{y} = -3\text{x}+2\\\\\text{x} = -3\text{y}+2\\\\\text{x}-2 = -3\text{y}\\\\\text{y} = \frac{\text{x}-2}{-3}\\\\\text{y} = \frac{2-\text{x}}{3}\\\\f^{-1}(\text{x}) = \frac{2-\text{x}}{3}\\\\[/tex]

The value of AB= 24 BD is congruent to BC. BD=BC

BD = 5x – 26, BC = 2x + 1, and AC = 43

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The inequalities that represent the situation are 15x + 10y ≥ 400 and x + y ≤ 35 and the viable solutions are (10, 25), (20, 12), (35, 0) and (20, 20)

From the question, we have the following parameters that can be used in our computation:

These parameters above mean that

15x + 10y = Total earnings

x + y = Number of hours

So, we have

15x + 10y ≥ 400

x + y ≤ 35

The above represent the inequalities of the situation

In (a), we have

15x + 10y ≥ 400

x + y ≤ 35

Next, we test the options

(10, 25)

15 * 10 + 10 * 25 ≥ 400 ⇒ 400 ≥ 400

10 + 25 ≤ 35 ⇒ 35 ≤ 35

True

(10, 20)

15 * 10 + 10 * 20 ≥ 400 ⇒ 350 ≥ 400

10 + 20 ≤ 35 ⇒ 30 ≤ 35

False

(20, 12)

15 * 20 + 10 * 12 ≥ 400 ⇒ 420 ≥ 400

20 + 12 ≤ 35 ⇒ 32 ≤ 35

True

(35, 0)

15 * 35 + 10 * 0 ≥ 400 ⇒ 525 ≥ 400

35 + 0 ≤ 35 ⇒ 35 ≤ 35

True

(20, 20)

15 * 20 + 10 * 20 ≥ 400 ⇒ 500 ≥ 400

20 + 20 ≤ 35 ⇒ 40 ≤ 35

False

Hence, the viable solutions are (10, 25), (20, 12), (35, 0) and (20, 20)

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Answer:yes

Step-by-step explanation:

yes

Answer:

Starbursts = 12 dollars

M&M = 2 dollars

1. Make Startbursts and M&Ms into "x"

(a) 1x + 2x = 12

2. 3*1= 3

4+2x=12

(b) 4 packs

The perimeter of ΔAEB is 18.5 feet.

A triangle's other two sides are divided in the same proportion if a line drawn parallel to either one of its sides intersects the other two sides at two different spots.

As,  side AB║DF hence we will use Triangle Proportionality theorem to calculate the length of side AE.

Since, AE/AD = EB/ BF

AE/2 = 7.2/8.4

AE = 6.5 feet

So, the perimeter of ΔAEB is

= AE + EB + BE

=6.5+4.2+7.8

=18.5 ft.  

Hence, the perimeter  is 18.5 feet.

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The distance between two points A and B is 5 units.

Pyramids have been a symbol of great architecture inventions of the mankind. Egyptian People were the first once to design pyramids on a very large level.

Let a be the Side of Area of base of Pyramid . The base of a pyramid is a square shaped quadrilateral

The Volume of the Pyramid with base area   and height  is given by the equation (1)

[tex]Volume = (1/3)Ah....(1)[/tex]

[tex]A = a^{2}[/tex]

[tex]a = Side[/tex]

A = [tex]a^{2}[/tex] = [tex]44cm^{2}[/tex]

[tex]h = 12.6[/tex]

Putting these values in equation (1) we get

[tex]Volume = (1/3)X44X12.6[/tex]

[tex]Volume[/tex] [tex]= 184.8cm^{3}[/tex]

Hence , the volume of the pyramid is 184.8cm³

a) The flowchart for the congruency of triangles is given below.

b) The triangles are congruent as DQ ≅ XZ, PQ ≅ XY, and ∠PQD ≅ ∠ZXY.

What is congruency?

The word 'congruent' means 'exactly equal' in terms of shape and size. Even when we turn, flip, or rotate the shapes, they remain equal.

In the given diagram, it can be seen that -

Line segment DQ = Line segment XZ.

Similarly, Line segment PQ = Line segment XY.

Since, the two sides of the triangle are equal, hence the third side must also be equal.

Line segment PD = Line segment ZY.

Also, in the diagram it can be seen that ∠PQD ≅ ∠ZXY.

Therefore, the triangles PDQ and ZXY are equal and congruent.

The flowchart for the same is given below.

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Answer:

y =x+3

y =-x²+[?]

Find the gradient;

What is the formula for the gradient of a graph?

Finding the gradient of a straight-line graph

X2 = 2

Y2 = 7

ΔX = 4

ΔY = 3

θ = 36.869897645844°

Equation of the line:

y = 0.75x + 5.5

When x=0, y = 5.5

When y=0, x = -7.3333333333333

OR

X2 = -6

Y2 = 1

ΔX = -4

ΔY = -3

θ = 216.86989764584°

Equation of the line:

y = 0.75x + 5.5

When x=0, y = 5.5

When y=0, x = -7.3333333333333

On solving the provided question, we can say that the mean that 6 tests including the last test brian took.

A dataset's mean is the sum of all values divided by the total number of values, often known as the arithmetic mean (as opposed to the geometric mean). Often referred to as the "mean," this is the most often used measure of central tendency. Simply dividing the dataset's total number of values by the sum of all of those values yields this result. Both raw data and data that have been combined into frequency tables can be used for calculations. Average refers to a number's average. It is straightforward to calculate: Divide by how many digits there are after adding up all the digits. the total divided by the count.

(91n + 98) / (n + 1)  =  92

    91n + 98  =  92(n + 1)

 91n + 98  =  92n + 92

 98  =  n + 92

6  =  n

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[tex]\cfrac{6a^2 + 21a}{3a^2}\implies \cfrac{6a^2 }{3a^2}+\cfrac{21a}{3a^2}\implies 2+\cfrac{7}{a}[/tex]

The fractions of the numbers are found as follows:

To divide a number by the nth power of 10, the decimal digit in the number is moved n units to the right.

The fraction that represents 10% of a number is given as follows:

10/100 = 1/10.

Meaning that the number is divided by 10, which is the first power of 10, and thus the decimal digit is moved one unit right in the number.

The fraction that represents 1% of a number is given as follows:

1/100.

Meaning that the number is divided by 100, which is the second power of 10, and thus the decimal digit is moved two units right in the number.

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Answer:

66

Step-by-step explanation:

a^5

Answer:

16 929

Step-by-step explanation:

a1 = 22      d = 11     a 54  = 605

Sum = 54 ( 22 + 605) /2 = 16 929  

In an amphitheater, seat are arranged in 50 semi-circular row-facing a dome Tate. The first row contain 23 seat, and each row contain 4 more eat than the previous row. Therefore, there are 6050 seat are in the amphitheater.

Given that:

There are total 50 semi circular row.

In the first row there are 23 seats.

The given problem is AP series:

Arithmetic Series:

An arithmetic sequence is defined as an algebraic sequence in which each successive term has an equal difference. It is obtained by adding a constant to any preceding term. For the first term 'a' and the tolerance 'd' of the AP, here is a list of commonly used arithmetic progressions to solve various AP related problems:

where l is the last term in the arithmetic progression.

Now,

Sum of n terms  = n/2 [ 2a + (n-1)d]

                          = 50/2 [ 2× 23 + (50 -1)4]

                          = 25 [46 + 49×4]

                          = 25 × 242

                          = 6050

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The height of the tree is closest to 40 feet.

We can use the proportion of similar triangles to determine the height of the tree.

Let h be the height of the tree and s be the length of the shadow cast by the tree.

We know that the height of the building is 60 feet and the length of its shadow is 21.5 feet. So:

(h/s) = (60/21.5)

We also know that the length of the shadow cast by the tree is 18 feet.

So we can substitute that into the proportion and solve for h:

h = (60/21.5) * 18

h = (60*18)/21.5

h ≈ 40

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Answer:

16/11

Step-by-step explanation:

Let x = 1.45454...

Multiply both sides by 100:

100x = 145.454...

Subtract both sides by x:

100x - x = 145.454... - x

100x - x = 145.454... - 1.45454...

99x = 144

x = 144/99

The term can be simplified in lowest as 16/11. Hence, 1.45454... can be written as the ratio of two integers of 16/11

Answer:

a₄ = - 2

Step-by-step explanation:

to calculate a term in the sequence subtract 5 from the previous term

give first term a₁  = 13 , then

a₂ = a₁ - 5 = 13 - 5 = 8

a₃ = a₂ - 5 = 8 - 5 = 3

a₄ = a₃ - 5 = 3 - 5 = - 2

Answer:

Hey there! So the square root of 38 can be simplified by breaking it down into simpler parts. We can use a technique called prime factorization. We need to find the largest perfect square that is a factor of 38 and extract it from the square root. So, we can express 38 as 219 and that means we have 2^2=4 as the largest perfect square. Therefore, we can simplify the square root of 38 as 2sqrt(19).

It can't be simplified anymore, but it gives us a more manageable form. Let me know if you have any more questions, I'm happy to help!

________________________________________________________

A perfect square is a number that can be written in the form of n^2 (n raised to the power of 2) for some whole number n. For example 4, 9, 16, 25 and 36 are perfect squares. A way to determine if a number is a perfect square is by taking the square root of that number, if the result is a whole number it's a perfect square. Also, you can check it by finding the prime factorization of the number, if each of the prime factors appears in even power, it's a perfect square.

There are a few techniques that can make simplifying square roots quicker and easier:

Recognizing perfect squares (as described above) and extracting them from the square root

Using prime factorization to simplify the square root

Recognizing when the square root can be written as a rational number (a fraction of two integers)

Another way to approach is to factorise the radicand and combine any pairs of identical factors out from the square root.

Please let me know if you have any other questions on this topic or any other topic I could help with.

√38 is the simplest representation of the square root of 38.

The square root of a number is a value that, when multiplied by itself, gives the original number.

In the case of √38, we are looking for a number that, when multiplied by itself, equals 38.

Since 6 multiplied by itself is 36 and 7 multiplied by itself is 49, we can conclude that the square root of 38 lies between 6 and 7.

We cannot find a whole number that satisfies the equation √38 = x, where x is an integer.

Therefore, we express the square root of 38 as an irrational number, meaning it cannot be expressed as a simple fraction or a terminating or repeating decimal.

Hence, √38 is the simplest representation of the square root of 38.

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Answer:

y-5=-1/7(x-4)

Step-by-step explanation:

The equation of the line in point-slope form is y-5=-1/7(x-4)

Answer:

y = -1/7 x + 39/7

Step-by-step explanation:

An equation in point slope form is

y = mx+b  where m is the slope and b is the y intercept

We are given the slope

y = -1/7 x + b

Substitute in the point for x and y and solve for b

5 = -1/7(4) +b

5 = -4/7 +b

5 + 4/7 = b

5*7/7 + 4/7 = b

35/7 + 4/7 = b

39/7 = b

The equation is

y = -1/7 x + 39/7

From the given graph, at Point B a total of 8000 fluffernutters & 500 blank books could be produced.

The study of graphs, which are mathematical constructions used to represent pairwise relationships between things, is known as graph theory in mathematics. In this context, a graph is made up of vertices connected by edges.

In discrete mathematics, a graph is made up of vertices—a collection of points—and edges—the lines connecting those vertices. In addition to linked and disconnected graphs, weighted graphs, bipartite graphs, directed and undirected graphs, and simple graphs, there are many other forms of graphs.

Straight line graphs called linear graphs are used to show the relationship between two quantities. This graph makes it easier to show a result as a collection of straight lines. The term "linear" simply means a straight line; curves, dots, bars, etc. are not used.

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Answer:

Step-by-step explanation:

1) 1,15,29,43,57,.....

here, 15-1 =14 and 29-15=14

therefore,it is an arithmetic sequence with common difference = 14

2) 3,6,9,15,17,....

here, 6-3=3, 9-6=3 , 15-9=6

therefore,it is not an arithmetic sequence.

3) 93,86,79,72,65,......

here, 86-93=-7 , 79-86=-7 and 72-79=-7

therefore,it is an arithmetic sequence with common difference = -7

4) 37,34,31,29,26,......

here, 34-37 = -3,31 - 34 = -3,29 - 31 =-2

therefore,it is not an arithmetic sequence.