What is the meaning of "[tex]x=a^{-1}b[/tex] is a solution since [tex]aa^{-1}b=1b=b[/tex]?

For the given functions f(x) and g(x), the inverse functions of f and g are:

a. f(g(x)) = x; g(f(x))  = x

ƒ and g are inverses of each other.

b. (g(x)) = x; g(f(x))  = x

ƒ and g are inverses of each other

The inverse functions of the functions f(x) and g(x),are determined as follows;

a. f(x) = 2x, g(x) = x/2

f(g(x)) = f(x/2)

f(g(x)) = 2(x/2)

f(g(x)) = x

g(f(x)) = g(2x)

g(f(x))  = (2x)/2

g(f(x))  = x

Since f(g(x)) = g(f(x)) = x, ƒ and g are inverses of each other.

b. f(x) = 2x + 1, g(x) = (x -1)/2

f(g(x)) = f((x-1)/2)

f(g(x)) = 2((x-1)/2) + 1

f(g(x)) = x - 1 + 1

f(g(x)) = x

g(f(x)) = g(2x + 1)

g(f(x)) = ((2x + 1) - 1)/2

g(f(x)) = 2x/2

g(f(x)) = x

Since f(g(x)) = g(f(x)) = x, ƒ and g are inverses of each other.

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Step-by-step explanation:

All you would need to do is to use the circumference of a circle equation, C = 2pi(r) or C=pi(d). Since we are given the radius we will use C= 2pi(r).
C=2pi(4.4)

C=8.8pi inches

Answer:

n(EnS)= 10

Step-by-step explanation:

n(U)= 55

n(EnS)= x

n"(EuS)"= 2x

n(EuS)= 35

We have,

n"(EuS)"= n(U) - n(EuS)

=55-35

=20

n(EnS) /n"(EuS)" = 1/2

x/20 = 1/2

2x=20

x=20/2

x=10

Therefore, n(EnS) = 10

Answer:

To find ƒ⁻¹(x), we need to solve for x in terms of ƒ⁻¹(x).

So, we start with the equation:

x = 2 (ƒ⁻¹(x) - 1)

We isolate ƒ⁻¹(x) by dividing both sides by 2 and adding 1 to both sides:

ƒ⁻¹(x) = (x/2) + 1

Therefore, the inverse of the given function f(x) is:

ƒ⁻¹(x) = (x/2) + 1

Answer:

4√3

Step-by-step explanation:

length > width

If width = y

length =4y

Length × width = area

y × 4y = 192

4 y^2 = 192

y^2 = 48

y = 4 √3

Answer:

pi/6

Step-by-step explanation:

the full arc of the circle is the circumference = 2pi×r.

since r = 1, we get circumference = 2pi.

the full arc of 2pi is for 360° or 2pi radians (hence the definition).

an angle given in radians for the norm circle with radius = 1 IS therefore already the arc length for that angle.

for 2pi the arc length is 2pi.

for pi/6 the arc length is ... pi/6.

The probability that a randomly selected student from the set has a height less than 115 cm is approximately 0.04, or 4%

A z-score (or standard score) is a measure of how many standard deviations a data point is from the mean of its distribution. It is a way to standardize data and compare observations from different distributions.

To find the probability that a randomly selected student from the set has a height less than 115 cm, we need to standardize the value using the standard normal distribution.

First, we calculate the z-score:

z = (115 - 150) / 20

  = -1.75

Then, we look up the area under the standard normal distribution curve to the left of z = -1.75 using a table or a calculator. The table or calculator gives us the value 0.0401.

Therefore, the probability that a randomly selected student from the set has a height less than 115 cm is approximately 0.04, or 4%, rounded to two decimal places:

P(X < 115) = 0.04

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A set of middle school student heights are normally distributed with a mean of 150150150 centimeters and a standard deviation of 202020 centimeters. Let XXX represent the height of a randomly selected student from this set. Find P(X<115)P(X<115)P, left parenthesis, X, is less than, 115, right parenthesis. You may round your answer to two decimal places.

Answer: To use Excel's NORM.INV() function to solve this problem, we can follow these steps:

Determine the z-score corresponding to the top 10% of the distribution. We can use the NORM.INV() function to find this value. Since we want the top 10%, we'll use a probability of 0.9 and a mean of 330lbs and a standard deviation of 15lbs. In Excel, we can use the formula:

=NORM.INV(0.9,330,15,TRUE)

This gives us a z-score of 1.281552.

Once we have the z-score, we can use the formula for a normal distribution to find the corresponding weight. The formula is:

z = (x - μ) / σ

where z is the z-score, x is the weight we're trying to find, μ is the mean weight (330lbs), and σ is the standard deviation (15lbs).

Rearranging this formula to solve for x, we get:

x = z * σ + μ

Substituting in the values we know, we get:

x = 1.281552 * 15 + 330

This gives us a weight of approximately 349lbs.

Therefore, the minimum weight of the sumo wrestlers at the highest weight of the league is about 349lbs, rounded to the nearest whole number.

Step-by-step explanation:

Dilation is a transformation that changes the size of an object while keeping its shape the same.

In this case, the scale factor of 3 tells us that the triangle will be three times larger than the original triangle.

To perform the dilation, you would take each vertex of the triangle and multiply the x-coordinate and y-coordinate by 3. For example, if one vertex of the original triangle had coordinates (x, y), the coordinates of that same vertex in the dilated triangle would be (3x, 3y).

It's also worth noting that the dilation is done with respect to a fixed point called the center of dilation. You can think of this point as the "pivot" around which the dilation occurs.

The center of dilation is typically represented by the letter "O".

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a. f(x) = x³

b. g(x) = -f(x-2)

Answer:

  a) parent: f(x) = x³

  b) g(x) = -f(x -2)

Step-by-step explanation:

You want the parent function and the given function written in terms of the parent function for g(x) = -(x -2)³.

The parent function is the function that remains after scale factors and translations are removed. In general, a transformed function will look like ...

  g(x) = c·f((x-a)/b) +d

which scales the function f(x) horizontally by a factor of b, vertically by a factor of c, and translates the result by (a, d).

Here, we recognize that ...

  g(x) = -(x -2)³

has c = -1, a = 2, b = 1, d = 0

and the parent function is ...

  f(x) = x³

Using the values for a, b, c, d that we recognized above, we have ...

  g(x) = -f(x -2)

__

Additional comment

When scale factors are negative, they reflect the function over the relevant axis. Here, the negative vertical scale factor reflects the function vertically over the x-axis. The translation is 2 units to the right.

Answer:

Step-by-step explanation:

50$ (Sorry for the mistake, it’s a little hard for me to give such assignments, if the translator would translate correctly, then there would be no mistake, sorry again)

Answer:

Step-by-step explanation:

A tire is sold at a discount of 10%. It is sold for $45. Find the usual price of the tire.

from the question you understand that $45 is 90% of the original price (100%-10%=90%), so we divide 45 by 90 and multiply by 100 and we  find the original price

45 : 90 x 100 =

0.5 x 100 =

50$

----------------------------

50$ - 10% = 45

When the probability of an event is high, we can say that the likelihood of that event occurring is also high. Conversely, when the probability of an event is low, the likelihood of that event occurring is also low.

What is probability?

In mathematics, probability is a measure of the likelihood that an event will occur. It is a way of quantifying the uncertainty or randomness associated with an event or set of events. The probability of an event is expressed as a number between 0 and 1, with 0 indicating that the event is impossible and 1 indicating that the event is certain to occur. For example, the probability of flipping a fair coin and getting heads is 0.5, or 50%, since there are two equally likely outcomes (heads or tails) and getting heads is one of them. Probability theory is a fundamental branch of mathematics that has applications in many areas, including statistics, finance, engineering, and computer science, among others.

Here,

The likelihood of selecting a green ribbon from Emilie's drawer is high, since the probability of randomly selecting a green ribbon is 84%. This means that out of all the ribbons in the drawer, 84% of them are green.

Another way to interpret this is that if Emilie were to randomly select a ribbon from her drawer without looking, there is an 84% chance that the ribbon she selects will be green.

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Correct question is "Emilie has a drawer full of color for ribbons the probability of randomly selecting a green ribbon is 84% which describes the likelihood by the example of selecting a green ribbon".

Answer:

Step-by-step explanation:

In a parallelogram, opposite sides are parallel and equal.

     MA // HT  & AH is transversal.

        ∠MAG ≅ ∠GHT        {Alternate interior angles are equal}    --Angle

     MA = HT   {MATH is a parallelogram}   ---> Side

    MA // HT & MT is transversal.

         ∠AMG ≅ ∠HTG    {Alternate interior angles area equal} ---->Angle

ΔMGA ≅ ΔTGH   { Angle side angle congruent}

 AG = HG           {CPCT}

MT bisects AH

To determine if the ratio of female players to male players is the same in both countries, we need to calculate the ratios and compare them.

For the first country,

let female players be f1,male players be m1

the ratio of female players to male players is:f1:m1 (i.e f1/m1)

11,200 / 14,000 = 0.8

For the second country,

let female players be f2,male players be m2

the ratio of female players to male players is:f2:m2 (i.e f2/m2)

10,400 / 13,000 = 0.8

As we can see, both ratios are the same, which means that the ratio of female players to male players is the same in both countries. Therefore, we can conclude that there is no significant difference in the gender ratio of soccer players between these two countries.

Answer:

The female-to-male ratios for both countries are the same (0.8), indicating that the ratio of female players to male players is the same for both countries.

Explanation:

To determine whether the ratio of female players to male players is the same for both countries, we need to calculate the ratio of female players to male players for each country and compare them.

Country 1: Female-to-male ratio

= 11,200 / 14,000

= 0.8

Country 2: Female-to-male ratio

= 10,400 / 13,000

= 0.8

Answer: 1/7

Step-by-step explanation:

d d x ( ∑ n = 0 ∞ c n ( x − a ) n ) = c 1 + 2 c 2 ( a − a ) + 3 c 3 ( a − a ) 2 + ⋯ = c 1/7

a) The amount of the insurance or the sum assured by Julia Carson on his brick home is $80,000.

b) The annual premium is $5,148.00.

The sum assured represents the fixed amount that the insurance company pays to Julio Carson (the policyholder) if the unpredictable event, such as fire, occurs and damages his brick home.

The purpose of the payment is to reimburse Julio not for the costs incurred but according to the risks undertaken and it is a fixed sum of money unlike the sum insured, which reimburses to cover the costs incurred.

Replacement value of the brick home = $100,000

Sum insured = 80%

= $80,000 ($100,000 x 80%)

Monthly premium = $429

Number of insured years = 10 years

Annual premium = $5,148 ($429 x 12)

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[tex]U(\stackrel{x_1}{4}~,~\stackrel{y_1}{-2})\qquad N(\stackrel{x_2}{1}~,~\stackrel{y_2}{2}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{2}-\stackrel{y1}{(-2)}}}{\underset{\textit{\large run}} {\underset{x_2}{1}-\underset{x_1}{4}}} \implies \cfrac{2 +2}{-3} \implies \cfrac{ 4 }{ -3 } \implies - \cfrac{4 }{ 3 }[/tex]

The quotient is 0.9x² + 0.17x - 0.259, and the remainder is 0.181.

Synthetic division is a simplified method of dividing a polynomial by a linear factor of the form (x-a). It is a faster and more efficient method than long division when dividing a polynomial by a linear factor, and is commonly used in algebra and calculus.

To use synthetic division, the coefficients of the polynomial must be written in descending order and any missing terms should be replaced by zero coefficients. The divisor must be a linear factor of the form (x-a), where 'a' is a constant.

Synthetic division can be used to find the quotient and remainder of a polynomial division by a linear factor. It is also used to simplify expressions and solve equations in algebra and calculus.

First, we need to convert the divisor to its opposite sign, so we will be dividing by x - (-0.7) = x + 0.7

The coefficients of the dividend are 0.9, 0.8, 0, 0, so we will set up our synthetic division table as follows:

-0.7 | 0.9 0.8 0 0

      | -0.63 -0.259 0.181

      |____________________

         0.9 0.17 -0.259 0.181

The quotient is 0.9x² + 0.17x - 0.259, and the remainder is 0.181.

Therefore, 0.9x³ + 0.8x = (x + 0.7)(0.9x² + 0.17x - 0.259) + 0.181.

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The quotient is 0.9x² + 0.17x - 0.259, and the remainder is 0.181.

A more straightforward way to divide a polynomial by a linear factor is by synthetic division. (x-a).

The polynomial's coefficients must be written in descending order and any missing terms should be substituted with zero coefficients in order to apply synthetic division. A linear factor of the type (x-a), where "a" is a constant, must be used as the divisor.

To determine the quotient and remainder of a polynomial division by a linear factor, utilise synthetic division. In algebra and calculus, it is also used to resolve equations and simplify statements.

We will divide by in order to first change the divisor to its opposite sign x - (-0.7) = x + 0.7

Since the dividend's coefficients are 0.9, 0.8, 0, 0, we will set up our artificial division table as follows:

-0.7 | 0.9 0.8 0 0

     | -0.63 -0.259 0.181

     |____________________

        0.9 0.17 -0.259 0.181

The quotient is 0.9x² + 0.17x - 0.259, and the remainder is 0.181.

Therefore, 0.9x³ + 0.8x = (x + 0.7)(0.9x² + 0.17x - 0.259) + 0.181.

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Answer:

Step-by-step explanation:

Since the triangle was rotated by 30 degrees, the angle AOA' is 30 degrees, as is the angle A'OC. Thus, m/AOC is equal to 60 degrees. To demonstrate that BD is perpendicular to CD, OB = OC, angles AOB and COA are both 60 degrees, and angles BOD and COD are both 60 degrees. BD and CD are both perpendicular to the diameter through A.

A triangle is said to be equilateral if all three of its sides are the same length and all three of its angles are exactly 60 degrees. It has three regular sides and is a polygon. When dividing an equilateral triangle into two congruent 30-60-90 triangles, the altitude, median, and angle bisector from any vertex are also the same line.

5. Let's identify the circle's intersection points as the equilateral triangle's A, B, and C. All angles in a triangle that is equilateral are 60 degrees. Point A will change locations when the triangle is rotated by 30 degrees; let's designate this new position A'. Since O is the centre of the circle, the angle AOC remains at 60 degrees. Since the triangle has been turned by 30 degrees, the angle AOA' is 30 degrees, as is the angle A'OC. Thus, m/AOC is equal to 60 degrees.

6. The center of the circle is O, and D is the point where the diameter through B intersects the diameter through C. To demonstrate that BD is perpendicular to CD, the center of the circle, OB = OC, and angles AOB and COA are both 60 degrees, so angle BOC is 120 degrees. Angle BOD is a right angle, and angle OBD is half of angle BOC, so it is 60 degrees. Angle COD is also a right angle, and angle OCD is half of angle BOC, so it is also 60 degrees. Since angles OBD and OCD are both 60 degrees, and they are angles in a triangle, the third angle (angle BDC) must also be 60 degrees.

Therefore, triangle BDC is equilateral, and BD = CD. Since BD and CD are both perpendicular to the diameter through A, BD is perpendicular to CD.

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Answer:

C

Step-by-step explanation:

The graph that shows the solution set of 7 ≤ n + 5 include the following: A. graph A.

In Mathematics and Geometry, an inequality simply refers to a mathematical relation that is typically used for comparing two (2) or more numerical data and variables in an algebraic equation based on any of the inequality symbols;

In this exercise, we would determine the solution set by evaluating the given equation 7 ≤ n + 5 as follows;

7 ≤ n + 5

7 - 5 ≤ n + 5 - 5

2 ≤ n

n ≥ 2

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Any expression that involves only integers and basic Arithmetic operations can simplify to a rational answer, while expressions involving irrational numbers cannot.

There are several expressions that simplify to a rational answer. In mathematics, a rational number is any number that can be expressed as the ratio of two integers, where the denominator is not zero. Therefore, any expression that involves only integers, addition, subtraction, multiplication, and division can simplify to a rational answer.

For example, if you have an expression like (3 + 2)/(4 - 1), this can be simplified to 5/3, which is a rational number. Similarly, expressions like 6/3 or 7 - 2 also simplify to rational answers.

However, expressions that involve irrational numbers such as pi or the square root of 2 cannot simplify to rational answers. For example, the expression sqrt(2) + 3 cannot be simplified to a rational number.

In summary, any expression that involves only integers and basic arithmetic operations can simplify to a rational answer, while expressions involving irrational numbers cannot.

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Answer:18 cents.

Step-by-step explanation: I did the math.

Hope this helps <3

She will save $0.18

To find the answer, you use the amount of what it usually costs and subtract the sale price to get the amount she saves.

$0.34 - $0.16 = $0.18

Answer: Let's use the variables "r" and "c" to represent the cost of one rose and one carnation, respectively.

From the first sentence, we know that 8 roses and 5 carnations cost $31.75. So we can write the equation:

8r + 5c = 31.75

From the second sentence, we know that 1 rose and 3 carnations cost $5.75. So we can write the equation:

r + 3c = 5.75

Now we have two equations with two variables. We can use substitution or elimination to solve for "c". Let's use substitution:

r = 5.75 - 3c (from the second equation)

Substitute this expression for "r" into the first equation:

8(5.75 - 3c) + 5c = 31.75

Simplify and solve for "c":

46 - 24c + 5c = 31.75

-19c = -14.25

c = 0.75

Therefore, the cost for a carnation is $0.75.

Step-by-step explanation:

Answer:

24 students turned in there homework assignments.

Step-by-step explanation:

80% = .8

30 x .8 = 24

we have successfully simplified the expression x⁵-x³ into x³(x + 1)(x - 1). This can be useful for further algebraic manipulation or simplification in larger expressions.

How to simplify?

To simplify the expression x⁵-x³, we can factor it using the distributive property of multiplication. First, we can factor out x³ from both terms:

x⁵ - x³ = x³(x² - 1)

Now, we can further simplify by recognizing that x² - 1 is a difference of squares, which can be factored into (x + 1)(x - 1). So, we have:

x⁵ - x³ = x³(x² - 1) = x³(x + 1)(x - 1)

This is the fully simplified form of the expression. We can also expand the expression to verify that it is equivalent to the original expression:

x³(x + 1)(x - 1) = (x³ * x) + (x³ * -1) + (x³ * 1) + (x³ * -1) = x⁴ - x³ + x³ - x³ = x⁴ - x³

So, we have successfully simplified the expression x⁵-x³ into x³(x + 1)(x - 1). This can be useful for further algebraic manipulation or simplification in larger expressions.

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Answer:

[tex]\frac{61}{66}[/tex]

Step-by-step explanation:

Freshmen:

1200 + 20 = 1220

Sophomores:

12(15) = 180  60/5 = 12

1500 - 180 = 1320

[tex]\frac{1220}{1320}[/tex] = [tex]\frac{61}{66}[/tex]  (to simplify divide the numerator and denominator by 20)

Helping in the name of Jesus.

Thus, the simplification of the given function using difference quotient formula is found as 4.

The difference quotient is the product of the difference between the function values, f(x + h) - f(x), and indeed the difference between the input values, (x + h) - x, given a function f(x) and two input values, x as well as x + h (where h is indeed the distance between x and x + h).

For the given question:

function F(x) = -4x+8

difference quotient formula:

[f(x + h) -f(x)] / h

Put x = x + h

F(x+ h) = -4(x + h) + 8

F(x+h) = -4x  -4h + 8

F(x) = -4x+8

Now,

[f(x + h) -f(x) ]/ h = [-4(x + h) + 8  + 4x - 8]/ h

On simplifying:

[f(x + h) -f(x) ]/ h = 4h/h = 4

Thus, the simplification of the given function using difference quotient formula is found as 4.

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