how do you write 2.22 in expanded form

Final answer:

The graph of f(x) = 1/x is a hyperbola with vertical and horizontal asymptotes at x = 0 and y = 0, respectively. The domain is all real numbers except 0, and the range is the same. Unlike linear, quadratic, or exponential functions, the hyperbolic shape of 1/x presents its unique characteristics needing careful domain consideration.

Explanation:

The graph of the basic rational function f(x) = 1/x is characterized by two distinct branches, one in the first quadrant (where x and y are both positive) and the other in the third quadrant (where x and y are both negative). This function is a hyperbola, and as the value of x approaches zero from either the positive or negative side, the value of f(x) increases without bound, referred to as a vertical asymptote at x = 0. Additionally, as the value of x becomes very large or very small (approaching positive or negative infinity), the value of f(x) approaches zero, creating a horizontal asymptote at y = 0. Importantly, this function is undefined at x = 0, so the domain is all real numbers except 0, (domain: ℝ \ {0}). The function takes on all real number values for the output, so the range is also all real numbers except 0 (range: ℝ \ {0}).

Comparing to other types of functions, unlike linear functions like f(x) = 7x which produces a straight line or quadratic functions such as f(x) = 2x² which graphs as a parabola, the rational function f(x) = 1/x leads to a non-linear graph with two separate branches. Exponential functions like f(x) = eⁿ¹, however, grow or decline at an increasing rate, which is a different behavior compared to the hyperbolic decline or rise of a rational function. It's vital to note that as with the function |√x|, there are points along the graph of 1/x where the function is not defined, highlighting the importance of considering the domain in function analysis.

The function 1/x² also represents a rational function but with a different characteristic as it exhibits positive blowup on both sides. This means that as x approaches zero, whether from the positive or negative side, the function tends to positive infinity, a behavior different from the reciprocal function 1/x.

107 adult tickets were sold for the school play.

Let's denote the number of adult tickets sold as x. According to the given information, the number of student tickets sold is three times the number of adult tickets sold. So, the number of student tickets sold would be 3x. The total number of tickets sold is 428, so we can set up the equation:

x + 3x = 428

Combining like terms, we have:

4x = 428

Dividing both sides of the equation by 4, we find:

x = 107

Therefore, 107 adult tickets were sold.

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To determine how many 5/16 are in 1 7/8, you convert 1 7/8 to an improper fraction, 15/8, and then divide it by 5/16, which yields 6.

To find out how many 5/16 are in 1 7/8, you first need to convert 1 7/8 to a improper fraction. Since 1 is equal to 8/8, adding 7/8 gives us 15/8. Now, to find how many 5/16 are in 15/8, you can divide 15/8 by 5/16. To do this, you multiply 15/8 by the reciprocal of 5/16, which is 16/5.

15/8 × 16/5 = (15×16)/(8×5) = 240/40. This simplifies to 6, meaning there are 6 segments of 5/16 in 1 7/8.

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1st digit - 0-9 = 10 numbers

2nd digit = 9 numbers

3rd digit = 8 numbers

4th digit = 7 numbers

5th digit = 6 numbers

10 * 9 * 8 * 7 * 6 = 30,240 zip codes

Answer:

The inverse statement is

"A number that does not has exactly two distinct factors is not prime"

Step-by-step explanation:

Given a statement we have to tell the inverse of that statement.

The given statement is

"A number that has exactly two distinct factors is prime".

We have to write the inverse of the above statement.

The inverse of the statement is

"A number that does not has exactly two distinct factors is not prime"

The expected net winnings are  -$36. That is you lose $36 according to chance.

We have given

A game of chance involves rolling a 14-sided die once. If a number from 1 to 3 comes up, you win 2 dollars.

We have to determine expected net winnings

The expected value is what the player can expect to win or lose if they were to play many times with the same game.

let's say that you have a 3/13 chance of winning $3 and a 2/13 chance of winning $10 and an 8/13 chance of winning $0. 

Therefore if the dice is rolled 13 times then the expected to win 3X$3 + 2X$10 = $29 

But you pay $5 per roll so 13 rolls would cost me 13X$5 = $65 

Therefore the expected net winnings are $29-$65 = -$36 

Therefore the expected net winnings are  -$36.

So you lose $36 according to chance.

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The mode of the given data set is $7.04, the median is $9.99, the mean is $9.634, and the IQR is $3.48.

The mode of the given set is $7.04, as it occurs the most number of times.

The median is $9.99, which is the middle value when the data is arranged in ascending order.

The mean can be calculated by finding the sum of all the values and dividing it by the total number of values. In this case, the mean is $9.634.

The IQR (Interquartile range) is the difference between the third quartile and the first quartile. To find the IQR, we need to first calculate the quartiles. First, we arrange the data in ascending order: $7.04, $7.04, $7.05, $8.51, $9.98, $9.99, $11.95, $11.99, $12.50, $12.99. The first quartile is the average of the 3rd and 4th values, which is $8.51. The third quartile is the average of the 8th and 9th values, which is $11.99. Therefore, the IQR is $11.99 - $8.51 = $3.48.

1 gallon = 8 pints

15/8 = 1.875 gallons per minute

60 minutes per hour

1.875 * 60 = 112.5 gallons per hour

To convert 18 gallons to quarts, multiply by the conversion factor of 4 quarts per gallon, giving a result of 72 quarts.

The gasoline tank of a minivan holds 18 gallons. To determine how many quarts it can hold, we need to use the conversion factor that 1 gallon is equivalent to 4 quarts. Since we are converting from a larger unit to a smaller unit, we will multiply the number of gallons by 4.

18 gallons x 4 quarts/gallon = 72 quarts

Therefore, the minivan's gasoline tank can hold 72 quarts of gasoline.

Answer:-9 degrees

Step-by-step explanation:The average drop in temperature per minute is 9 degrees.

The symbol μ is used for population mean, while x is used for sample mean in statistics.

The symbol μ (pronounced "mew") is used to represent the population mean.

The population mean is a parameter that represents the average value of a variable in the entire population.

On the other hand, the symbol x is used to represent the sample mean, which is the average value calculated from a subset of the population called a sample.

The symbol μ (pronounced "mew") is used to represent the population mean.

It denotes the mean of all values within an entire population, which is a parameter of that population.

To illustrate why μ represents the population mean, consider a dataset: 1, 1, 1, 2, 2, 3, 4, 4, 4, 4, 4.

If this were to represent an entire population, we would calculate the mean and then denote it with μ.

Answer: He traveled 54.28 miles per hour as an average.

Step-by-step explanation:

Since we have given that

Number of miles traveled in first hour = 52 miles

Number of miles traveled in second hour = 64  miles

Number of miles traveled in third hour = 49 miles

Average per hour is given by

[tex]=\dfrac{3}{\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}}\\\\\\=\dfrac{3}{\dfrac{1}{52}+\dfrac{1}{64}+\dfrac{1}{49}}\\\\\\=\dfrac{3}{\dfrac{784+637+832}{40768}}\\\\\\=\dfrac{3}{\dfrac{2253}{40768}}\\\\\\=\dfrac{40768\times 3}{2253}\\\\\\=\dfrac{122304}{2253}\\\\\=54.28\ miles/hr[/tex]

Hence, he traveled 54.28 miles per hour as an average.

Answer:

The statements that are correct are:

c) [tex](in+a)(in+a^{-1})=2in+a+a^{-1}[/tex]

d) [tex]a^2b^7[/tex] is invertible.

e) [tex]a+b[/tex] is invertible.

Step-by-step explanation:

We are given that:

a and b are invertible n×n matrices.

We have to tell which of the following statements are true.

a)

[tex](ab)^{-1}=a^{-1}b^{-1}[/tex]

This statement is false.

Since:

[tex](ab)^{-1}=b^{-1}a^{-1}[/tex] and it may not be equal to the term [tex]a^{-1}b^{-1}[/tex]

b)

[tex]aba{-1}=b[/tex]

This expression could also be written as:

[tex]ab=ba[/tex]

Since on Post multiplying by a on both the sides.

But here we don't know whether the matrices are commutative or not.

Hence, the statement is false.

c)

[tex](in+a)(in+a^{-1})=2in+a+a^{-1}[/tex]

This statement is true.

since,

[tex](in+a)(in+a^{-1})=in(in+a^{-1})+a(in+a^{-1})\\\\=in^2+in.a^{-1}+a.in+aa^{-1}\\\\=in+a^{-1}+a+in\\\\=in+a^{-1}+a[/tex]

where in denote the identity matrix.

and we know that:

[tex]in^2=in[/tex]

d)

[tex]a^2b^7[/tex] is invertible.

This statement is true.

Since we know that prodct of invertible matrices is also invertible.

As [tex]a[/tex] is invertible so is [tex]a^2[/tex].

Also [tex]b[/tex] is invertible so is [tex]b^7[/tex].

Hence Product of  [tex]a^2[/tex] and  [tex]b^7[/tex] is also invertible.

i.e. [tex]a^2b^7[/tex] is invertible.

e)

[tex]a+b[/tex] is invertible.

This statement is true as sum of two invertible matrices is invertible.

f)

[tex](a+b).(a-b)=a^2-b^2[/tex]

This statement is false.

Since,

[tex](a+b).(a-b)=a(a-b)+b(a-b)\\\\=a.a-a.b+b.a-b.b\\\\=a^2-ab+ba-b^2[/tex]

Now as we are not given that:

[tex]ab=ba[/tex]

Hence, we could not say that:

[tex](a+b).(a-b)=a^2-b^2[/tex]

The correct statements which are true for all invertible [tex]\left({n\times n}\right)\cdot\left({n\times n}\right)[/tex] are:

(c). [tex]\left({in+a}\right)\left({in+{a^{-1}}}\right)=2in+a+{a^{-1}}[/tex]

(d). [tex]{a^2}{b^7}[/tex] is invertible.

(e). [tex]a+b[/tex] is invertible.

Further Explanation:

Given:

The matrix [tex]a[/tex] and [tex]b[/tex] are invertible [tex]\left({n\times n}\right)[/tex] matrices.

Calculation:

(a)

The statement is [tex]{\left({ab}\right)^{-1}}={a^{-1}}{b^{-1}}[/tex] false.

[tex]\begin{aligned}{\left({ab}\right)^{-1}}&={a^{-1}}{b^{-1}}\\\left({ab}\right){\left({ab}\right)^{-1}}&=\left({ab}\right){a^{-1}}{b^{-1}}\\I&\ne ab{a^{-1}}{b^{-1}}\\\end{aligned}[/tex]

The statement is [tex]{\left({ab}\right)^{-1}}={a^{-1}}{b^{-1}}[/tex] false.

(b)

The statement is [tex]ab{a^{-1}}=b[/tex].

Now multiply by a both the side.

[tex]\begin{aligned}ab{a^{-1}}a&=ba\\ab&\ne ba\\\end{aligned}[/tex]

The statement is [tex]ab{a^{-1}}=b[/tex] is false.

(c)

The statement is [tex]\left({in+a}\right)\left({in+{a^{-1}}}\right)=2in+a+{a^{-1}}[/tex].

Solve the above equation to check whether it is invertible.

[tex]\begin{aligned}\left({in+a}\right)\left({in+{a^{-1}}}\right)&=in\left({in+{a^{-1}}}\right)+a\left({in+{a^{-1}}}\right)\\&=i{n^2}+in\cdot{a^{-1}}+a\cdot in+a{a^{-1}}\\&=in+{a^{-1}}+a+in\\&=in+{a^{-1}}+a\\\end{aligned}[/tex]

The statement is true.

(d)

The statement is [tex]{a^2}{b^7}[/tex].

The product of invertible matrices is always invertible.

As [tex]a[/tex] is invertible so [tex]{a^2}[/tex] is also invertible.

As [tex]b[/tex] is invertible so [tex]{b^7}[/tex] is also invertible.

Hence, the product of [tex]{a^2}[/tex] and [tex]{b^7}[/tex] is also invertible.

The statement is true.

(e)

The statement [tex]a+b[/tex] is true as [tex]a+b[/tex] is always invertible.

(f)

The statement is [tex]\left({a+b}\right)\cdot\left({a-b}\right)={a^2}-{b^2}[/tex].

Solve the equation to check the inevitability.

[tex]\left({a+b}\right)\times\left({a-b}\right)={a^2}-ab+ba-{b^2}[/tex]

The statement is not true as [tex]ab=ba[/tex].

Hence, the correct statements which are true for all invertible [tex]\left({n\times n}\right)\cdot\left({n\times n}\right)[/tex] are:

(c). [tex]\left({in+a}\right)\left({in+{a^{-1}}}\right)=2in+a+{a^{-1}}[/tex]

(d). [tex]{a^2}{b^7}[/tex] is invertible.

(e). [tex]a+b[/tex] is invertible.

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Answer details:

Grade: High School

Subject: Mathematics

Chapter: Linear equation

Keywords: Invertible, matrices, matrix, statement, function, true, determinants, elements, inverse.