3. Solve. 5 < x - 2 < 11

A. 5 < x < 11
B. 7 < x < 13
C. 7 < x < 11
D. 5 < x < 16

The inverse function of f(x) = 6x^4 is f^(-1)(x) = (x/6)^(1/4), and f^(-1)(x) is a function.

To find the inverse function of f(x) = 6x^4, we can follow these steps:

1. Replace f(x) with y:
y = 6x^4

2. Swap x and y:
x = 6y^4

3. Solve for y:
Divide both sides by 6:
x/6 = y^4

4. Take the fourth root of both sides:
(x/6)^(1/4) = y

Now we have found the inverse function:
f^(-1)(x) = (x/6)^(1/4)

To determine if f^(-1)(x) is a function, we need to check if it passes the vertical line test. The vertical line test states that if a vertical line intersects a graph in more than one point, then the graph is not a function.

Let's imagine graphing f^(-1)(x) = (x/6)^(1/4). We see that for every x-value, there is only one corresponding y-value. Hence, each vertical line will intersect the graph at most once, confirming that f^(-1)(x) is indeed a function.

In conclusion, the inverse function of f(x) = 6x^4 is f^(-1)(x) = (x/6)^(1/4), and f^(-1)(x) is a function.

Answer:

1

Step-by-step explanation:

[tex]27 \cdot ((3^3)^{-1})[/tex]

We need to evaluate the given expression

3^3 is 3 times 3 times 3= 27

[tex]27 \cdot ((3^3)^{-1})[/tex]

[tex]27 \cdot ((27)^{-1})[/tex]

Now we apply exponential property

[tex]a^{-m}= \frac{1}{a^m}[/tex]

[tex]27^{-1}= \frac{1}{27}[/tex]

[tex]27 \cdot ((27)^{-1})[/tex]

[tex]27 \cdot \frac{1}{27}=1[/tex]

To find out how much shorter Noam's walk would be if there were a direct path from the school to his house, we need to calculate the displacement using the Pythagorean theorem. Noam walks 8 blocks north and then 6 blocks east. The displacement is the straight-line distance from the starting point to the endpoint. Therefore, the displacement is the square root of (8² + 6²), which is the square root of 64 + 36, which is the square root of 100, which is 10 blocks. So, if there was a direct path, Noam's walk would be 10 blocks shorter.

To find out how much shorter Noam's walk would be if there were a direct path from the school to his house, we need to calculate the displacement using the Pythagorean theorem. Noam walks 8 blocks north and then 6 blocks east. The displacement is the straight-line distance from the starting point to the endpoint.

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Therefore, the displacement is the square root of (8² + 6²), which is the square root of 64 + 36, which is the square root of 100, which is 10 blocks. So, if there was a direct path, Noam's walk would be 10 blocks shorter.

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The square root of 7 is an irrational number because it cannot be expressed as a ratio of two integers and its decimal form is infinite and non-repeating.

The square root of 7 is indeed an irrational number. An irrational number is one that cannot be expressed as the ratio of two integers, or as a predictable or repeating decimal. Unlike the square roots of perfect squares like 4 or 25, which are 2 and 5 respectively, the square root of 7 does not result in a whole number or a rational fraction. Additionally, this number cannot be exactly represented by a finite or repeating decimal sequence, which means it continues infinitely without repetition.

When we use a calculator to determine the square root of 7, we might see a value such as 2.646, but this is only an approximation because the actual decimal expansion goes on infinitely without repeating. Furthermore, the square root of a number like 7 does not solve a polynomial with rational coefficients, which further proves its irrationality.

Rational numbers, on the other hand, include integers and fractions where the numerator and the denominator are both integers. Numbers like 2/3 are rational because they have a clear ratio. However, roots of numbers that are not perfect squares, such as √7, cannot be expressed in such a ratio and are therefore irrational.

Final answer:

The student made an error in describing the expression 5/n as "the quotient of n and 5." The correct phrase to describe this expression would be "the quotient of 5 and n."

Explanation:

The student made an error in describing the expression 5/n as "the quotient of n and 5." The correct phrase to describe this expression would be "the quotient of 5 and n."

When describing division, it is important to remember that the dividend (the number being divided) comes before the divisor (the number dividing the dividend).

The required number is 27.

A linear equation only has one or two variables. No variables in a linear equation is raised to power greater than 1 or used as denominator of a fraction.

Let x be the number,

Then

Twenty-four is subtracted from a number = x - 24

and number divided by nine = x/9

Now according to the question,

x - 24 = x/9

Multiplying whole equation by 9 we get,

9x - 216 = x

Adding 216 both the sides we get

9x  = x + 216

Substracting x both the side we get

8x = 216

Dividing whole equation by 8 we get

x = 27

which is the required number.

Hence,the required Number is 27.

More about linear equation:

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divide 36 by 8

36 / 8 = 4.5

 so he had 4 pieces that were 8 feet long and he had 1 piece 4 feet long

 so he had 5 pieces of wire

12  x 10  = 120

12   X 100  = 1200

Take for instance in 6 x 2, 6 x 20 and 6 x 200. Find the factors and these are the computations. We can also say: 

6 x 20 = 6 x 2 x 10 = 12 x 10 = 120 
60 x 20 = 6 x 10 x 2 x 10 = 12 x 10 x 10 = 1200

It must be remembered that in the commutative and associative properties of multiplication, the order of the factors does not matter when solving this kind of problem

greg score = x

 kara score = 3/4x-1

x + 3/4x-1 = 153

1 3/4x -1 = 153

1 3/4x = 154

x = 154 / 1 3/4 = 88

gregs score was 88

 kara s score was 153 -88 = 65

Michael made 48 cookies and is bringing 75% of them to a school party. To calculate this, multiply 48 (the total number of cookies) by 0.75, resulting in 36 cookies that he will bring to school.

Michael made 4 dozen chocolate chip cookies. He wants to bring 75% of them to school for a party. To find out how many cookies he will bring, we need to calculate 75% of 4 dozen cookies. Since each dozen is 12 cookies, 4 dozen would be 4 times 12, which equals to 48 cookies.

To find 75% of 48, you can either multiply 48 by 0.75 or calculate 75/100 of 48:

Therefore, Michael will bring 36 cookies to the school for the party.

Michael brings 36 cookies to school for the party.

To find out how many cookies Michael brings to school, we need to calculate 75% of the total number of cookies he made.

Michael made 4 dozen cookies, and a dozen is equal to 12, so he made a total of 4 x 12 = 48 cookies.

75% of 48 is calculated by multiplying 48 by 0.75. 48 x 0.75 = 36.

Therefore, Michael brings 36 cookies to school for the party.

Jupiter, the Roman king of the gods, has no mythical records signifying that he had a pet dog. He is mostly associated with the eagle and the bull.

Jupiter, the Roman king of the gods, The question you're asking seems to be based on the mythology of the Roman god Jupiter. However, there is no record in Roman mythology of any pet dog belonging to Jupiter. Jupiter, also known as Jove, was the king of the gods and the god of the sky and thunder in Ancient Rome. Animals often associated with him are the eagle and the bull, but not a dog.

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The base of the triangle is calculated to be 12 inches using the area and height. Then, through the Pythagorean theorem, the hypotenuse is found to be approximately 36.06 inches.

The subject of the question is related to the calculation of a triangle's hypotenuse using the area and height of a right triangle. Given that the area of a triangle is calculated by halving the product of the base and the height (Area = 1/2 * base * height), we can use the given area to find the base of the right triangle.

First, we call the base of the triangle 'x'. The area is 210 square inches and the height is 35 inches.

We rearrange the formula for the area of a triangle to find 'x' as follows:  base = (2*Area) / height.

Our calculation then becomes x = (2*210) / 35. This simplifies to x = 12 inches.  

Next, we use the Pythagorean Theorem, that states: a² + b² = c², where 'c' represents the hypotenuse. Here, 'a' is the base found (12 inches), 'b' is the height (35 inches), and 'c' is the hypotenuse we're seeking.

Plugging the values into the theorem, we get c = √(12² + 35²). Hence, the hypotenuse of the right triangle is approximately 36.06 inches.

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Answer:

Option C. [tex]BC=31.2\ units[/tex]

Step-by-step explanation:

step 1

In the right triangle ABD find the length side AB

Applying the Pythagoras Theorem

[tex]AB^{2}=5^{2}+12^{2}[/tex]

[tex]AB^{2}=169[/tex]

[tex]AB=13\ units[/tex]

step 2

In the right triangle ABD

we know that

[tex]m<ABD=m<BCD[/tex]

[tex]sin(<ABD)=\frac{5}{13}[/tex] -------> equation A

step 3

In the right triangle ABC

[tex]sin(<BCD)=\frac{13}{5+DC}[/tex] ------> equation B

Remember that

[tex]m<ABD=m<BCD[/tex]

so equate equation A and equation B solve for DC

[tex]\frac{5}{13}=\frac{13}{5+DC}[/tex]

[tex]5(5+DC)=13*13[/tex]

[tex]25+5DC=169[/tex]

[tex]5DC=169-25[/tex]

[tex]5DC=144[/tex]

[tex]DC=144/5=28.8\ units[/tex]

step 4

In the right triangle BDC find the length side BC

Applying the Pythagoras Theorem

[tex]BC^{2}=28.8^{2}+12^{2}[/tex]

[tex]BC^{2}=973.44[/tex]

[tex]BC=31.2\ units[/tex]

x = -4, so use the first equation, x^2 = -4^2 = -16

x = 4, use 3rd equation, sqrt(x) = sqrt(4) = 2

Answer:

(0, –2); maximum

Step-by-step explanation:

[tex]y = -x^2 - 2[/tex]

WE are given with the quadratic equation.

Vertex form of quadratic equation is in the form of [tex]y=a(x-h)^2+k[/tex]

where (h,k) is the vertex

[tex]y = -(x-0)^2 - 2[/tex]

The value of a=-1

when the value of 'a' is negative, then the vertex is maximum

To find the vertex we look at the value of h  and k

h=0 and k= -2

Vertex is (0,-2)

(0, –2); maximum