Find the domain and range for the following functions:
(3 Marks)
a) y=1/√ x2-4
y=(2x^2)-2x+5
y=
1/(1-1/(x-1))

1 gallon of milk is equal to 16 cups of milk (1 gallon = 4 quarts = 16 cups).

Mario poured 4 cups of milk for each of the 4 people, which is a total of 16 cups of milk.

So the amount of milk left is:

16 cups (total) - 16 cups (poured) = 0 cups

Therefore, there is no milk left.

The height οf the clοset is x ft = 8 ft.

The vοlume οf a rectangular prism is the amοunt οf space οccupied by the three-dimensiοnal οbject. It is calculated by multiplying the length, width, and height οf the prism.

In this prοblem, we are given that the clοset has a vοlume οf 220 ft^3 and its dimensiοns are x+3 ft. wide, x-5.5 ft. deep, and x ft. high. Sο, we can write the equatiοn:

V = (x+3)(x-5.5)(x)

where V is the vοlume οf the clοset.

Tο sοlve fοr x, we can simplify the equatiοn by multiplying the factοrs:

[tex]V = (x^2 - 2.5x - 16.5)(x) = x^3 - 2.5x^2 - 16.5x[/tex]

Nοw, we can substitute the given value οf V and sοlve fοr x:

[tex]220 = x^3 - 2.5x^2 - 16.5x[/tex]

We can rearrange this equatiοn tο get:

[tex]x^3 - 2.5x^2 - 16.5x - 220 = 0[/tex]

This is a cubic equatiοn that can be sοlved by factοring οr using numerical methοds. By using numerical methοds, we can find that οne sοlutiοn οf this equatiοn is x = 8.

Therefore, the height of the closet is x ft = 8 ft.

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Answer:

144

Step-by-step explanation:

The desired solution is to the nearest quarter of a unit, the approximate solution is x=2.25.

A quarter is a fraction that represents one-fourth of a whole number or quantity. It can be written as a ratio, a decimal, or a percentage. A quarter is one of the simplest fractions, and it can be used to divide an object or number into four equal parts. It is also used to calculate percentages and proportions.

A table of values is a great way to approximate a solution to an equation. To approximate a solution to the equation f(x)=g(x) to the nearest quarter of a unit, the table of values must contain the x-values and the corresponding f(x) and g(x) values.

For example, if the equation was f(x)=2x+3 and g(x)=x+2, the table of values would look like this:

x f(x) g(x)

0 3 2

1 5 3

2 7 4

3 9 5

4 11 6

To find the approximate solution to f(x)=g(x), the table must be examined for the first x-value where the f(x) and g(x) values are equal. In this case, the x-value is 2 and the corresponding f(x) and g(x) values are both 7. Since the desired solution is to the nearest quarter of a unit, the approximate solution is x=2.25.

This method of using a table of values to approximate a solution to an equation is very simple and quick. It is a great way to get a rough answer to an equation without having to solve it. It is also a useful tool for double-checking the exact solution to an equation.

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Given:

5yd

To find:

The area of the hexagon

Solution:

The expression that represents the width of the rectangle is 6x^5y^2

We know that the area of a rectangle is given by the formula:

Area = Length x Width

We can rearrange the formula for the area to solve for the width:

Width = Area / Length

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

Area = 54x^8y^6

Length = 9x^3y^4

Substituting the given values, we get:

Width = (54x^8y^6) / (9x^3y^4)

Evaluate the quotient expression

Width = 6x^5y^2

Hence, the width expression is 6x^5y^2.

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Answer:

Step-by-step explanation:

Im not really sure what the heck you doing because im only in 8th grade but right now im in central and i'm kind of good at math but i'll try. Lets see........ I think they would all just be $11.50, but like i said im only in 8th grade and not that good at this kind of math so before you answer it please check with someone else to see if i got it right for you, cause i don't want to give you the wrong answer and you fail it.

Answer:

Let's start by using algebra to solve the problem.

Let x be the total number of entries at the fair.

Given that 60% of the entries are livestock, we can write:

0.6x = number of livestock entries

Of those livestock entries, 70% are sold at the auction, which means:

0.7(0.6x) = 130

Simplifying the second equation, we get:

0.42x = 130

Solving for x, we can divide both sides by 0.42:

x = 309.52

Since we can't have a fractional number of entries, we can round up to the nearest whole number, giving us:

x = 310

Therefore, there were a total of 310 entries at the fair.

Step-by-step explanation:

Therefore, the predicted population of Alaska in 2040 is approximately 1,038,428.

Given by the question.

P₀ = 410,851

P = 739,818

t = 2017 - 1980 = 37

r = ln (739818/410851) / 37

r ≈ 0.0136

Therefore, the equation that models the population of Alaska t years after 1980 is:

P(t) = 410,851 * [tex]e^{(0.0136*t)}[/tex]

To predict the population of Alaska in 2040, we need to find the value of P (60), since 2040 is 60 years after 1980:

P (60) = 410,851 * [tex]e^{(0.0136*60)}[/tex]

P (60) ≈ 1,038,428

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The value of x in the triangle based on Pythagoras Theorem will be 13

Pythagoras Theorem is the way in which you can find the missing length of a right angled triangle. The triangle has three sides, the hypotenuse (which is always the longest), Opposite (which doesn't touch the hypotenuse) and the adjacent (which is between the opposite and the hypotenuse).

Pythagoras is in the form of;

a²+b²=c². However, it can also be written in the form of c²=a²+b²

This will be:

x² = 12² + 5²

x² = 144 + 25

x² = 169

x = ✓169

x = 13

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The length of the base of the parallelogram is 10.25 inches.

given that,

the area of parallelogram=102 1/2 square inches

height=10 inches

The following formula determines the area of a parallelogram:

Base + height = area.

The parallelogram's area and height are provided. By entering these values in the above formula we obtain:

102 1/2 = base × 10

We can divide both sides by 10 to find the base:

102 1/2 ÷ 10 = base

By dividing 1 by 2, we may convert the mixed number 1/2 to a decimal:

102 1/2 ÷ 10 = 102.5 ÷ 10 = 10.25

therefore, the length of the base of the parallelogram is 10.25 inches long.

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If the length of the tile is x and the breadth of the tile is y then the area of each tile is equal to xy units.

Given that we can assume the length of tile be x and breadth of tile be y. We are required to find the area of the each tile and draw a figure showing the tile whose length is x and the breadth is y.

Tile is in the shape of the rectangle because the length and breadth are different and we know that the area of the rectangle is equal to product of the length and breadth of that rectangle.

If the length of tile is x and the breadth of tile is y then the area of the tile is equal to x*y which is xy units. Figure is attached with the solution.

Hence if the length of the tile is x and the breadth of the tile is y then the area of each tile is equal to xy units.

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Triangle T prime is similar to triangle T with a scale factor of 5/9.

What is the scale factor?

In mathematics, a scale factor is a number that scales, or multiplies, a quantity or geometric figure. In other words, it is the ratio of any two corresponding lengths in two similar figures.

To determine the scale factor of the dilation, we can compare the corresponding side lengths of the two triangles.

For example, we can compare the longest sides of each triangle (also known as the "hypotenuse" in a right triangle):

scale factor = length of corresponding side in T prime / length of corresponding side in T

scale factor = 20 / 36

scale factor = 5/9

This means that triangle T prime is a reduction (since the scale factor is less than 1) of triangle T by a factor of 5/9. In other words, the side lengths of triangle T prime are 5/9 times the corresponding side lengths of triangle T. Therefore, we can find the lengths of the other sides of triangle T prime by multiplying the corresponding side lengths of triangle T by the scale factor:

10 = 18 * 5/9

20 = 36 * 5/9

So, the side lengths of triangle T prime are 20, 10, and (36*5/9) = 20.

Therefore, triangle T prime is similar to triangle T with a scale factor of 5/9.

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The amount of cost for the fabric is given by A = $ 13

Equations are statements in mathematics that have two algebraic expressions on either side of the equals (=) sign.

It displays the similarity of the connections between the phrases on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are examples of the parts of an equation. When creating an equation, the "=" symbol and terms on both sides are necessary.

Given data ,

Let the total cost of red fabric be represented as A

Now , the amount of red fabric be = 6 1/2 yards

And , the cost of red fabric per yard = $ 2

So , the total cost of red fabric A = amount of red fabric x cost of red fabric per yard

On simplifying the equation , we get

The total cost of red fabric A = ( 6 1/2 ) x 2

The total cost of red fabric A = ( 13/2 ) x 2

The total cost of red fabric A = $ 13

Therefore , the value of A is $ 13

Hence , the total cost of red fabric is $ 13

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Answer:

70 and 2.91 days

hope this helped

The final answer is: \[ C = 40.32^{\circ} \]Therefore, the indicated angle is 40.32 degrees.

To determine the indicated angle, we can use the Law of Cosines, which states that: \[ c^2 = a^2 + b^2 - 2ab\cos(C) \]We can rearrange this equation to solve for the cosine of the indicated angle: \[ \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} \]Plugging in the given values for a, b, and c, we get: \[ \cos(C) = \frac{69.01^2 + 39.28^2 - 42.65^2}{2(69.01)(39.28)} \]Simplifying the expression, we get: \[ \cos(C) = 0.7664 \]Now, we can use the inverse cosine function to find the indicated angle: \[ C = \cos^{-1}(0.7664) \]This gives us an angle of 40.32 degrees. However, the question asks us to round our answer to two decimal places, so the final answer is: \[ C = 40.32^{\circ} \]Therefore, the indicated angle is 40.32 degrees.

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The nearest value of x is 56°. We solve this question using Pythagoras theorem for that we also find one of its side which is not given.

The square on the hypotenuse of a right-angled triangle has the same area as the sum of the squares on the other two sides, according to a Pythagorean theorem.

Given AB = 5 and AC = 9

By Using Pythagoras theorem,

AB² + BC² = AC²

5² + BC² = 9²

BC² = 9² - 5²

BC² = 81 - 25

BC = 7.48 (Approx.)

Now, we will find the value of x°

[tex]So, tan\ x^{\circ} = \frac{BC}{AB}[/tex]

[tex]tan \ x^{\circ} = \frac{7.48}{5}[/tex]

[tex]tan\ x^{\circ} = 1.496[/tex]

[tex]x^{\circ} =tan^{-1} 1.496[/tex]

[tex]x^{\circ} = 56.24^{\circ}[/tex] , which is close to 56°

So the nearest value of x is 56°.

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The solutions to the given trigonometric equations are:
i. θ = π/4, 3π/4, 5π/4, 7π/4
ii. θ = π/2, 3π/2
iii. θ ≈ 76.96º, 13.04º

To solve the given trigonometric equations using algebra, we need to rearrange the equations and use trigonometric identities to find the values of θ that satisfy the equations.

i. 2 sin^2 θ = 1, 0 < θ < 2π
First, we rearrange the equation to isolate sin^2 θ:
sin^2 θ = 1/2
Next, we take the square root of both sides:
sin θ = ±√(1/2)
Using the identity sin θ = cos (π/2 - θ), we can find the values of θ that satisfy the equation:
θ = π/4, 3π/4, 5π/4, 7π/4

ii. sin 1/2 θ = 1, - π < θ < 2π
First, we rearrange the equation to isolate θ:
sin θ = 2
Next, we use the identity sin θ = 1/csc θ to find the values of θ that satisfy the equation:
θ = π/2, 3π/2

iii. 4 cos(θ-45º) + 7 = 10, 0 < θ < 360 º
First, we rearrange the equation to isolate cos(θ-45º):
4 cos(θ-45º) = 3
cos(θ-45º) = 3/4
Next, we use the identity cos θ = sin (π/2 - θ) to find the values of θ that satisfy the equation:
θ = 45º + cos^-1(3/4), 45º - cos^-1(3/4)

Using desmos, we can verify our solutions and find the approximate values of θ:
θ ≈ 76.96º, 13.04º

Therefore, the solutions to the given trigonometric equations are:
i. θ = π/4, 3π/4, 5π/4, 7π/4
ii. θ = π/2, 3π/2
iii. θ ≈ 76.96º, 13.04º

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Answer:

8

Step-by-step explanation:

Let's call the unknown side [tex]c[/tex].

The Pythagorean Theorem states that in any right triangle, [tex]a^{2} + b^2 = c^2[/tex]

Now plug in the known values.

[tex]6^2 + (2\sqrt 7)^2 = c^2\\36 + 4(7) = c^2\\36 + 28 = c^2\\64 = c^2\\c = \sqrt{64} = 8[/tex]

There are three recipes provided to mention the fractional amounts of ingredients:

Recipe for Chicken Curry:

2 teaspoons of turmeric

1/2 teaspoon of cumin

1/4 spoon of any type of pepper

If you were to double this recipe, you would use:

4 teaspoons of turmeric

1 teaspoon of cumin

1/2 teaspoon of pepper

To double the amount of the ingredients, you would multiply each fraction by 2.

Chocolate chip cookies:

1/4 teaspoon of baking soda

1/2 teaspoon of salt

double the quantities if u want to make more or huge quantities

Tomato soup recipe

half a spoonful of dried basil

1/4 teaspoon of dried thyme

for this recipe, we will need

1 spoon of basil

1/2 teaspoon of dried thyme

Recipe for Pasta Salad:

fourth cup of Olive oil

1/3 cup of red wine vinegar

1/2 teaspoon of dried oregano

1/4 teaspoon of salt

1/8 teaspoon of black pepper

We can multiply by 2 if we increase the quantity.

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The expressiοn (B) 2 × 4 × 5 × 1/3 × 1/3 × 1/3 can be used tο find the vοlume οf the rectangular prism is cubic yards.

A mathematical equatiοn links twο statements and utilises the equals sign (=) tο indicate equality. In algebra, an equatiοn is a mathematical assertiοn that prοves the equality οf twο mathematical expressiοns. Fοr instance, in the equatiοn 3x + 5 = 14, the equal sign separates the numbers by a gap. A mathematical fοrmula may be used tο determine hοw the twο sentences οn either side οf a letter relate tο οne anοther. The lοgο and the particular piece οf sοftware are usually identical. like, fοr instance, 2x - 4 = 2.

Vοlume is calculated using the fοrmula LWH

Here each cube has side 1/3 yd

Sο vοlume = (2 × 1/3) × (4 × 1/3) × (5 × 1/3)

It can be written as 2 × 4 × 5 × 1/3 × 1/3 × 1/3

Thus, οptiοn B is cοrrect.

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Complete question:

The function eˣ is considered as a function of real numbers has domain (−∞,∞) and have value between 0 to ∞ . That is the reason why "a" cannot be zero.

Considered as a function of real numbers, the function ex has domain (−∞,∞) and range (0,∞). Therefore, only strictly positive values ​​can be assumed. Considering ex as a function of complex numbers, we know that there is a domain C and a range C\{0}. In other words, the only value eˣ cannot have is 0. 

Thus, the base cannot be equal to 0.

A formula, rule, or law that defines the relationship between one variable (the independent variable) and another (the dependent variable). Functions are ubiquitous in mathematics and essential in the natural sciences for formulating physical relationships. 

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a cannot be equal to zero in an exponential function because a must be greater than 0 in order for the equation to work, and because the base of an exponential function must be a positive number for it to be an exponential function.

An exponential function is a type of mathematical function that increases or decreases by a consistent rate over time. It is usually expressed in the form of y = abx, where a and b are constants and x is the independent variable. The graph of an exponential function is a curved line that starts off flat and then increases or decreases in a curved manner. The rate of change of the function is determined by the value of b, with higher values of b resulting in a steeper curve.

In this situation, a must be greater than 0 because an exponential function cannot have a base of 0. This is because any number raised to the power of 0 is equal to 1, so if the base of the exponential function is 0, then the entire equation would be equal to C1, which is simply C. Therefore, the only way to make the equation work is to make sure that a is greater than 0.

Therefore, a cannot be equal to zero in an exponential function. This is because a must be greater than 0 in order for the equation to work, and because the base of an exponential function must be a positive number in order for it to be an exponential function.

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The result when the number 72 is decreased by 50% is 36.

We can use the formula:

Result = Original Number - (Original Number x Percent Decreased)

In this case, the original number is 72 and the percent decreased is 50%.

So, we can plug in the values into the formula:

Result = 72 - (72 x 0.50)

Result = 72 - 36

Result = 36

Therefore, the result when the number 72 is decreased by 50% is 36.

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The original equations for these two transformed equations can be found by reversing the transformations that have been applied to the standard sine function y = sin(x).

For Equation 1: [tex]y = 3sin(2x-2)[/tex], the original equation is y = sin(x). The transformations that have been applied are a vertical stretch by a factor of 3, a horizontal compression by a factor of 2, and a horizontal shift to the right by 2 units. To reverse these transformations, we would divide the y-values by 3, divide the x-values by 2, and shift the graph to the left by 2 units.

For Equation 2: [tex]y = 4sin(x-1)+3[/tex], the original equation is also y = sin(x). The transformations that have been applied are a vertical stretch by a factor of 4, a horizontal shift to the right by 1 unit, and a vertical shift up by 3 units. To reverse these transformations, we would divide the y-values by 4, shift the graph to the left by 1 unit, and shift the graph down by 3 units.

Therefore, the original equations for these two transformed equations are both y = sin(x).

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The value that best approximates the radius of the circle is 25.38 inches.

The measure of the central angle in degrees is related to the length of the arc intercepted by the central angle and the radius of the circle by the formula:

Arc length = (Central angle measure / 360°) x (2πr)

where r denotes the circle's radius.

Substituting the given values, we have:

16 inches = (37° / 360°) x (2πr)

Simplifying and solving for r, we get:

r = 16 inches / [(37/360) x (2π)]

r ≈ 25.38 inches (rounded to two decimal places)

Therefore, the value that best approximates the radius of the circle is 25.38 inches.

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The value shown on the number line is 3 8/10

The number line represents the given parameter

On the number line, we have the point located between 3 and 4

Such that the point is at 0.8 between 3 and 4

When the above expression is represented as a number, we have

Number = 3 + 0.8

Express the decimal as a fraction

So, we have

Number = 3 + 8/10

Evaluate

Number = 3 8/10

Hence, the solution is 3 8/10

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The original piece of cardboard should be approximately 32.86 inches by 16.43 inches.

To find the size of the original piece of cardboard, we can use the formula for the volume of the box, which is V = l × w × h. Since the box is open-topped, the height will be the size of the square that was removed from each corner, which is 3 inches. Also, given that the length of the original piece of cardboard is twice the width, we can substitute l = 2w in the formula. Therefore, the formula becomes:

V = (2w) × w × 3

Now, we can plug in the given volume of 1,620 in3 and solve for w:

1,620 = (2w) × w × 3

540 = 2w^2

270 = w^2

w = √270

w ≈ 16.43 inches

Now, we can use the value of w to find the length of the original piece of cardboard:

l = 2w = 2(16.43) = 32.86 inches

Therefore, the original piece of cardboard should be approximately 32.86 inches by 16.43 inches.

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Rows of chairs that are needed to make 48 musicians sit is 8 rows.

The division is the process of repetitive subtraction. It is the inverse of the multiplication operation. It is defined as the act of forming equal groups. While dividing numbers, we break down a larger number into smaller numbers.

Given,

Number of musicians in orchestra

18 string musicians, 9 percussion musicians,

15 brass musicians, and 12 woodwind musicians

Total number of musicians

= 18 + 9 + 15 + 12

= 54

Six of the musicians cannot play in the next performance

Number of musicians playing in next performance

= 54 - 6

= 48

48 musicians will sit on 48 chairs

Number of chairs in one row = 6

Number of rows for 48 chairs = 48/6 = 8

Hence, 8 rows of chairs are needed to make 48 musicians sit.

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For the given function we have:

domain  [0, 76,867] and range [$0, $12,375,587]

We know that x represents the number of people in atendance, then:

R(x) = 161*x

Represents the revenue.

The domain is the set of possible inputs, and it will go between 0 and 76,867 (total number of seats).

The range will go between:

R(0) = 161*0 = 0

R(76,867) = 76,867*161 = 12,375,587

Then the domain is all the integers in [0, 76,867] and the range is [$0, $12,375,587]

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