What are the domain and range of each relation?

Drag the answer into the box to match each relation.

1.  Consider the expression [tex]-5a^4(2a^2+4)=-10a^6-20a^4.[/tex]

Start with left side and use dustributive property :

[tex]-5a^4(2a^2+4)=-5a^4\cdot 2a^2-5a^4\cdot 4=-10a^6-20a^4.[/tex]

This option is true.

2. Consider the expression [tex]-4x^2(2x^2+5)=-8x^4-20x^2.[/tex]

Start with left side and use dustributive property :

[tex]-4x^2(2x^2+5)=-4x^2\cdot 2x^2-4x^2\cdot 5=-8x^4-20x^2.[/tex]

This option is true.

3. Consider the expression [tex]-6y^4(4y^2+2)=-24y^8-12y^4.[/tex]

Start with left side and use dustributive property :

[tex]-6y^4(4y^2+2)=-6y^4\cdot 4y^2-6y^4\cdot 2=-24y^6-12y^4\neq -24y^8-12y^4.[/tex]

This option is false.

4. Consider the expression [tex]-4b^3(5b^2+3)=-20b^6-12b^3.[/tex]

Start with left side and use dustributive property :

[tex]-4b^3(5b^2+3)=-4b^3\cdot 5b^2-4b^3\cdot 3=-20b^5-12b^3\neq -20b^6-12b^3.[/tex]

This option is false.

Answer: A, B - true, C, D - false.

Answer

−5a⁴(2a²+4)=−10a⁶−20a⁴ is correct.

−4x²(2x²+5)=−8x⁴−20x²  is correct.

−6y⁴(4y²+2)=−24y⁸−12y⁴ is NOT correct.

−4b³(5b²+3)=−20b⁶−12b³  is NOT correct.

Explanation

Equation 1

−5a⁴(2a²+4)=−10a⁶−20a²    ⇒ −5a⁴(2a²+4) = ( −5a⁴×2a²)+ ( −5a⁴×4)

                                                                      = -10a⁶ - 20a⁴

−5a⁴(2a²+4)=−10a⁶−20a² is correct

Equation 2

−4x²(2x²+5)=−8x⁴−20x²        ⇒   −4x²(2x²+5) = (-4x²×2x²) + (-4x²×5)

                                                                           = -8x⁴ - 20x²

−4x²(2x²+5)=−8x⁴−20x²  is correct.

Equation 3

−6y⁴(4y²+2)=−24y⁸−12y⁴     ⇒ −6y⁴(4y²+2) =  (−6y⁴×4y²) + (-6y⁴×2)

                                                                       = -24y⁶ - 12y⁴

−6y⁴(4y²+2)=−24y⁸−12y⁴ is NOT correct.

Equation 4

−4b³(5b²+3)=−20b⁶−12b³       ⇒ −4b³(5b²+3) = (−4b³×5b²) + (-4b³×3)

                                                                           = -20b⁵ - 12b³

−4b³(5b²+3)=−20b⁶−12b³  is NOT correct.                                        

Answer:

first we will multiply -3/10 and 60,

-3/10 x 60 = -18    

(one-tenth of 60 is 6, and because we are multiplying -3/10 and not -1/10 we can multiply 6 by -3 to get 18.)

then multiply the exponents

x(x) = x^2

y(y^6) = y^7

we can then multiply all terms together to form one single term

-18(x^2)y^7=-18x^2y^7

our final answer is:

-18x^2y^7

The answer to the given question is [tex]\[ \boxed{-18x^2y^7} \][/tex]

First, we need to multiply the numerical coefficients and the literal coefficients (variables) separately. The numerical coefficients are -3 and 60, and the literal coefficients are 1/10, x, y, x, and [tex]y^6[/tex].

Multiplying the numerical coefficients:

[tex]\[ -3 \times 60 = -180 \][/tex]

Multiplying the literal coefficients (variables):

[tex]\[ \frac{1}{10} \times x \times y \times x \times y^6 \][/tex]

Since we have two x's and one y to the power of 6, we can simplify this as:

[tex]\[ \frac{1}{10} \times x^2 \times y^7 \][/tex]

Now, we combine the numerical and literal coefficients:

[tex]\[ -\frac{180}{10} \times x^2 \times y^7 \][/tex]

Simplifying the numerical fraction:

[tex]\[ -18 \times x^2 \times y^7 \][/tex]

Therefore, the final simplified expression is:

[tex]\[ -18x^2y^7 \][/tex]

The answer is: [tex]-18x^2y^7[/tex].

Given

One month julia collected 8.4 gallons of rainwater.

she used 5.2 gallons of rainwater to water her garden

6.5 gallons of rainwater to water flowers

Find out how much was the supply of rainwater increased or decreased by the end of the month.

To proof

As given in the question

One month julia collected 8.4 gallons of rainwater

she used 5.2 gallons of rainwater to water her garden and 6.5 gallons of rainwater to water flowers

Total water she used in the month = 5.2 gallons + 6.5gallons

                                                         = 11.7 gallons

Let the supply of rainwater increased or decreased by the end of the month

be x .

Than the equation become in the form

x + 8.4 = 11.7

x = 3.3 gallons

Therefore the supply of rainwater increased or decreased by the end of the month is 3.3 gallons.

Hence proved

Total rainwater collected by Julia = 8.4 gallons

Water used for watering garden = 5.2 gallons

Water used for watering flowers = 6.5 gallons

Hence, total water used by Julia = [tex]5.2+6.5=11.7[/tex] gallons

11.7 gallons were used and only 8.4 gallons were collected , so supply of rainwater decreased by [tex]11.7-8.4=3.3[/tex] gallons

Jon is selling tickets for the school talent show. On the 1st day, he sold 3 senior tickets and 12 child tickets for $195. On the 2nd day he sold 13 senior tickets for $299. Find the price of a senior citizen ticket.

Create a system of equations to help you solve this problem. The system of equations will look like: 3s + 12c = 195 and 13s = 299. The variable s represents the cost of senior tickets and the variable c represents the cost of children tickets.

[tex]\left \{ {{3s~+~12c~=~195} \atop {13s~=~299}} \right.[/tex]

Solve the second equation for the variable s as this is the easiest way to solve the problem. Solve the second equation for s by dividing both sides of the equation by 13 to isolate the variable s.

s = 23

Since the question was only asking for the price of a senior citizen ticket, you are technically done. The first equation was only put there to confuse you or allow you to check your work if you needed to. The price of a senior citizen ticket (variable s) is $23.

Answer:

5.13 feet

Step-by-step explanation:

Engine is driving the propeller with a radius of 8 feet and its centerline 13 feet above the ground.

And the speed is 700 revolutions per minute, the height of one propeller tip as a function of time is given by:

[tex]h=13+8 \sin(700t)[/tex]

We have been asked to find the value of height, 'h', when t=4 minutes.

Plugging the value of time, 't', in the equation, we already know that we need to use degrees (not radians) we get:

[tex]h=13+8 \sin (700 \times 4)[/tex]

[tex]h=13+8 \sin (2800)[/tex]

[tex]h=13+8 \times (-0.984)[/tex]

[tex]h=13+(-7.872)[/tex]

[tex]h=13-7.872[/tex]

[tex]h=5.128\approx 5.13[/tex]

So the height of one propeller tip at t=4 minutes is 5.13 feet.

Find how far the car can travel on one gallon of gas, by dividing total miles by number of gallons:

105 miles / 7 gallons = 15 miles per gallon.

Now multiply that by the number of gallons to find total miles:

15 miles per gallon x 9 gallons = 135 total miles.

The solution to ax² + bx + c = 0  is [tex]x=-\frac{b}{2a}\pm\sqrt{\frac{b^2-4c}{4a} }[/tex]

A quadratic function is an equation of degree 2.  

Given the equation:

ax² + bx + c = 0

Subtracting c from both sides to get:

ax² + bx = -c

Dividing through by a:

x² + (b/a)x = -c/a

Add to both sides the square of half of the coefficient of x that is (b²/4a):

x² + (b/a)x + (b²/4a)= -c/a + (b²/4a)

(x + b/2a)² = (b²-4c/4a)

[tex]x=-\frac{b}{2a}\pm\sqrt{\frac{b^2-4c}{4a} }[/tex]

The solution to ax² + bx + c = 0  is [tex]x=-\frac{b}{2a}\pm\sqrt{\frac{b^2-4c}{4a} }[/tex]

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Final answer:

Next, Florence should complete the square on the quadratic equation by dividing by 'a' (if applicable), moving the constant term to the other side, and adding the square of half the coefficient of 'b' to both sides. This is a precursor step toward factoring and ultimately deriving the quadratic formula.

Explanation:

The step Florence should do next in deriving the quadratic formula from the equation ax² + bx + c = 0 is to complete the square. To do this, Florence first needs to divide the entire equation by a to normalize the coefficient of x² to 1, assuming a≠0. Then she should move the constant term to the other side of the equation to get x² + (b/a)x = -c/a. The next step is to add the square of half of the coefficient of x to both sides of the equation, which is (b/2a)². This process sets the stage for factoring the left side of the equation as a perfect square.

Once the perfect square is created, the equation can be written in the form (x + b/2a)² = (b² - 4ac)/4a². Then, by taking the square root of both sides and isolating x, Florence will arrive at the quadratic formula: x equals minus ‘b’, plus-or-minus the square root of ‘b’ squared minus four ‘a’ ‘c’, all over two 'a'.

y = - [tex]\frac{3}{4}[/tex] x  - [tex]\frac{7}{2}[/tex]

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

to calculate m use the gradient formula

m = ( y₂ - y₁ ) / (x₂ - x₁ )

with (x₁, y₁ ) = (- 6, 1 ) and (x₂, y₂ ) = (2, - 5 )

m = [tex]\frac{-5-1}{2+6}[/tex] = [tex]\frac{-6}{8}[/tex] = - [tex]\frac{3}{4}[/tex]

the partial equation is

y = - [tex]\frac{3}{4}[/tex] x + c

to find c substitute either of the 2 given points into the partial equation

using (- 6, 1 ), then

1 = [tex]\frac{9}{2}[/tex] + c ⇒ c = 1 - [tex]\frac{9}{2}[/tex] = - [tex]\frac{7}{2}[/tex]

y = - [tex]\frac{3}{4}[/tex] x - [tex]\frac{7}{2}[/tex] ← equation of line

Answer:

The correct option is:  B

Step-by-step explanation:

The x-axis is labeled Time driving, and the y-axis is labeled Distance driven.

Segment B is a horizontal line from x equals 0.5 until 1.75.

So, the value of [tex]y[/tex] is constant in the interval [tex]x=0.5[/tex] to [tex]x=1.75[/tex] is 0.

Thus, the distance traveled in the time interval 0.5 to 1.75 will be 0, which means the car was in the same position in that time interval.

So, the interval B on the graph indicates she is stuck in a traffic jam.

Answer:

B

Step-by-step explanation:

Answer:

The required number of bowls are 5.  

Step-by-step explanation:

Given : A sugar bowl holds 237 grams. You have a one kilogram bag of sugar.

To find : Estimate how many bowls of sugar you can fill from the bag?

Solution :

1 bowl can hold 237 gram of sugar.

We have, 1 kg of sugar or 1000 gram of sugar.                

According to question,

237 gram of sugar can hold in 1 bowl.

So, 1 gram of sugar can hold in [tex]\frac{1}{237}[/tex] bowl.

1000 gram of sugar can hold in [tex]\frac{1000}{237}[/tex] bowl.      

1000 gram of sugar can hold in [tex]4.21[/tex] bowl.  

Which means, The required number of bowls are 5.

As 4 bowls have [tex]237\times 4=948[/tex] grams of sugar.

Sugar left is 1000-948=52 grams

That 52 grams is filled into 5th bowl.                  

answer is (-7,2)

y = -x -5

y= x+9

Both equations have y on the left hand side

So we equate both equations

We replace -x-5 for  y in the second equation

-x -5 = x+9

Subtract x on both sides

-2x -5 = 9

Now add 5 on both sides

-2x = 14

Divide by -2 from both sides

x = -7

Now plug in -7 for x in the first equation

y = -x -5

y = -(-7)  -5= 7-5 = 2

So answer is (-7,2)

1. You have the following system of equations:

[tex]\left \{ {{y=-x-5} \atop {y=x+9}} \right.[/tex]

2. Therefore, you have that [tex]y=y[/tex], then:

[tex]-x-5=x+9[/tex]

3. Solve for [tex]x[/tex]:

[tex]-5-9=x+x\\2x=-14\\x=-7[/tex]

3. Now, substitute this value into one the original equations:

[tex]y=x+9\\y=-7+9\\y=2[/tex]

The answer is: (-7,2)

Step 1. Expand

q + 12 - 2q + 44 > 0

Step 2. Simplify q + 12 - 2q + 44 to -q + 56

-q + 56 > 0

Step 3. Regroup terms

56 - q > 0

Step 4. Subtract 56 from both sides

-q > -56

Step 5. Multiply both sides by -1

q < 56

To solve the inequality q + 12 – 2(q – 22) > 0, distribute the negative sign, isolate the variable, and divide by -1 with a flipped inequality sign to find q < 56.

To solve the inequality q + 12 – 2(q – 22) > 0, we can start by distributing the negative sign to the terms inside the parentheses: q + 12 - 2q + 44 > 0. Simplifying, we have -q + 12 + 44 > 0, which becomes -q + 56 > 0. Next, we isolate the variable by subtracting 56 from both sides: -q > -56. Finally, we divide both sides by -1, remembering to flip the inequality sign when dividing by a negative number: q < 56.

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Answer:

5

Step-by-step explanation:

The total number of bags of mixed nuts with Carmela are 5.

Equation modelling is the process of writing a mathematical verbal expression in the form of a mathematical expression for correct analysis, observations and results of the given problem.

Given is Carmela who mixes 3/4 kilogram of walnuts, 1/2 kilogram of almonds, and 1/4 kilogram of pecans together. She divides the mixed nuts into 3/10 kilogram bags.

We can write the equation as -

(3/4 + 1/2 + 1/4) x 1000 = (3/10) x 1000 x n

1.5 x 1000 = 3 x 100 x n

1.5 x 10 = 3n

15 = 3n

n = 5

Therefore, the total number of bags of mixed nuts with Carmela are 5.

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To find the height of the original oil tank from the sample, you set up a proportion using the scale given. The calculation shows that the original tank is 25 meters tall, corresponding to answer A.

The question is asking us to find the height of the original oil tank given that the sample tank is [tex]\frac{1}{25}th[/tex] the size of the original and that the sample is 1 meter tall. Since the sample is a scaled-down version of the original, we can set up a proportion to find the original height.

To set up the proportion, we assume the scale is such that 1 meter on the sample represents 25 meters on the original. Therefore, we can write the proportion as:

[tex]\frac{1\ meter}{ x\ meters} = \frac{1}{25}[/tex]

We can then cross-multiply to solve for x:

1 * x = 25 * 1

x = 25 meters

Thus, the height of the original oil container tank is 25 meters, which corresponds to answer choice A.

The question centers on mathematics, where we calculate the number of cookies Tovah needs and explore probability and trading scenarios related to assorted cookies and resource allocation.

The subject of this question is Mathematics, particularly focusing on basic arithmetic, probability, and problem-solving. Tovah needs a total of 700 cookies and already has 300 cookies. To determine how many more cookies Tovah needs, we can subtract the number she already has from the total number needed: 700 - 300 = 400. Hence, she needs to obtain 400 more cookies.

Additionally, when discussing assorted cookies, we can delve into probability and combinatorics. For instance, if we consider a scenario with cookies containing chocolate, nuts, or both, we can calculate the probability that a certain combination is selected. This involves understanding percentages, probability trees, and independence of events.

Lastly, we can explore resource allocation and trading as seen in examples where individuals barter items like chocolate bars or Halloween candy. This introduces concepts like gains from trade and distribution of resources, which are essential to economic mathematics.

A Pythagorean triplet is a set of 3 positive integer numbers which may be the sides of a right triangle, i.e. they meet the Pythagorean theorem c² = a² + b².

You can check that the numbers on your table are Pythagorean triplets by substituting them in the Pythagorean equation:

Now, lets look for the pattern:

x-value     Pythagorean

                 triple    

3               (6, 8, 10)              6/2 = 3

                                            3² - 1 = 9 - 1 = 8

                                            3² + 1 = 9 + 1 = 10

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

4               (8, 15, 17)              8/2 = 4

                                             4² - 1 = 16 - 1 = 15

                                             4² + 1 = 16 + 1 = 17

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

5               (10, 24, 26)              10/2 = 5

                                                 5² - 1 = 25 - 1 = 24

                                                 24² + 1   = 25 + 1 = 26

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

6               (12, 35, 37)              12/2 = 6

                                                6² - 1 = 36 - 1 = 35

                                               6² + 1   = 36 + 1 = 37

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

From which you find the pattern: the first number is 2x, the second number is x² - 1, and the third number is x² + 1

                                            ⇒ (2x)² + (x² - 1)² = (x² + 1)², or

                                                (x² - 1)² + (2x)² = (x² + 1)².

Other example of a Pythagorean triple is (3, 4, 5). You migth think that it does not follow the pattern, but if you do x = 2, you end with:

Hence, (3, 4, 5) also follows the pattern.

Only  right triangles with non-integer sides do not form Pythagorean triples.

Of course you may proof that (x² - 1)² + (2x)² = (x² + 1)² is an identity (always true):

Left hand side: (x⁴ - 2x² + 1) + 4x² = x⁴ + 2x² + 1

Right hand side: x⁴ + 2x² + 1

∴ The equation is always true.

At the end, the pattern is true for any Pythagorean triplet, but a more formal proof is beyond the scope of this question.

ANSWER

The correct answers are option B and C

EXPLANATION

A linear function with slope [tex]m=1[/tex] and y - intercept [tex]0[/tex] has equation, [tex]f(x)=x[/tex]

Option A

[tex]f(-1)=-1[/tex]

[tex]g(-1)=(-1)^2=1[/tex]

Therefore [tex]f(-1) \ne g(-1)[/tex]

Option B

[tex]f(1)=1[/tex]

[tex]g(1)=(1)^2=1[/tex]

Therefore [tex]f(1) = g(1)[/tex]

Option C

[tex]f(0.5)=0.5[/tex]

[tex]g(0.5)=(0.5)^2=0.25[/tex]

Therefore [tex]f(x) > g(x)[/tex]

on [tex](0,1)[/tex] See graph also.

Option D

At y-intercept, [tex]x=0[/tex]

This implies that,

[tex]f(0)=0[/tex]

[tex]g(0)=(0)^2=0[/tex]

Therefore g(x) does not have a greater y-intercept.

The function f(x) with a slope of 1 and a y-intercept of 0 is compared with the quadratic function g(x)=x^2. f(1) is equal to g(1) and f(x) is greater than g(x) on the interval (0,1).

The function g(x)=x^2 is a quadratic function, and the function f(x) with a slope of 1 and a y-intercept of 0 is a linear function. Let's evaluate the given statements:

In summary, the true statements are: f(1) is equal to g(1), and f(x) is greater than g(x) on the interval (0,1).

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This isn't even the full question

Answer:

42.7716% of the variation in the number of bags of trash collected can be explained by differences in the number of volunteers.

Step-by-step explanation:

The correlation coefficient (r) between the number of volunteers x and the number of bags of trash collected y is 0.654

For finding the percent of the variation in one variable explained by the other variable, we just need to take square of the correlation coefficient.

Here,  [tex]r=0.654[/tex]

So,  [tex]r^2 = (0.654)^2 = 0.427716[/tex]

Now, for converting it into percentage, we will multiply it by 100.

So, [tex]0.427716*100\% = 42.7716 \%[/tex]

Thus, 42.7716% of the variation in the number of bags of trash collected can be explained by differences in the number of volunteers.

Final answer:

About 42.8% of the variation in the bags of trash collected can be attributed to the number of volunteers, as indicated by the squared correlation coefficient (0.654²).

Explanation:

The correlation coefficient (r) measures the strength and direction of a linear relationship between two variables. When squared to calculate the coefficient of determination (r²), it represents the proportion of the variance in the dependent variable that is predictable from the independent variable. In the given scenario, with a correlation coefficient of 0.654, the coefficient of determination would be 0.654², which calculates to approximately 42.8%. This percentage indicates that about 42.8% of the variation in the number of bags of trash collected (y) can be explained by the variation in the number of volunteers (x).

I think that the answer is the 4th triangle hope this helped.

That would be triangle DEF and the last one (M - -)

The corresponding sides  compared with triangle ABC are not in same ratio

6 is your answer good luck

Answer:

x ≤ 8

Step-by-step explanation:

Will first used 16 of the 144 ounces for his own water bottle.  This leaves 144-16 = 128 ounces.

Dividing that equally amount 16 plastic cups, each cup would have at most

128/16 = 8 ounces.

This means each one would have x ≤ 8 ounces.

The correct inequality is 8 > 6 .

To solve this problem, we need to figure out how many ounces of water Will could pour into each cup. He started with a cooler filled with 144 ounces of water.

Then, he took out 16 ounces for his own water bottle. So, he had 144 - 16 = 128 ounces left.

Since he poured the same amount of water into each of the 16 cups, we can divide the total remaining ounces by the number of cups to find out how many ounces each cup contains:

[tex]\[ \frac{128}{16} = 8 \][/tex]

So, Will could have poured 8 ounces of water into each cup.

Therefore, the correct inequality is:

8 > 6

I think the answer is 2.7 hours

Using the formula Time = Distance ÷ Speed, we find that the driver would spend approximately 1.67 hours on the first 100 miles and 0.91 hours on the last 50 miles. Adding these two times gives a total travel time of approximately 2.58 hours.

The first thing you need to do is calculate the time spent in each part of the trip. To calculate time, we use the formula Time = Distance ÷ Speed. For the first 100 miles at 60 mi/h, it takes: Time = 100 miles ÷ 60 mi/h = 1.67 hours. Moving on to the next 50 miles at 55 mi/h, it takes: Time = 50 miles ÷ 55 mi/h = 0.91 hours.

Adding these two times together gives us the total time for the trip: 1.67 hours + 0.91 hours = 2.58 hours. So, the driver would take approximately 2.58 hours to reach his destination if he traveled 100 miles at 60 mi/h and the remaining 50 miles at 55 mi/h.

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I think it's degrees or like something to do with translating?!

It is function composition. If you have one function f(x), and another function g(x), then we can create a new function named g∘f (read as: "gg composed with ff") that is defined as(g∘f)(x)=g(f(x))(g∘f)(x)=g(f(x))For example, if f(x)=x+1f(x)=x+1, and g(x)=2x−1g(x)=2x−1, then(g∘f)(x)=g(f(x))=g(x+1)=2(x+1)−1=2x+1

5/22 is the fraction of girls to boys and as a decimal it is 2.3 rounded and not rounded is 2.27 repeated.

Answer:

Unit rate of driving (speed) is 50 miles per hour

Step-by-step explanation:

Michael drove 350 miles in 7hrs

Speed = Distance ÷ time

Speed= 350 miles ÷ 7 hours = 50 miles per hour.

The difference would be:

1,517.