1. Use the graph of the rational function
to complete the following statement.
As x → -3*, f(x) →
N

Answer:

It will take 13 years to Ishaan age will be 5 times William age.

Step-by-step explanation:

Given : Ishaan is 72 years old and William is 4 years old.

To find : How many years will it take until Ishaan is only 5 times as old as William?

Solution :

Let x be years since today.

According to question,

Ishaan's age is I=72+x

William's age is W=4+x

Now, We want the time for Ishaan age will be 5 times William's age:

i.e. [tex]I=5W[/tex]

[tex]72+x=5(4+x)[/tex]

[tex]72+x=20+5x[/tex]

[tex]72-20=5x-x[/tex]

[tex]52=4x[/tex]

[tex]x=13[/tex]

It will take 13 years to Ishaan age will be 5 times William age.

By this time William will be 17 years old and Ishaan will be 85 years old.

It will take 13 years to Ishaan age will be 5 times William age.

Given that,  

Based on the above information, the calculation is as follows:

Let x  be years since today.

So,  

Ishaan's age is I=72+x

William's age is W=4+x

Now

I = 5W

72 +x = 5(4 + x)

72 + x = 20 + 5x

52 = 4x

x = 13

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Answer:

Therefore the lengths of the opposite side pairs, AB and BC are 12 units and 6 units .

Step-by-step explanation:

Given:

[] ABCD is a Parallelogram.

∴ pairs of opposite sides are congruent

∴ AB = DC and

  BC = AD

To Find:

Length of AB = ?

Length of BC = ?

Solution:

[] ABCD is a Parallelogram. .............Given

∴ pairs of opposite sides are congruent

∴ AB = DC        and          BC = AD

On substituting the values we get

For 'x' i.e AB = DC  

[tex]4x=x+9\\\\4x-x=9\\\\3x=9\\\\x=\frac{9}{3}\\ \\x=3[/tex]

For 'y' i.e BC = AD

[tex]2y=6\\\\y=\frac{6}{2} \\\\y=3\\[/tex]

Now substituting 'x' and 'y' in AB and BC we get,

[tex]Length\ AB=4\times 3 =12\ units\\\\Length\ BC=2\times 3 =6\ units[/tex]

Therefore the lengths of the opposite side pairs, AB and BC are 12 units and 6 units .

C) 12, 6

Set opposite sides equal to each other and solve for x or y.

AB = DC → 4x = x + 9 → 3x = 9 → x = 3

So, AB = DC = 12

And,

AD = BC → 6 = 2y → y = 3

So, AD = BC = 6

Answer:

[tex]g(n)=g(n/2)+n/2[/tex]

Step-by-step explanation:

[tex](2k+1)!=1*3*5*7*...*(2k-1)*(2k+1)*2*4*6*8*...*(2k-2)*(2k)[/tex]

Notice that the odd factors of above equality don't contribute to the largest power of 2 that divides (2k+1)!

Therefore, we can conclude that [tex]g(2n+1)=g(2n)[/tex].

[tex](2k)!=1*3*5*7*...*(2k-3)*(2k-1)*2*4*6*8*...*(2k-2)*(2k)=\\= (1*3*5*...*(2k-3)*(2k-1))*((2*1)*(2*2)*(2*3)*...*(2*(k-1))*(2*k))=\\= (1*3*5*...*(2k-3)*(2k-1))*2^k*k![/tex]

Notice that the odd factors of above equality don't contribute to the largest power of 2 that divides (2k)!

Hence, [tex]g(2k)-g(k)=k[/tex]

So for [tex]k = n/2[/tex],

[tex]g(n)-g(n/2)=n/2\\g(n)=g(n/2)+n/2[/tex]

Answer:

The Function formula is given by [tex]c = 150 -8s[/tex].

Step-by-step explanation:

Given:

Number of candies eaten each time he has snacks = 8

Number of snacks ate = 18

Now Number of candies eaten will be equal to number of snacks he ate multiplied by number of candies eaten each time he ate snacks

Hence. Number of candies he ate in 18 snacks = 18* 8 = 144 candies

Also given that 6 candies are remaining after eating 18 snacks;

Total Number of candies at the beginning will be = Number of candies he ate in 18 snacks + Number of candies remaining after 18 snacks

Total Number of candies at the beginning will be = 144 + 6 = 150 candies

Now, let The number of pieces of candy remaining in the bag be 'c'

Also Let the number of snacks Tyrone eats be 's'

Since it has been given that in number of candies in each snack is 8 candies.

Therefore,In 's' snacks , candies will be eaten by him = [tex]8s[/tex]

Total number of candies = 150

Now Remaining candies 'c' is equal to Total number of candies minus number of candies eaten in 's' snacks.

framing in equation form we get;

Remaining candies [tex]c = 150 -8s[/tex]

Hence the Function formula is given by [tex]c = 150 -8s[/tex].

The equation that represents the scenario is:

Option 1: n + (n + 1) + (n + 2) = 9

Step-by-step explanation:

We have to convert the given statement in an equation. As the variable used in the choices is n, so we will also use n.

Let n be an integer

then

The next integer will be n+1

and next integer to n+1 will be n+1+1 = n+2

So,

putting it in equation form

[tex]n + (n+1) + (n+2) = 9[/tex]

Hence,

The equation that represents the scenario is:

Option 1: n + (n + 1) + (n + 2) = 9

Keywords: Variables, equations

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Answer:

N+(N+1)+(N+2)=9

Step-by-step explanation:

just did it on edge

Yee Yee MERICA

Answer: There is decrease in percent of 26.47% from 2001 to 2008.

Step-by-step explanation:

Since we have given that

Number of runs that chipper had batted in 2001 = 102

Number of runs that chipper had batted in 2008 = 75

So, the year gap = 7 years

Difference in runs = 102- 75 =27

So, the percent change in runs batted from 2001 to 2008 is given by

[tex]\dfrac{27}{102}\times 100\\\\=\dfrac{2700}{102}\\\\=26.47\%[/tex]

Hence, there is decrease in percent of 26.47% from 2001 to 2008.

Yes, you are correct.

From the image, we can see 3 triangles, the large triangle and the two triangles that make the larger triangle.

All 3 of these triangles are similar since they're angle measurements are the same (can be proven by geometry).

We can gather the following information from the given information.

We know the hypotenuse to corresponding leg ratio must be equal since these two triangles are similar. Thus, we have the equation:

12/x = x/9

Using the cross product property gives us:

x² = 9 × 12

x² = 108

Taking the square root of both sides gives us:

x = √108

= 6 √3

Thus, your answer is correct. Great job!

Let me know if you need any clarifications, thanks!

Answer:

15

sweatshirts were large and red.

Step-by-step explanation:

Hope this helps

Answer:the total number of sweatshirts in the shipment that were both large and red is 15

Step-by-step explanation:

The total number of sweatshirts received by Dans sporting goods is 120. Half of the sweatshirts were size large. This means that the number of sweatshirts that were size large is 120/2 = 60.

One-fourth of the large sweatshirts were red. This means that the number of large sweatshirts that were also red would be

1/4×60 = 15

Final answer:

The average rate of change of the function on the interval from x = 0 to x = 5 is 1/8.

Explanation:

The average rate of change of a function on an interval is calculated by finding the slope of the secant line between two points on the interval. In this case, the interval is from x = 0 to x = 5.

The function is [tex]f(x) = (1/2^(3))x.[/tex]

To find the average rate of change, we need to calculate the slope between the points (0, f(0)) and (5, f(5)).

Using the formula for slope, [tex]m = (y2 - y1) / (x2 - x1)[/tex], we substitute the values from the given function and interval:

[tex]m = ((1/2^(3))(5) - (1/2^(3))(0)) / (5 - 0)[/tex]

m = (5/8 - 0) / 5 = 5/40 = 1/8

Therefore, the average rate of change of the function on the interval from x = 0 to x = 5 is 1/8.

Since in the question, there are no ordered pairs given so you can consider the below explanation to check which ordered pairs satisfies the inequality 3x-4y>12

Step-by-step explanation:

We need to identify which ordered pair(x,y) satisfies the inequality 3x-4y>12

The ordered pair (x,y) which satisfies the inequality will be the values of x and y that satisfies the inequality given 3x-4y>12

So, if x= 2 and y = 2 then the inequality will be:

3x-4y>12

3(2)-4(2)>12

6-8>12

-2>12 is false because -2 is not greater than 12

So, ordered pair (2,2) doesn't satisfy the inequality 3x-4y>12

Now if x = 10 and y = 2 then the inequality will be:

3x-4y>12

3(10)-4(2)>12

30-8>12

22>12 is true because 22 is greater than 12.

So, ordered pair (10,2) satisfies the inequality 3x-4y>12

Since in the question, there are no ordered pairs given so you can consider the above explanation to check which ordered pairs satisfies the inequality 3x-4y>12

Keywords: Solving inequalities

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To find the order pair (x,y) satisfying the inequality 3x - 4y > 12, substitute potential order pairs for x and y and evaluate. If the result is greater than 12, then the order pair satisfies the inequality.

The inequality given is 3x - 4y > 12. To find the order pair (x,y) that satisfies this inequality, let's plug in some potential solutions and see which one works.

For example, let's try the order pair (6, 1): 3(6) - 4(1) = 18 - 4 = 14, which is greater than 12, so (6,1) does satisfy the inequality.

But if we try the order pair (2, 3): 3(2) - 4(3) = 6 - 12 = -6, which is not greater than 12, so (2,3) does not satisfy the inequality.

Through these examples, you can see how we can find potential solutions to the inequality by substituting different order pairs for x and y.

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To find the standard form of the circle's equation, one must determine the center and radius by solving a system of equations derived from the circle passing through the points (0,0), (2.8,0), and (5,2).

The question asks to write the standard form of the equation of the circle that passes through three given points, namely the origin, (2.8,0), and (5,2). The standard form for the equation of a circle is (x-h)² + (y-k)² = r², where (h,k) is the center of the circle, and r is the radius.

To find the equation, we need to determine the center and the radius. Since the circle passes through the origin, we can establish a system of equations based on the other two points which the circle passes through. We would end up with two equations:

By solving this system of equations, we can calculate the values of h, k, and r, and then plug these into the standard form equation of a circle.

The required cost of a computer Suzie will pay is $545.69.

The percentage is the ratio of the composition of matter to the overall composition of matter multiplied by 100.

Here,
The original price of a computer is $599.99.
Suzie has a 15%-off coupon for the computer.
Discounted price = (1 - 0.15)×599.99

Discounted price = 0.85×599.99

A sales tax of 7% will be added to the sale price of the computer.
Price of the computer after taxes = (1 + 0.07)0.85×599.99
                                                   = $545.69

Thus, the required cost of a computer Suzie will pay is $545.69.

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Final answer:

To find how much Suzie will pay for a computer with a 15% discount and a 7% sales tax, first calculate the discount, then subtract it from the original price, calculate the sales tax on the sale price, and add it to the sale price. The total payment is the original price reduced by the discount and increased by the sales tax, rounded to the nearest cent would be approximately $545.69.

Explanation:

Calculation of Total Payment for the Computer with a Discount and Sales Tax

To calculate how much Suzie will pay for the computer after applying a 15%-off coupon and adding a 7% sales tax, you will need to perform two main calculations. First, calculate the discount amount by converting the percentage to a decimal and multiplying it by the original price. Second, calculate the sales tax on the discounted price by again converting the percentage to a decimal and multiplying it. Finally, add the sales tax to the discounted price to arrive at the total cost.

Here's a step-by-step breakdown of the calculations:

Add the sales tax to the sale price to find the total cost: $509.9915 + $35.699405 = $545.690905.

Therefore, the expression to calculate Suzie's total payment for the computer is ($599.99  times (1 - 0.15))  times (1 + 0.07). When actually performing the calculation, the total payment, rounded to the nearest cent, would be approximately $545.69.

Answer:

In 9 days will the second warehouse have 1.5 times more coal than the first one

Step-by-step explanation:

The coal left in warehouses after x days can be found using the equation:

Let the second warehouse have 1.5 times more coal than the first one after n days then  

Answer:

Beneficial terms of trade or trading price between two parties are opportunity costs lower than the cost to manufacture them locally which benefits both parties. They must be less enough to cover the freight charges or additional service charges that may arise.

========================================

Work Shown:

Let P and Q be two concentric circles

P = area of circle with radius 19

P = pi*r^2

P = pi*19^2

P = 361pi

-------

Q = area of circle with radius 29

Q = pi*r^2

Q = pi*29^2

Q = 841pi

-------

R = area of shaded region between circle P and circle Q

R = region outside circle P, but inside circle Q

R = Q - P

R = 841pi - 361pi

R = 480pi

------

S = area of circle with area equal to region R's area

S = 480pi

S = pi*r^2

pi*r^2 = 480pi

r^2 = 480

r = sqrt(480)

r = sqrt(16*30)

r = sqrt(16)*sqrt(30)

r = 4*sqrt(30)

Answer:

x = -0.5.

Step-by-step explanation:

x^2 = 16^x

One way to do this is to use 2 graphs  y = x^2 and y = 16^x  .

The solution  is where the 2 graphs intersect.

The answer is x = -0.5.

Answer:

The probability of getting a four of a kind is 29/23030 = 0.00126.  

Step-by-step explanation:

The poker deck has 52 cards, divided in 4 suits, and each suit has 13 different denominations.

In this exercise you are given that the 2 cards you are dealt have different denominations, lets call them A and B, so in order to make a 4 of a kind, you need to get from the 5 remaining community cards, 3 cards that have the denomination of one of A or B.

You cant get a four of a kind with two different denominations at the same time, because you need to reserve the majority of the community cards for just one of them. So each four of a kind is a disjoint event from the others.

Recall that, apart from the obvius four of a kind that you can get with A and B, you can still get four of a kind with any other denomination, because 4 of the community cards can still have the same denomitation.

Lets first calculate the probability to get a four of a kind with A. With B it is the same computation and the result will, therefore, be the same.

First, note that the order in which the cards are given doesnt matter, so we can calculate the probability of getting triple A in the flop (in other words, the first 3 given cards), and then multiply by all possible permutations.

Since you start with one A in your hand, there are 3 in the deck of 50 cards (remove the two cards given to you). The probability of drawing the first A is 3/50. The probability to draw the second A once you drew the first one is 2/49 (because you removed another A from the deck). The probability for the third A to be drawn is 1/48. This gives us a probability of

3/50 * 2/49 * 1/48 = 1/19600

of drawing the 3 A in the first 3 given cards. The other 2 cards can be anything.

To draw 3 aces in 5 cards we need to multiply be the total amount of ways to put 3 cards in a set of 5. That number is equivalent to the number of ways to pick places for the three A ignoring the order for them. That is, [tex] {5 \choose 3} = 10 . [/tex] So, in total, the probability to get a four of the kind A is

10 * 1/19600 = 1/1960

Similarly, the probability to get a four of a kind with B is 1/1960.

To get a four of a kind with any other kind, we need first to specify the kind; we have 13-2 = 11 possibilities to choose (we substract the 2 kinds A and B). Then we need to specify how the cards of that kind will be placed among the community cards. We have as many possibilities as the total amount of ways to pick where the other card will be, that is, 5 possibilities. Hence, we have 55 possibilities to make a four of a kind with a kind different than A or B.

We need to calculate now what is the probability of getting 4 cards of a specific kind in a specific way, for example, the first four cards. That probability is

4/50 * 3/49 * 2/48 * 1/47 = 1/230300

because we start with 50 cards on the deck with 4 cards of them being of the kind we need, and we remove them one by one.

As a result, the probability of getting a four of a kind with a different kind than A and B is 55/230300 = 11/46060.

By summing disjoint cases, We conclude that that the probability to get a four of a kind is

11/46060 + 1/1960 + 1/1960 = 29/23030 = 0.00126  

Answer:

50 miles per hour

Step-by-step explanation:

Given:

Distance planned to travel each day is, [tex]D=500\ mi[/tex]

Hours of driving for covering the above distance is, [tex]t=10\ h[/tex]

The question asks to find the number of miles per hour Anderson plan to drive.

In other words, we need to calculate the average speed with which Anderson has to drive in order to cover 500 miles in 10 hours daily.

Therefore, the formula to calculate speed when distance covered and time taken are known is given as:

Average speed = [tex]\frac{Distance}{Time}[/tex]

Average speed = [tex]\frac{D}{t}[/tex]

Plug in 500 miles for 'D', 10 h for 't' and solve for speed. This gives,

[tex]\textrm{Average speed}=\frac{500}{10}\ mi/h\\\textrm{Average speed}=50\ mi/h[/tex]

Therefore, Anderson need to drive at an average of 50 miles per hour to cover the distance of 500 miles in 10 hours.

Answer:D

Step-by-step explanation:

Given

There are 52 cards out which we have to choose 3 cards

and starting 2 will have same denomination

For first card we can withdraw any card of any denomination and for second card we have to choose card of same denomination

There are total of 4 cards of same denomination and after 1 st card is drawn we need to choose 1 card out of  3 card

Probability[tex]=\frac{^3C_1}{^{51}C_1}[/tex]

[tex]=\frac{3}{51}[/tex]  

Answer:

V (max)  = 2 ft³

and x side of the base is  x  =  0,5 feet

Step-by-step explanation:  See annex ( two different cubes)

We have a square piece of cardboard of  3 inches wide

Let  x be lenght of side to cut in each corner

Then the base of the (future cube) is    3  -  2x,  and the area is  

( 3 - 2x )²

And  the height   is x  Then volume of the cube as a function of x is:

V(x)  =  ( 3 - 2x )² *x   or       V(x)  =    ( 9 + 4x² - 12x )*x

V(x)  =  4x³  -  12x²  + 9x

Taking derivatives on both sides of the equation

V´(x)  =  12x² - 24x  + 9

V´(x)  =  0       12x² - 24x  + 9  =  0   simplifying   4x² - 8x  + 3  =  0

Second degree equation solving for x

x₁,₂ =  [ 24 ± √( 576) - 432  /24

x₁,₂ =  [24 ±√144 ]/24

x₁,₂ = ( 24 ± 12) /24         x₁  =  1.5 feet        x₂  = 0,5 feet

Of these two values we have to dismiss x₁  because if  x = 1.5 we don´t have a cube ( 0 height )

Then we take x  = 0,5  feet

And

V (max)  =  (2)²*0,5   =  4*0,5

V (max)  = 2 ft³

=========================

Explanation:

Let's label each pair of outcomes as an ordered pair (x,y)

x = first spinner outcome

y = second spinner outcome

We have these 12 different possible combos

(1,3),   (1,4),   (1,5),   (1,6)

(2,3),  (2,4),  (2,5),  (2,6)

(3,3),  (3,4),   (3,5),  (3,6)

Focus on the third column where y = 5 for each (x,y) pair. Of this column only (2,5) has x be an even number. The other results in that column have x be odd.

Basically (1,5) is the only possible outcome if we want the first spinner to lan on an even number, and the second spinner to land on 5.

In these conditions, there is only one outcome where the first spinner does not show an odd number and the second spinner shows a 5.

The problem involves two spinners. The first spinner has three sections labeled as 1, 2, and 3, while the second spinner has four sections labeled as 3, 4, 5, and 6. We are looking for cases where the result from the first spinner is not an odd number and the result from the second spinner is 5.

On the first spinner, the only non-odd digit is '2'. On the second spinner, the number '5' is what we seek. Therefore only one outcome will satisfy both conditions - when the first spinner lands on 2 and the second spinner on 5.

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Answer: c. ​10

Step-by-step explanation:

Given : An independent-measures research study compares three treatment conditions using a sample of n = 5 in each treatment.

For this study, the three sample means are, M1 = 1, M2 = 2, and M3 = 3.

Grand mean = [tex]\overline{X}=\dfrac{M1+M2+M3}{3}[/tex] [All sample have equal elements]

[tex]=\dfrac{1+2+3}{3}=2[/tex]

For ANOVA,

[tex]SS_{between}=n\sum(M_i-\overline{X})^2[/tex] , where i= 1, 2 , 3

[tex]SS_{between}=5((1-2)^2+(2-2)^2+((3-2)^2)[/tex]

[tex]SS_{between}=5((-1)^2+(0)^2+((1)^2)= 5(1+1)=10[/tex]

Hence,the value would be obtained for [tex]SS_{between}=10[/tex] .

hence, the correct answer = c. ​10

Final answer:

For an independent-measures research study comparing three treatment conditions with n = 5 in each treatment, the SSbetween value calculated using ANOVA is 10.

Explanation:

To calculate the SSbetween (Sum of Squares Between) for an independent-measures research study using a one-way ANOVA, we first need the mean of the group means, also known as the grand mean (GM), and then use the formula to find SSbetween: SSbetween = ∑ n(M - GM)², where n is the number of subjects in each group, M is the group mean, and GM is the grand mean.

In this case, with three sample means of M1 = 1, M2 = 2, and M3 = 3, each with n = 5, the grand mean is GM = (1+2+3) / 3 = 2. Using the formula, we have:

SSbetween = 5(1 - 2)² + 5(2 - 2)² + 5(3 - 2)²
SSbetween = 5(-1)² + 5(0)² + 5(1)²
SSbetween = 5(1) + 5(0) + 5(1)
SSbetween = 5 + 0 + 5
SSbetween = 10.

Therefore, the value obtained for SSbetween in this study would be 10.

Answer:

  4π -8

Step-by-step explanation:

Consider the attached diagram. The purple areas are those contained within 1 or 3 circles. The red areas are fully equivalent to the purple areas. The area of interest is the sum of the purple and red areas, or twice the red area.

Twice the red area is the area of the circle of radius 2 less the area of a square of diagonal 4.

For a circle of radius 2, its area is ...

  A = πr² = π·2² = 4π

For a square of diagonal 4, its area is ...

  A = (1/2)d² = (1/2)4² = 8

The area of interest is the difference of these, ...

  area of interest = circle area - square area

  = 4π -8

Answer:

The answer to your question is   -3x² + 29x -150

Step-by-step explanation:

                                -3x² + 29x -150

  x³ + 6x² - 3x - 5     -3x⁵ + 11x⁴ + 33x³ - 26x² - 36x - 6

                                +3x⁵ +18x⁴ -   9x³  - 15x²

                                   0     29x⁴ +24x³ - 41x² - 36x

                                         -29x⁴ - 174x³ +87x²+145x

                                            0     -150x³ +46x² +109x - 6

                                                  +150x³ +900x²-450x -750

                                                       0       946x² -341x -756

Quotient =   -3x² + 29x -150

Remainder = 946x² -341x -756

Answer: the number of adult tickets sold is 150

the number of student tickets sold is 210

Step-by-step explanation:

Let x represent the number of adult tickets sold.

Let y represent the number of student tickets sold.

The adult tickets cost $12 each, and the student tickets cost $8 each. The total cost of tickets sold $3,480. This means that

12x + 8y = 3480 - - - - - - - - - -1

The total number of tickets sold is 360. This means that

x + y = 360 - - - - - - - - - -2

Multiplying equation 1 by 1 and equation 2 by 12, it becomes

12x + 8y = 3480

12x + 12y = 4320

Subtracting

-4y = -840

y = -840/-4 = 210

Substituting y = 210 into equation 2, it becomes

x = 360 - y = 360 - 210

x = 150

To find the number of adult and student tickets sold, we can set up a system of equations and solve for the variables. Using the given information, we find that 150 adult tickets and 210 student tickets were sold.

To solve this problem, we can set up a system of equations. Let's let x represent the number of adult tickets and y represent the number of student tickets. From the given information, we can set up the following equations:

x + y = 360

12x + 8y = 3480

We can solve this system using the substitution method or the elimination method. Let's solve it using the substitution method:

From the first equation, we can express x in terms of y: x = 360 - y

Substituting this expression for x in the second equation, we get: 12(360 - y) + 8y = 3480

Simplifying the equation, we get: 4320 - 12y + 8y = 3480

Combining like terms, we have: -4y = -840

Dividing both sides by -4, we find: y = 210

Substituting this value of y back into the equation x + y = 360, we get: x + 210 = 360. Solving for x, we find: x = 150

Therefore, 150 adult tickets and 210 student tickets were sold.

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what's the answer choices

Answer:

A) 1525

B) 4636

C) 8479

Step-by-step explanation:

Lest say that The book had been sold x copies in the period of June 1990-June 1995. And we know that the book had been sold 152*x/100 copies in the period of June 1995–June 2000. By June 2000 in total 3843 book had been sold. Therefore number of the copies that sold in the period of June 1990-June 1995 is:

[tex]252*x/100=3843\\x=1525[/tex]

number of the copies that sold in the period of June 1995-June 2000 is:

[tex]152*x/100=2318[/tex]

For the period June 2000–June 2005 2318*2=4636 book had been sold.

Answer: [tex]B.\ g(x)=\frac{1}{4}(x+3)^3[/tex]

Step-by-step explanation:

Some transformations for a function f(x) are shown below:

1. If [tex]f(x-k)[/tex], the function is shifted right "k" units.

2. If [tex]f(+k)[/tex], the function is shifted left "k" units.

3. If [tex]bf(x)[/tex] and [tex]b>1[/tex], the function is stretched vertically by a factor of "b".

4. If [tex]bf(x)[/tex] and [tex]0<b<1[/tex], the function is compressed vertically by a factor of "b".

In this case, you know that the parent function f(x) is:

[tex]f(x)=x^3[/tex]

If the graph of the function g(x) is obtained by compressing vertically and shifting the function f(x) to the left, then:

[tex]g(x)=bf(x+k)[/tex] and [tex]0<b<1[/tex]

Based on this, you can identify that the following function could be the equation of the g(x):

[tex]g(x)=\frac{1}{4}(x+3)^3[/tex]

Answer:

See explanation below

Step-by-step explanation:

In fact, the interval has to be closed (just as [a,b], for b>a). Otherwise, this is not neccesarily true. For example, consider the function f(x) = -1/x defined on the open interval (0.1). f is continuous (quotient of continuous functions) but it does not have a minimum value: it decreases infinitely near zero.

To show this result on the interval [a,b], the idea is the following:

We can use a previous theorem. If f is continuous on [a,b], there exists some N>0 such that N≤f(x) (that is, f is bounded below). Now, we take the biggest N such that N≤f(x) for all x∈[a,b] (this is known as the greatest lower bound)

The number N is the candidate for the minimum value of f. Next, we have to show that there exists some p∈[a,b] such that f(p)=N. To do this, we must use the continuty of f on [a,b]. There are many ways to do it, and usually they require the epsilon-delta definition of continuity.

This is just a description of the ideas involved, but of course, a rigorous proof would need more technical details, depending on the theorems you are allowed to use.