DETERMINE IF THE SEQUENCES ARE ARITHMETIC, IF SOFIND THE COMMONDIFFERENCE, THE 52 TERM, AND EXPLICIT FORMULA

31, -69, -169, -269

the third term of the expansion of (11x+3y)⁶ is 798,720x³y³.we can use the binomial theorem to solve this .

what is  binomial theorem ?

The binomial theorem is a theorem in algebra that describes the algebraic expansion of powers of a binomial. A binomial is a polynomial with two terms, such as (a + b) or (x - y).

In the given question,

To find the third term of the expansion of (11x+3y)⁶, we can use the binomial theorem, which states that:

(a + b)ⁿ = C(n,0)aⁿb⁰ + C(n,1)*aⁿ⁻¹*b¹+ C(n,2)*aⁿ⁻¹*b² + ... + C(n,n)a⁰bⁿ

where C(n,k) denotes the binomial coefficient, which is the number of ways to choose k items from a set of n items.

In this case, we have a = 11x and b = 3y, and n = 6. We want to find the third term, which corresponds to k = 3 in the expansion. Therefore, we need to compute the binomial coefficient C(6,3), as well as the powers of 11x and 3y that appear in the third term.

The binomial coefficient C(6,3) can be computed as follows:

C(6,3) = 6! / (3! * (6-3)!) = 20

To find the powers of 11x and 3y that appear in the third term, we need to look at the general pattern in the expansion. Each term in the expansion consists of a product of powers of a and b, with the powers of a decreasing by 1 from n to 0, and the powers of b increasing by 1 from 0 to n. Therefore, the kth term in the expansion will have powers of a and b given by aⁿ⁻ᵇand bᵃ, respectively.

In the third term, we have k = 3, so we need to find the coefficient of a³ = a³ = (11x)³ and b³ = (3y)³. The third term is given by:

C(6,3)(11x)³(3y)³

Plugging in the values for C(6,3), (11x)³, and (3y)³, we get:

20*(11x)³*(3y)³ = 2011³x³3³y³ = 798,720x³y³

Therefore, the third term of the expansion of (11x+3y)⁶ is 798,720x³y³.

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The lower limit is 189.331 milligrams per one hundred milliliters, and the upper limit is 195.469 milligrams per one hundred milliliters.

To find the 90% confidence interval for the mean serum cholesterol level of all middle-aged women, we can use the formula:

CI = x' ± t(α/2, n-1) × (s/√n)

where:

x' is the sample mean serum cholesterol level

t(α/2, n-1) is the critical t-value for a two-tailed test with a 90% confidence level and n-1 degrees of freedom

s is the sample standard deviation of the serum cholesterol level

n is the sample size

Substituting the given values, we get:

CI = 192.4 ± t(0.05, 11) × (5.9/√12)

To find the critical t-value, we can use a t-distribution table or calculator. For a two-tailed test with 11 degrees of freedom and a 90% confidence level, the critical t-value is approximately 1.796.

Substituting this value and simplifying, we get:

CI = 192.4 ± 1.796 × 1.707

CI = 192.4 ± 3.069

Therefore, the 90% confidence interval for the mean serum cholesterol level of all middle-aged women is (189.331, 195.469).

This means that we can be 90% confident that the true mean serum cholesterol level for all middle-aged women falls within this interval.

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Here are the answers to the above questions:

To calculate the locator and percentile, we need to first arrange the scores in ascending order:

6, 8, 11, 12, 14, 17, 20, 34, 36, 38, 47, 48, 49

a) To calculate the locator for the 52nd percentile, we use the following formula:

Lp = (p/100) * (N + 1)

where Lp is the locator for the pth percentile, p is the percentile we are interested in, and N is the total number of scores.

For the 52nd percentile, p = 52 and N = 13. Substituting these values into the formula, we get:

L52 = (52/100) * (13 + 1) = 6.72

Therefore, the locator for the 52nd percentile is 6.72.

b) To determine the 52nd percentile, we need to find the score corresponding to the locator L52. Looking at the ordered list of scores, we see that the score at position 6 is 17 and the score at position 7 is 20. Since the locator L52 falls between positions 6 and 7, we can use linear interpolation to estimate the 52nd percentile:

P52 = (7 - L52) / (7 - 6) * 50 + (L52 - 6) / (7 - 6) * 50

P52 = (7 - 6.72) / (7 - 6) * 50 + (6.72 - 6) / (7 - 6) * 50

P52 = 38 + 0.48 * 50

P52 = 61.4

Therefore, the 52nd percentile is approximately 61.4.

c) To find the percentage of scores that are above the 52nd percentile, we subtract the 52nd percentile from 100:

100 - 52 = 48

Therefore, approximately 48% of the scores in the dataset are above the 52nd percentile.

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Answer:

x=4

y=5

Step-by-step explanation:

Thus, the statement  -3 to the -3 power written without using exponents is -1/27.

A number's power or exponent, in other words, tells how many times it must be multiplied by itself. Any integer, fraction, or decimal can serve as the basis in this situation. Moreover, the exponent might have either a positive or negative value.

Given that-

-3 to the -3 power

This can written as:

= (-3)⁻³

Now using the property of exponents,taking reciprocal of the number will remove the negative number.

= (1 /-3)³

= (1)³ / (-3)³

= 1/(-27)

= - 1/ 27

Thus, the statement  -3 to the -3 power written without using exponents is -1/27.

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The monthly payment and total price paid by Antony at interest rate of 6.8% for vehicle is $362.59 and $21204.4 respectively.

Cost price of the car Antony decided to purchase = $19,000

Down payment =  20% of purchase value

Interest rate of amount to be finance = 6.8%

Time period of finance amount = 4 years

Down payment = 0.2 × $19,000

                         = $3,800

Amount that Anthony needs to finance,

Amount to finance = $19,000 - $3,800

                               = $15,200

The monthly payment , use the formula for the monthly payment on a loan,

M = P × r × (1 + r)ⁿ / ((1 + r)ⁿ - 1)

where M is the monthly payment,

P is the principal the amount to finance

r is the monthly interest rate

n is the total number of payments which is the number of years multiplied by 12

The monthly interest rate is 6.8% / 12 = 0.00567,

and the total number of payments is 4 × 12 = 48.

Substituting these values into the formula, we get,

⇒M = $15,200 × 0.00567 × (1 + 0.00567)⁴⁸ / ((1 + 0.00567)⁴⁸ - 1)

⇒M = $15,200 × 0.00567 × 1.3118 / (1.3118 -1)

⇒M = $15,200 × 0.00567 × 1.3118 / 0.3118

⇒M = 113.056 / 0.3118

⇒M ≈ $362.59

Anthony's monthly payment will be about $362.59

Total price that Anthony paid for his vehicle,

add up the down payment and the total amount of payments over the 4-year period.

Total price = $3,800 + ( $362.59 × 48)

                  = $21204.4

Therefore, the monthly payment and the total price at interest rate of 6.8% that Anthony paid for vehicle is $362.59 and $21204.4 respectively.

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The final amount with interest earned at a rate of 3.4% compounded semi-annually would be approximately $7,758.

Let's calculate the values based on the given information.

Given:

Initial deposit (P) = $7,500

Interest rate (r) = 3.4% or 0.034 (decimal)

Compounding frequency (n) = 2 (semi-annually)

We can use the compound interest formula to calculate the values:

A = P[tex](1 + r/n)^(nt)[/tex]

Where:

A = the final amount (including interest)

P = the principal amount (initial deposit)

r = the interest rate (in decimal)

n = the number of times interest is compounded per time period

t = the number of time periods

Let's calculate the final amount (A) after one year (2 semi-annual periods):

A =[tex]$7,500(1 + 0.034/2)^(2*1)[/tex]

A =[tex]$7,500(1.017)^2[/tex]

A = $7,500(1.034344) [rounded to 6 decimal places]

A ≈ $7,758 [rounded to the nearest dollar]

So, after one year, the final amount with interest earned at a rate of 3.4% compounded semi-annually would be approximately $7,758.

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Answer:

Angle 1 = 51.6°

Angle 2 = 38.4°

Step-by-step explanation:

All complimentary angles add up to 90°.

If two complimentary angles were congruent then their measures would both be 45°.

Since one angle's measure is 6.6° more than the other's measure, it's measure is 6.6° more than 45°

Vice versa for the second angle, its measure will be 6.6° less than 45°

Answer:

10*8y2

Step-by-step explanation:

The appropriate domain for the function h(t) = −3t^2 + 9t + 12 is [0, 3.08].

The domain of a function is the set of all possible input values for which the function is defined. In this case, the function h(t) gives the height of the ball above the ground as a function of time t.

The ball is thrown from the tree house at some initial time, and it continues to travel until it hits the ground. The height of the ball is defined for all values of t between the time it is thrown and the time it hits the ground.

To find the domain of h(t), we need to determine the range of values of t for which the function is defined. The ball cannot have a negative height, so the function is undefined for values of t that would make h(t) negative.

To find the time at which the ball hits the ground, we need to set h(t) equal to zero and solve for t:

h(t) = -3t^2 + 9t + 12 = 0

Using the quadratic formula, we get:

t = (-9 ± √(9^2 - 4(-3)(12)))/(2(-3))

t = (-9 ± √(105))/(-6)

t ≈ -0.41 or t ≈ 3.08

Since the ball cannot have a negative height, the domain of the function is the interval [0, 3.08]. Therefore, the appropriate domain for the function h(t) = −3t^2 + 9t + 12 is [0, 3.08].

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The total possible points in the course is approximately 470.

Let's assume that the total possible points in the course is "x".

Since you needed 90% of the course total for an A, that means you needed to earn 90% of "x" to get an A. This is equivalent to 0.90x.

You missed an A by only 8 points, so your actual earned points must be 90% of "x" minus 8. This can be expressed as 0.90x - 8.

Given that your actual earned points for the course is 415, we can set up an equation:

0.90x - 8 = 415

Now, we can solve for "x" to determine the total possible points in the course:

0.90x = 415 + 8

0.90x = 423

x = 423 / 0.90

x ≈ 470

So, the total possible points in the course is approximately 470.

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The effective annual yield of the investment is 0.1501

To find the effective annual yield, we need to use the formula:

(1 + r/n)^n - 1

where r is the annual interest rate and n is the number of times the interest is compounded per year.

In this case, r = 0.14 (14% expressed as a decimal), and n = 52 (since interest is compounded weekly).

Plugging these values into the formula, we get:

(1 + 0.14/52)^52 - 1 = 0.1501

Therefore, the effective annual yield is 0.1501, which is equivalent to 15.01% (rounded to two decimal places).

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On solving the provided question we can say that As a result, set D denotes a linear function.

The slope of a line indicates how steep it is. The gradient's mathematical equation is known as "gradient overflow" (the change in y divided by the change in x). The slope is defined as the ratio of the vertical change (rise) between two places to the horizontal change (run). The slope-intercept form of an equation is used to express a straight line's equation, which is written as y = mx + b. The y-intercept is found where the slope of the line is m, b is b, and (0, b). For example, consider the slope and y-intercept of the equation y = 3x - 7.The slope of the line is m. b is b at the y-intercept, and (0, b).

The set of ordered pairs expressing a linear function must meet the requirement that any two locations in the set have a constant rate of change (slope).

We can determine the rate of change between the two supplied positions using set D:

slope = (y2 - y1) / (x2 - x1)

slope = (1 - 2) / (2 - (-1))

slope = -1/3

We may use the slope-intercept form of a linear equation to see if the remaining points in set D fulfil the same rate of change: y = mx + b, where m is the slope and b is the y-intercept.

We can solve for b using the first point in D, (-1,2):

2 = (-1/3)(-1) + b

b = 7/3

So the linear equation for set D is: y = (-1/3)x + 7/3

Now we can check if the remaining points in set D satisfy this equation:

(2, 1): 1 = (-1/3)(2) + 7/3 (satisfies the equation)

As a result, set D denotes a linear function.

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Thus, equation shows the connection between the quantity of weeks, w, and the quantity of books gathered, b: b = 10w.

Simply put, the slope-intercept form is the way to write a line's equation so that the y-intercept (at which line crosses a vertical y-axis) and slope (steepness) are instantly visible. This form is frequently known as the y = mx + b form.

The table shows an increase of 10 volumes every week in the total number of books acquired.

Weeks (w) Books Collected (b)

1                   10

2                  20

3                  30

4                   40

As a result, the quantity of books gathered and the number of weeks (w) have a linear relationship (b). The following equation can be used to illustrate this relationship:

b = mw + c

where w is the number of weeks, m denotes the slope (rate of change), and c denotes the number of books aquired at week 0.

The weekly shift in the amount of books gathered, which is 10, is the slope (m). By adding one of the points from table to the equation, we can now get the y-intercept (c). Use the first and tenth points:

10 = 10 * 1 + c

c = 0

Currently, the following equation reflects the relationship among the number of weeks (w) as well as the quantity of books gathered (b):

b = 10w + 0

b = 10w

Thus, equation shows the connection between the quantity of weeks, w, and the quantity of books gathered, b: b = 10w.

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Complete question:

The data in the form of a table, which shows the total number of books collected (b) for different number of weeks (w).

Weeks (w) Books Collected (b)

1                   10

2                  20

3                  30

4                   40

Trisha is collecting books to donate. The table below shows the total number of books collected, b, for different number of weeks, w. Which equation represents the relationship between the number of weeks, w, and the number of books collected, b?

The ratio of the areas is given as follows:

1:5.8.

The ratio of the areas is obtained applying the proportions in the context of the problem.

When a prism is dilated by a scale factor of k, we have that:

Hence the ratio of the areas is the cubic root of the ratio of the volumes squared, thus, as the ratio of the volumes is of 1:14, we have that:

(14²)^(1/3) = 5.8.

Hence:

1:5.8.

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The equilibrium tuition price p and the consumers' and producers' surpluses at this tuition level is 11400

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

q = 20,000 − 2p

q = 8,500 + 0.5p

The equilibrium tuition price p and the consumers' and producers' surpluses at this tuition level is when q = q

So, we have

20,000 − 2p = 8,500 + 0.5p

Evaluate the like terms

2.5p = 28500

Divide by 2.5

p = 28500/2.5

Evaluate

p = 11400

Hence, the equilibrium tuition price p is 11400

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Answer:

266.6 ft

Step-by-step explanation:

If a person stands at a point and looks up at an object, the angle between their horizontal line of sight and the object is called the angle of elevation.

To find how far away from the building you are standing, we need to find the distance labelled "x" on the attached diagram. We can do this by modelling the given scenario as a right triangle and solving for x by using the tangent trigonometric ratio.

[tex]\boxed{\tan \theta=\sf \dfrac{O}{A}}[/tex]

where:

Given information:

As your eyes are 5 ft above the ground, we have to model the line of sight as 5 ft above ground level.  Therefore, the side of the right triangle opposite the angle of elevation is the height of the building less 5 ft:

[tex]\implies \sf 1000 \; ft - 5\; ft=995 \; ft[/tex]

Let "x" be the horizontal distance between you and the building.

Therefore, the values to substitute into the tangent ratio are:

Substitute these values into the ratio and solve for x:

[tex]\tan 75^{\circ}=\dfrac{995}{x}[/tex]

[tex]x=\dfrac{995}{\tan 75^{\circ}}[/tex]

[tex]x=266.609446...[/tex]

[tex]x=266.6\; \sf ft\; (nearest\;tenth)[/tex]

Therefore, you are standing 266.6 ft from the building.

Hence ,the sum of ∠1 and ∠2  is 90 degrees this is a relationship between ∠1 and ∠2.

In Euclidean geometry, an  angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles  formed by two rays lie in the plane that  contains the rays. Angles are also formed by the  intersection of two planes. These are called dihedral angles.

complementary angles are two angles that add up to 90 degrees, supplementary angles are two angles that add up to 180 degrees,

vertical angles are opposite angles at an intersection of two straight lines, and adjacent angles are two angles that are next to each other.

According to figure , the sum of ∠1 and ∠2  is 90 degrees and If the sum of the two angles is equal to the measurement of a right angle .

So, the pair of angles ∠1 and ∠2  is said to be complementary angles.

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The function y = √x+2 is undefined for any value of x that makes the expression inside the square root negative, which in this case is x ≤ -2.

What is an inequality equation?

An inequality equation is a mathematical statement that compares two expressions using an inequality symbol such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to).

The function that is undefined for x = 0 is y = √x+2.

When x = 0, we have:

y = √0+2

y = √2

Since there is no real number whose square is negative, the square root of a negative number is undefined in real numbers.

Therefore, the function y = √x+2 is undefined for any value of x that makes the expression inside the square root negative, which in this case is x ≤ -2.

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The required value of z such that 0.6318 of the area lies between −z and z is 0.89.

If 0.6318 of the area lies between −z and z, then the area outside of this range is (1-0.6318) = 0.3682.

Since the normal distribution is symmetric, the area to the left of −z is equal to the area to the right of z. So, the area to the right of z is (0.3682 / 2) = 0.1841.

Using a standard normal distribution table, we can find the z-score corresponding to an area of 0.1841 to the right of z:

z = invNorm(0.1841) ≈ 0.89

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This shows that the two circles intersect at a point P, and that the distance between their centers, OP1OP2, is less than the sum of their radii, r1 + r2. Thus, the two circles touch each other internally at point P.

You must calculate each circle's radius and centre in order to accomplish this. The circles will externally touch if the total of the radii and the space between the centres are identical. The circles will internally touch if the variations in radii and the separation between their centres are equal.

Let C1 and C2 stand for the two circles, with O1 and O2 serving as their respective centres. Let r1 and r2 represent the radius of C1 and C2, respectively.

Let's say that point P is where the two circles converge. The Pythagorean Theorem thus gives us:

OP1² = r1²

OP2² = r2²

OP1² + OP2² = (r1 + r2)²

Simplifying the third equation, we get:

OP1² + OP2² = r1² + 2r1r2 + r2²

The left half of this equation can be rewritten as (OP1 + OP2)2 - 2OP1OP2. When we simplify the equation above and substitute this, we get:

(OP1 + OP2)² = (r1 + r2)² + 2OP1OP2

Since OP1 and OP2 are both positive, we have:

(OP1 + OP2)²> (r1 + r2)²

Therefore, we must have:

OP1OP2 < r1 + r2

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Answer: 480 in²

Step-by-step explanation:

    To calculate the surface area, we will find the area of each side and add it together.

Area of the rectangular sides can be found with A = LW:

    A = LW

    A = (12 in)(10 in)

    A = 120 in²

    A = LW

    A = (12 in)(10 in)

    A = 120 in²

    A = LW

    A = (12 in)(12 in)

    A = 144 in²

All the rectangular sides added together:

    120 in² + 120 in² + 144 in² = 384 in²

    Now, we will find the triangle sides. The formula for a triangle is [tex]A=\frac{BH}{2}[/tex], but since we have two congruent triangles I will not be dividing by 2.

    A = BH

    A = (12 in)(8 in)

    A = 96 in²

Lastly, I will add them both together:

384 in² + 96 in² = 480 in²

A)By applying permutation and combination we can select committee in  1782 ways.

B) The probability cannot be determined with the given information, as the calculated probability is greater than 1, which is not possible.

Permutation and combination are two concepts in mathematics that deal with counting the number of possible outcomes or arrangements in a given situation. Permutation refers to the arrangement of objects in a specific order, while combination refers to the selection of objects without regard to the order in which they are chosen.

Probability is a measure of the likelihood or chance that an event will occur. It is a number between 0 and 1.

According to the given information:

A) To find the number of ways to select a committee of 5 members, where exactly one of Mr Lan and Mrs Lan must be on the committee, we can use the principle of inclusion-exclusion i.e. Permutation and combination:

First, we find the total number of ways to select a committee of 5 members from 14 members (6 men and 8 women), which is given by:

C(14,5) = 2002

Next, we find the number of ways to select a committee of 5 members that includes both Mr Lan and Mrs Lan, which is given by:

C(12,3) = 220

Note that we choose 3 members from the remaining 12 members after Mr and Mrs Lan have been chosen.

Finally, we subtract the number of committees that include both Mr Lan and Mrs Lan from the total number of committees, since we only want committees that include exactly one of them:

2002 - 220 = 1782

Therefore, there are 1782 different ways to select a committee of 5 members where exactly one of Mr Lan and Mrs Lan must be on the committee.

B) To find the probability that there are more women than men on the committee, given that Mrs Lan was selected, we need to consider the number of committees that satisfy this condition and divide by the total number of committees that include Mrs Lan.

If Mrs Lan is selected, there are two cases to consider:

Case 1: The committee has more women than men.

In this case, we need to select 4 members from the 8 women and 1 member from the 5 men, which can be done in C(8,4) x C(5,1) = 560 ways.

Case 2: The committee has an equal number or more men than women.

In this case, we need to select 3 members from the 8 women and 2 members from the 5 men, which can be done in C(8,3) x C(5,2) = 560 ways.

Therefore, the total number of committees that include Mrs Lan and have more women than men or an equal number of women and men is 560 + 560 = 1120.

The total number of committees that include Mrs Lan is C(13,4) = 715, since we choose 4 members from the remaining 13 members after Mrs Lan has been chosen.

Therefore, the probability of selecting a committee with more women than men or an equal number of women and men, given that Mrs Lan was selected, is:

1120/715 = 1.5664

Note that this probability is greater than 1, which is not possible. Therefore, there is an error in the calculation, and the correct probability cannot be determined with the given information.

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Answer:

To improve my math skills, I plan on doing the following:

1. Practice, practice, practice: I will practice math problems regularly, starting with simple problems and gradually increasing the difficulty level.

2. Use online resources: There are many online resources available that provide tutorials, practice problems, and videos to help with math skills. I will utilize these resources to supplement my learning.

3. Work with a tutor: I will consider working with a tutor if I need additional help with math concepts or if I'm struggling with specific problems.

4. Study regularly: I will set aside time each day to study math, even if it's just for a few minutes.

5. Ask questions: I will ask my teacher or tutor for help if I don't understand a concept or if I'm struggling with a problem.

6. Stay positive: I will maintain a positive attitude towards math and believe in my ability to improve my skills.

By following these steps, I'm confident that I can improve my math skills and become more confident in my abilities.

Answer:

"My plan for improving my math skills is to focus more on understanding and going over new concepts and practice problems. I also will try studying math concepts online and using a method of solving extra problems and applying math in the real world to better understand."

Step-by-step explanation:

Answer:

66 because 10 percent of 60 is 6 so add 60 and 6

The only option given that has a variable raised to the third power is D. -2x³, so the correct answer is D.

In algebra, a monomial is an expression that consists of a single term, which is a product of a constant coefficient and one or more variables raised to non-negative integer powers. In each of these expressions, there is only one term, and each term is a product of a constant coefficient and one or more variables raised to non-negative integer powers. Monomials are important in algebra because they can be combined using various operations such as addition, subtraction, multiplication, and division, to form more complex expressions.

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Graph cookies and coffee, slope -2. Ben buys 2 cookies and 2 coffees. To optimize, allocate budget where MU per dollar is equal.

A budget is a financial plan that outlines expected income and expenses over a certain period. It helps individuals or organizations manage their money and make informed decisions about spending and saving.

Consumption refers to the use of goods and services by individuals, households, or organizations. It is an essential part of the economy and can be influenced by factors such as income, preferences, and prices.

According to the given information:

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Answer:

Therefore, Frank's monthly credit card payment will be $454.31.

Step-by-step explanation:

Monthly payment = (PV * r) / (1 - (1 + r)^-n)

PV = $2,500 + $1,000 + $3,000 + $1,500

= $8,000

r = 0.18 / 12

= 0.015

Monthly payment = ($8,000 * 0.015) / (1 - (1 + 0.015)^-24)

= $454.31

Thus, Riley's maximum distance from the starting position is 55 inches.

When an input value has a defined output value in reality, functions can be applied. For instance, how long a car has been driving affects how far it has travelled (the output) (the input).

Riley is swaying on a playground swing. As stated in the table, let t denote the passing of time in seconds and let f(t) denote Riley's horizontal displacement from her starting location in inches.

t:   0 ,0.75, 1.5, 2.25, 3,  3.75, 4.5,  5.25,   6,  6.75

f(t): 0 ,38.9, 55, 38.9, 0, -28.9, -55, -38.9, 0, 38.9

Recall that f(t) is positive when Riley swings closest to the starting position and negative when Riley swings in behind starting position.

So,

started from 0 and travelled distance of 55 inches in 1.5 seconds.

She is advancing by up to 1.5 seconds.

Then she begins to move backward, returning to her starting location after 3 seconds, and continuing to do so for an additional 4.5 seconds at a distance of -55.

Then begin to advance and return to the starting position after six seconds

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The piecewise function graphed is defined as follows:

C)

A piecewise function is a function that is defined by multiple equations, each of which applies to a specific interval or set of inputs. The equations are pieced together to form a single function that describes the behavior of the function over its entire domain.

The intervals for this problem are given as follows:

For the first interval, we have a linear function with a slope of 1/2 and an intercept of 2, hence:

y = 1/2x + 2, x ≤ -1.

For the second interval, we have a quadratic function with a vertex of -1, hence:

y = 1/2x² - 1, x > -1.

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