Question
Gerard worked the same number of hours, x, on Monday, Tuesday, and Wednesday, but on Thursday, he worked 5 hours less than the previous day. The total number of hours Gerard worked can be found using the expression, 4x−5 .

What does the "4" represent in the expression, 4x−5 ?

--------------------------------------------------------------------------------
the number of hours Gerard worked on Thursday


the number of hours Gerard worked each day


the total number of hours Gerard worked

the number of days Gerard worked

Answer:

C. Exchange the $100 bill for ten $10 bills.

Since the money is to be divided among 4 people, it is easier to divide it if the money is in equal denominations. Therefore, the first step should be to exchange the $100 bill for ten $10 bills, so that each person can receive an equal share of $25. This way, we can avoid having to deal with different denominations while dividing the money among the four people.

  All zeros of polynomial functions  -3/2 , 5i and -5i

What is zeros of polynomial ?

Zeros of polynomial are the points where the polynomial equals zero on the whole. In simple words, we can say that zeros of polynomial are values of the variable such that the polynomial equals 0 at that point. Zeros of a polynomial are also referred to as the roots of the equation and are often designated as α, β, γ respectively. Some of the methods used to find the zeros of polynomial are grouping, factorization, and using algebraic expressions.

2x^3+3x^2+50x+75 equal to 0

2x^3+3x^2+50x+75 = 0

⇒ x^2 (2x + 3) + 25 (2x + 3) = 0

⇒ (2x + 3) (x^2 + 25 ) = =

If any individual factor on the left side of the equation is equal to  0, the entire expression will be equal to 0

2x + 3 = 0

x^2 + 25 = 0

so, x = -3/2  and x = 5i and -5i

The final solution is all the values that make  2x^3+3x^2+50x+75 = 0

x = -3/2 , 5i and -5i

Hence, all zeros of polynomial functions  -3/2 , 5i and -5i

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Answer:When we choose a dime from the box, there are two coins left, one penny and one nickel. Since we do not replace the dime, the probability of choosing a penny next is 1/2.

Therefore, the probability of choosing a dime first and a penny second is:

P(dime first and penny second) = P(dime first) * P(penny second | dime first)

= (1/3) * (1/2)

= 1/6

So, the answer is A. 1/6.

Step-by-step explanation:

Answers
∠ OKL = 45

∠ MOL = 90

Explanation

All interior angles of a square equal to 90 degrees. All right angles are 90 degrees.

∠ OKL is a right angle (90 degrees) bisected in half, so it is 45.

∠ MOL is an interior angle so it is 90 degrees

The height οf the step is apprοximately 3.106 feet. Thus, οptiοn A is cοrrect.

What is the trigοnοmetry?

Trigοnοmetry is a branch οf mathematics that deals with the relatiοnships between the sides and angles οf triangles.

Tο determine the height οf the step, we need tο use trigοnοmetry. Let's assume that the height οf the step is h feet.

Since the ramp is elevated at an angle οf 15 degrees, we knοw that the sine οf this angle is equal tο the οppοsite side (h) divided by the hypοtenuse (12 feet).

sin(15) = h/12

Tο sοlve fοr h, we can multiply bοth sides by 12:

12 * sin(15) = h

Using a calculatοr, we can apprοximate sin(15) tο be 0.2588. Therefοre:

h ≈ 12 * 0.2588

h ≈ 3.106 feet

Hence, the height οf the step is apprοximately 3.106 feet.

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Complete question:

A 12-foot ramp to be elevated at an angle measuring 15 degrees to be level with a step. Approximately how far from the step should the ramp start?

3.13.911.614.5

Answer:

333 ft^2

Step-by-step explanation:

To find the area of the new dance floor, use the trapezoid area formula.

Area of a Trapezoid: [tex]\frac{(base 1+base 2)h}{2}[/tex]

[(18 1/2 + 18 1/2) · 18]/2 - Plug in values

(37x18)/2 - Simplify

333 ft^2 - Solve

A) The probability of the sample obtained having a mean altruism score lower than 72 is 0.9474. B) The probability of the sample obtained having a mean altruism score between 67 and 73 is 0.3584. C) The sample size (n) should be at least 207 in order to have a probability of 0.8 for a sample mean no greater than 67.

A. We can use the Central Limit Theorem to calculate the probability of the sample obtained having a mean altruism score lower than 72.

The Central Limit Theorem states that the mean of a sample of size n will be approximately normally distributed with a mean of μ and a standard deviation of σ/√n.

In this case, the mean of the sample will be approximately normally distributed with a mean of 66 and a standard deviation of 11.5/√27 = 2.214. Using the z-score formula, we can calculate the z-score for a sample mean of 72:
z = (72 - 66)/(11.5/√27) = 1.62
Using a z-table, we can find the probability of the sample mean being lower than 72:
P(z < 1.62) = 0.9474
Therefore, the probability of the sample obtained having a mean altruism score lower than 72 is 0.9474.

B. We can use the Central Limit Theorem to calculate the probability of the sample obtained having a mean altruism score between 67 and 73. The mean of the sample will be approximately normally distributed with a mean of 66 and a standard deviation of 11.5/√34 = 1.971. Using the z-score formula, we can calculate the z-scores for a sample mean of 67 and 73:
z1 = (67 - 66)/(11.5/√34) = 0.338
z2 = (73 - 66)/(11.5/√34) = 2.374
Using a z-table, we can find the probability of the sample mean being between 67 and 73:
P(0.338 < z < 2.374) = P(z < 2.374) - P(z < 0.338) = 0.9909 - 0.6325 = 0.3584
Therefore, the probability of the sample obtained having a mean altruism score between 67 and 73 is 0.3584.

C. We can use the Central Limit Theorem to calculate the sample size (n) such that the probability of randomly drawing a sample of children of the sample size (n) from this target population will have a probability of 0.8 for a sample mean no greater than 67. The mean of the sample will be approximately normally distributed with a mean of 66 and a standard deviation of 11.5/√n. Using the z-score formula, we can calculate the z-score for a sample mean of 67:
z = (67 - 66)/(11.5/√n) = 0.8
Solving for n, we get:
n = (11.5/(0.8*(67 - 66)))^2 = 206.13
Therefore, the sample size (n) should be at least 207 in order to have a probability of 0.8 for a sample mean no greater than 67.

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The multiplication equation that can be used to determine the number of people who contributed to the cost of the season ticket package is: $745 = $186.25 x 4.

To write a multiplication equation that can be used to determine the number of people who contributed to the cost of the season ticket package, we can use the following equation:

Total Cost = Cost per Person x Number of People

In this case, the total cost is $745 and the cost per person is $186.25. We can plug these values into the equation to get:

$745 = $186.25 x Number of People

To solve for the number of people, we can divide both sides of the equation by $186.25:

Number of People = $745 / $186.25

Number of People = 4

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Assuming the lines, which appear to be diameters, are actual diameters. The value of x is 10.63.

What is the diameter?

Any straight line that connects two points on a circle's circumference and goes through the circle's center. The length of such a line in geometry. The greatest separation possible between any two points in a metric space (geometry).

Exterior angles are 128, 32, and 45

Interior angles are 11x+4 and 85

Exterior angles = interior angles

128 + 32 + 45 = 11x+4 + 85

205 = 11x + 88

11x = 205 - 88 = 117

x = 117/11

x = 10.63

Therefore, the value of x is 10.63.

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Answer:

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Traffic problems can be solved by implementing various measures such as improving public transportation, promoting carpooling and ridesharing, creating pedestrian-friendly cities, and implementing smarter traffic management systems.

We proceed to explain some actions to solve traffic problems:

Improving public transportation can help reduce the number of cars on the road, thus reducing traffic congestion. This can be done by increasing the number of public buses and trains, as well as improving their reliability and efficiency.Promoting carpooling and ridesharing can also help reduce traffic problems by encouraging people to share rides instead of driving alone. This can be done through incentives such as discounted parking fees for carpoolers and ridesharing apps. Creating pedestrian-friendly cities can also help reduce traffic problems by encouraging people to walk or bike instead of driving. This can be done by implementing bike lanes, pedestrian-only zones, and safer crosswalks. Implementing smarter traffic management systems can help reduce traffic problems by using technology to monitor and control traffic flow. This can be done through the use of sensors, cameras, and smart traffic lights.

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The minimum amount of money Camille could have is 0 dollars if she doesn't buy anything, and the maximum amount of money she could have is 166 dollars if she buys the maximum quantity of each item.

How to obtain the maximum value of a function?

To find the maximum of a continuous and twice differentiable function f(x), we can firstly differentiate it with respect to x and equating it to 0 will give us critical points.

Putting those values of x in the second rate of function, if results in negative output, then at that point, there is maxima. If the output is positive then its minima and if its 0, then we will have to find the third derivative (if it exists) and so on.

We are given that;

The prices of different items at a garage sale: pants, shirts, shorts, and belts.  

Camille can buy up to 5 shirts.

Now,

If Camille buys 5 shirts, she would spend 5 * 6 = 30 dollars on shirts. Since she can buy up to 5 shirts, she may spend less than 30 dollars if she buys fewer shirts.

She can buy up to 10 pants, which would cost her 10 * 8 = 80 dollars.

She can buy up to 5 shorts, which would cost her 5 * 4 = 20 dollars.

She can buy up to 12 belts, which would cost her 12 * 3 = 36 dollars.

30 + 80 + 20 + 36 = 166 dollars

Therefore, by the  maxima and minima the answer will be 166 dollars if she buys the maximum quantity of each item.

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Maria has $22,000 invested in account A and $8,000 invested in account B.

Let's begin by setting up a system of equations to solve for the amount Maria has invested in each account. Let x be the amount invested in account A and y be the amount invested in account B. Since Maria has a total of $30,000 invested in both accounts, we can write the first equation as:

x + y = 30,000

Next, we can write an equation to represent the total annual interest earned from both accounts:

0.045x + 0.038y = 1,294

Now, we can use the substitution method to solve for x and y. We'll rearrange the first equation to solve for x:

x = 30,000 - y

Then, we'll substitute this value of x into the second equation:

0.045(30,000 - y) + 0.038y = 1,294

Simplifying the equation gives:

1,350 - 0.045y + 0.038y = 1,294

Combining like terms and isolating y gives:

0.007y = 56

y = 8,000

Now, we can substitute this value of y back into the first equation to solve for x:

x = 30,000 - 8,000

x = 22,000

So, Maria has $22,000 invested in account A and $8,000 invested in account B.

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Two famous structures/ inventions showing angles formed by secants and tangents are: The London Eye and The Eiffel Tower.

1. The London Eye: The London Eye is a giant Ferris wheel located on the South Bank of the River Thames in London. It is a popular tourist attraction and an iconic symbol of London. The London Eye has 32 oval-shaped capsules, each of which can carry up to 25 people. The structure is made up of several secants and tangents, which form angles at various points along the wheel.

2. The Eiffel Tower: The Eiffel Tower is a wrought-iron lattice tower located on the Champ de Mars in Paris, France. It is one of the most recognizable structures in the world and a symbol of France. The Eiffel Tower is made up of several secants and tangents, which form angles at various points along the structure.

Both of these structures are examples of how secants and tangents can be used in the design of famous structures and inventions. These angles play a crucial role in the stability and strength of the structures, allowing them to withstand the weight of the people and objects they hold.

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To solve the given linear system, we can use the Gaussian elimination method. This method involves reducing the given matrix to a row echelon form and then solving for the variables using back substitution.

Step 1: Reduce the given matrix to a row echelon form by performing elementary row operations.
\[ \left[\begin{array}{rrr|r} 6 & 2 & -3 & 0 \\ -5 & 3 & 9 & 0 \\ 2 & -7 & -1 & 0 \end{array}\right] \]
We can start by dividing the first row by 6 to get a leading 1:
\[ \left[\begin{array}{rrr|r} 1 & \frac{1}{3} & -\frac{1}{2} & 0 \\ -5 & 3 & 9 & 0 \\ 2 & -7 & -1 & 0 \end{array}\right] \]
Next, we can add 5 times the first row to the second row and subtract 2 times the first row from the third row:
\[ \left[\begin{array}{rrr|r} 1 & \frac{1}{3} & -\frac{1}{2} & 0 \\ 0 & \frac{16}{3} & \frac{17}{2} & 0 \\ 0 & -\frac{23}{3} & 0 & 0 \end{array}\right] \]
Finally, we can multiply the second row by $\frac{3}{16}$ and the third row by $-\frac{3}{23}$ to get a leading 1 in each row:
\[ \left[\begin{array}{rrr|r} 1 & \frac{1}{3} & -\frac{1}{2} & 0 \\ 0 & 1 & \frac{51}{32} & 0 \\ 0 & 1 & 0 & 0 \end{array}\right] \]

Step 2: Use back substitution to solve for the variables.
From the third row, we have:
$x_2 = 0$
Substituting this into the second row gives us:
$x_3 = 0$
And substituting these values into the first row gives us:
$x_1 = 0$
Therefore, the solution to the given linear system is:
$x_1 = 0$, $x_2 = 0$, and $x_3 = 0$

This means that the given linear system has a unique solution at the origin.

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Point c means that Danny spent $80 dollars on tickets for 5 games . then the unit rate can be represented by the point (5, 80) on the graph.

Explain about the rate of change?

The term "rate of change" (ROC) describes the rate at which something evolves over an extended period.

So, it is not the amount of specific features themselves but rather the deceleration or acceleration of changes (— in other words, the rate).Technically speaking, the Price Rate of Change marker calculates the percentage difference in price between the present price and the price from a predetermined number of periods ago.Since it enables investors to identify trends and other trends, rate of change is a crucial financial term.

Danny purchases the same number of tickets for every baseball game he attends during the summer, and the graph below shows how much was spent on the X Games in dollars.

Thus, point c means that Danny spent $80 dollars on tickets for 5 games . then the unit rate can be represented by the point (5, 80) on the graph.

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the complete question is attached.

The polynomial g(x)=x^(3)+3x^(2)-x-3 is a cubic polynomial with 4 terms.

To prove its degree using successive differences, we first need to find the successive differences of the polynomial's y-values for consecutive x-values. We can do this by substituting x-values into the polynomial and finding the differences between the resulting y-values.

For x=0, g(x)=0^(3)+3(0^(2))-0-3=-3
For x=1, g(x)=1^(3)+3(1^(2))-1-3=0
For x=2, g(x)=2^(3)+3(2^(2))-2-3=11
For x=3, g(x)=3^(3)+3(3^(2))-3-3=30

The first successive difference is 0-(-3)=3
The second successive difference is 11-0=11
The third successive difference is 30-11=19

Since the successive differences are not constant, we need to find the successive differences of the successive differences.

The first successive difference of the successive differences is 11-3=8
The second successive difference of the successive differences is 19-11=8

Since the successive differences of the successive differences are constant, the degree of the polynomial is 3, which is one less than the number of times we had to find successive differences. This confirms that g(x)=x^(3)+3x^(2)-x-3 is a cubic polynomial.

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An EM wave is written line E = E0 sin(kx - ωt) and EM wave equation is written as ∂²E/∂x² = μ₀ε₀∂²E/∂t²

An electromagnetic wave (EM wave) is a type of wave composed of an electric field and a magnetic field that propagate at the speed of light. The equation for an EM wave is given by E = E0 sin(kx - ωt), where E0 is the peak electric field, k is the wavenumber, ω is the angular frequency, x is the position, and t is time.

An ELECTROMAGNETIC WAVE, or EM wave, is a type of wave that travels through space and carries energy from one point to another. EM waves are created when electric and magnetic fields interact with one another. These waves are categorized by their frequency, which determines their energy and their wavelength.

The EM wave equation, also known as the wave equation, is used to describe how EM waves propagate through space. The equation is typically written in the form:

∂²E/∂x² = μ₀ε₀∂²E/∂t²

Where E is the electric field, μ₀ is the permeability of free space, ε₀ is the permittivity of free space, x is the position, and t is the time. This equation shows how the electric field changes over time and space, and is used to predict the behavior of EM waves.

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Given vectors are, u = (6,1,6), v = (1,-1,4) and w = (1,4,7) and a vector (23, 10, 23).We need to express (23, 10, 23) as a linear combination of u, v and w.

To find the coefficients, we use matrix notation.
Let A be a matrix containing u, v and w as its column matrices, and X be a matrix containing the coefficients.
Then,AX = B Where A = [u | v | w]X = [a | b | c]B = (23, 10, 23)
Therefore, [u | v | w][a | b | c] = (23, 10, 23)⇒ a(u1) + b(v1) + c(w1) = 23a(6) + b(1) + c(1) = 23⇒ 6a + b + c = 23⇒ a(u2) + b(v2) + c(w2) = 10a(1) - b(1) + c(4) = 10⇒ a - b + 4c = 10⇒ a(u3) + b(v3) + c(w3) = 23a(6) - b(4) + c(7) = 23⇒ 6a - 4b + 7c = 23
The above system of linear equations can be solved using Gaussian elimination method.
The row echelon form of the augmented matrix [A | B] is, [6 1 1 | 23][1 -1 4 | 10][0 0 2 | 9]
The solution is, c = 9/2, b = -3, and a = 1/2.
Therefore, (23, 10, 23) = (1/2)u - 3v + (9/2)w can be written as a linear combination of u, v and w.

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Answer:

Z= 6

Step-by-step explanation:

Z + 9 = 15

- do the inverse (opposite) operation

15 -9 = 6

9 -9 * cancel out *

Z = 6

CHECK:

6 +9 = 15

TRUE

Answer:

x=55°, y=35°

Step-by-step explanation:

The angle between x and 35° is a right angle.

A right angle equals 90°.

Therefore. 90=x+35

x=55°

Now that x is known, we can find y.

The angle made between the x and y axis is 90°.

Therefore, 90=x+y

90=55+y

y=35°

Answer:

X=55° and Y=35° ................................

The box plot for given minimum, maximum, median, Q1 and Q3  is plotted below.

What is a box and whisker plot?

A box and whisker plot—also called a box plot—displays the five-number summary of a set of data. The five-number summary is the minimum, first quartile, median, third quartile, and maximum. In a box plot, we draw a box from the first quartile to the third quartile.

The given data set is,

14, 13, 13.5, 16, 14, 15.5, 14.5.

Hence, We get;

Order of data is,

⇒ 13, 13.5, 14, 14, 14.5, 15.5, 16.

Here, minimum is 13

Maximum is 16

Median is 14

Lower quartile range(Q1) is 13.5

Upper quartile range(Q3) is 15.5

Therefore, The box plot for given minimum, maximum, median, Q1 and Q3  is plotted below.

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3 1/5 + 1 3/7

To add these two fractions, we need to find a common denominator. The least common multiple of 5 and 7 is 35.

3 1/5 = 16/5

1 3/7 = 10/7

16/5 + 10/7 = (16/5) * (7/7) + (10/7) * (5/5) = 112/35 + 50/35 = 162/35

We can simplify this fraction by dividing both the numerator and denominator by their greatest common factor (GCF), which is 1.

162/35 = 4 22/35

This expression does not equal 4 4/12.

1 11/12 + 2 1/4

Again, we need to find a common denominator to add these two fractions. The least common multiple of 4 and 12 is 12.

1 11/12 = 23/12

2 1/4 = 9/4

23/12 + 9/4 = (23/12) * (1/1) + (9/4) * (3/3) = 23/12 + 27/12 = 50/12

We can simplify this fraction by dividing both the numerator and denominator by their GCF, which is 2.

50/12 = 4 2/12

This expression does equal 4 4/12.

The standard deviation of the given data set is 825.87.

How to find standard deviation

The standard deviation of a data set is a measure of how spread out the data is from the mean (average) of the data set. It is calculated using the following formula:

Standard Deviation = √(Σ(x - mean)^2 / n)

Where:

- x is each individual data point

- mean is the average of the data set

- n is the number of data points in the data set

- Σ is the sum of all the values

To calculate the standard deviation of the given data set, we first need to calculate the mean:

Mean = (200 + 1200 + 300 + 350 + 500 + 55 + 10 + 500 + 800 + 850 + 1300 + 2000 + 2100 + 340 + 760 + 830 + 2670 + 9 + 3000) / 20 = 833.45

Next, we need to calculate the sum of the squared differences between each data point and the mean:

Σ(x - mean)^2 = (200 - 833.45)^2 + (1200 - 833.45)^2 + ... + (3000 - 833.45)^2 = 1.365E7

Finally, we can calculate the standard deviation:

Standard Deviation = √(1.365E7 / 20) = 825.87

Therefore, the standard deviation of the given data set is 825.87.

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The axis of symmetry of the parabola defined by the equation x=(1)/(36)y^(2)-(1)/(18)y+(37)/(36) is the vertical line x = (37)/(36). This is because the equation is in the form of y = ax^2 + bx + c, and the axis of symmetry is x = -b/2a. In this case, a = (1)/(36), b = -(1)/(18) and c = (37)/(36).

Therefore, the axis of symmetry is x = -(1)/(36) x -(1)/(18) = (37)/(36). This axis of symmetry divides the parabola into two symmetric halves and it is also the x-coordinate of the vertex of the parabola.

The vertex is the highest or lowest point of the parabola and it is the point of intersection between the axis of symmetry and the parabola. To find the y-coordinate of the vertex, we can substitute the x-coordinate of the vertex into the equation, which gives us y = (1)/(72) + (37)/(36) = (109)/(72). Therefore, the coordinate of the vertex is (37/36, 109/72).

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Epidemiology would include option (a), (c), and (d).

Epidemiology is the study of the distribution and determinants of health-related states or events in specified populations, and the application of this study to the control of health problems (JM. A Dictionary of Epidemiology, 2001). According to this definition, Epidemiology would include the following activities:

Option (a): Describing the demographic characteristics of persons with acute aflatoxin poisoning in Chittagong: This activity fits with the definition of Epidemiology because it involves studying the characteristics of a specified population, in this case the demographic characteristics of persons with acute aflatoxin poisoning in Chittagong, in order to better understand the determinants and distribution of the health-related state (acute aflatoxin poisoning).

Option (c): Comparing the family history, amount of exercise, and eating habits of those with and without newly diagnosed diabetes: This activity fits with the definition of Epidemiology because it involves studying the determinants of a health-related state, in this case newly diagnosed diabetes, in order to better understand the distribution and control of this health problem.

Option (d): Recommending that 'Kashundi restaurant' in IUB be closed after implicating it as the source of a hepatitis A outbreak: This activity fits with the definition of Epidemiology because it involves applying the study of the distribution and determinants of health-related states to the control of health problems, in this case the control of a hepatitis A outbreak.

In summary, Epidemiology would include option (a), (c), and (d).

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The distance from where you launched the ball to the goal is 52.6 feet.

The goal of this problem is to find the distance x from where you launched the ball to the goal. To do this, we can use the trigonometric function tangent. The tangent of an angle in a right triangle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the distance from where you scored to the sideline (88.1 feet) and the adjacent side is the distance we are trying to find (x).

So, we can set up the equation:

tan(58.828) = 88.1/x

Next, we can cross multiply to get:

x*tan(58.828) = 88.1

Then, we can divide both sides by tan(58.828) to get:

x = 88.1/tan(58.828)

Using the built-in calculator in the right sidebar, we can plug in the values and make sure the setting is on Deg for degrees. This gives us:

x = 88.1/1.6767

Finally, we can round our answer to the nearest tenth of a foot to get:

x = 52.6 feet

So, the distance from where you launched the ball to the goal is 52.6 feet.

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The value of x for the given equations are a. 10, b. -7, c = -8, d. 5.

What are equations and its solution?

Finding an equation's solutions, which are values (numbers, functions, sets, etc.) that satisfy the equation's condition and often consist of two expressions connected by an equals sign, is known as solving an equation in mathematics. One or more variables are identified as unknowns when looking for a solution. An assignment of values to the unknown variables that establishes the equality in the equation is referred to as a solution. Particularly but not exclusively for polynomial equations, the solution of an equation is frequently referred to as the equation's root.

Given that,

2(x-3) = 14

2x - 6 = 14

2x = 20

x = 10

b. -5(x - 1) = 40

-5x + 5 = 40

-5x = 35

x = -7

c. 12(x + 10) = 24

12x + 120 = 24

12x = 24 - 120

x = -8

d. (x + 6) = 11

x = 11 - 6

x = 5

Hence, the value of x for the given equations are a. 10, b. -7, c = -8, d. 5.

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Answer:

x < 5

Step-by-step explanation:

2x - 3 < x + 2 ≤ 3x + 5

2x - 3 < x + 2 and x + 2 ≤ 5

x < 5 and x ≤ 3

Answer: x < 5

The domain of f⁻¹(x) is [0, ∞) and the range of f⁻¹(x) is [-3, ∞).

The function f(x)=(x+3)² is not one-to-one, as it is a quadratic function and has a parabolic shape. However, we can restrict the domain of the function so that it is one-to-one and increasing. One way to do this is to restrict the domain to be greater than or equal to the x-coordinate of the vertex of the parabola. The vertex of the parabola is at (-3, 0), so we can restrict the domain to be x ≥ -3. In interval notation, this is [-3, ∞).

The inverse function of f(x) can be found by switching the x and y values and solving for y. This gives us:

x = (y+3)²

√x = y+3

y = √x - 3

So the inverse function is f⁻¹(x) = √x - 3. The domain of the inverse function is the range of the original function, and the range of the inverse function is the domain of the original function. Therefore, the domain of f⁻¹(x) is [0, ∞) and the range of f⁻¹(x) is [-3, ∞).

In summary:

f(x)'s domain: [-3, ∞)

f⁻¹(x)'s domain: [0, ∞)

f(x)'s range: [0, ∞)

f⁻¹(x)'s range: [-3, ∞)

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The solutions to the equations are;

1) {-2, 1/2}

2) {0, -2}

What is zero product property?

The zero product property is a fundamental property of algebra that states that if the product of two or more factors is equal to zero, then at least one of the factors must be zero. In other words, if a × b = 0, then either a = 0, b = 0, or both a and b are equal to zero.

Using the zero product property;

2x - 1 = 0

x = 1/2

x + 2

x = -2

The solution is;

{-2, 1/2}

Again for x(x + 2) = 0

x = 0

x + 2 = 0

x = -2

The solution is;

{0, -2}

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