Write the equation of the piecewise function ƒ that is represented by its graph.
Piecewise function with 3 pieces: [tex]x^2+1, 4x-8, 2[/tex]
The piecewise function ƒ that is represented by the graph in the image is:
ƒ(x) =
[tex]x^2+1, & \text{if } 0 \le x < 4 \\[/tex]
[tex]4x-8, & \text{if } 4 \le x < 5 \\[/tex]
[tex]5^2+1 = 26, & \text{if } x = 5[/tex]
This is because the graph of the function consists of three distinct pieces:
For 0≤x<4, the graph is a parabola with vertex at (0,1) and opening upwards. This suggests that the function is of the form [tex]ax^{2} +bx+c.[/tex]
We can find the values of a, b, and c by substituting the points (0,1), (1,1), and (4,16) into the equation.
This gives us the system of equations:
\begin{cases}
[tex]a \cdot 0^2 + b \cdot 0 + c = 1[/tex]
[tex]a \cdot 1^2 + b \cdot 1 + c = 1 \\[/tex]
[tex]a \cdot 4^2 + b \cdot 4 + c = 16[/tex]
\end{cases}
Solving this system gives us a=1, b=0, and c=1, so the equation of the function for this interval is [tex]x^{2} +1[/tex].
For 4≤x<5, the graph is a line with slope 4 and y-intercept −8.
This suggests that the function is of the form mx+b.
We can find the values of m and b by substituting the points (4,2) and (5,25) into the equation.
This gives us the system of equations:
\begin{cases}
[tex]4m+b = 2 \\[/tex]
5m+b = 25
\end{cases}
Solving this system gives us m=4 and b=−8, so the equation of the function for this interval is 4x−8.
For x=5, the graph is a horizontal line at y=26.
This suggests that the function is of the form c.
We can find the value of c by simply looking at the graph.
This gives us c=26, so the equation of the function for this interval is 26.
Therefore, the complete piecewise function is:
ƒ(x) =
\begin{cases}
[tex]x^2+1, & \text{if } 0 \le x < 4 \\[/tex]
[tex]4x-8, & \text{if } 4 \le x < 5 \\[/tex]
[tex]5^2+1 = 26, & \text{if } x = 5[/tex]
\end{cases}
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A corporation employs 2000 male and 500 female engineers. a stratified random sample of 200 male and 50 female engineers gives each engineer 1 chance in 10 to be chosen. this sample design gives every individual in the population the same chance to be chosen for the sample. is it an srs? explain your answer.
No, the sample design is not a simple random sample (SRS), but a stratified random sample. In an SRS, every individual in the population has an equal chance of being chosen for the sample.
Explanation:The given sample design is not a simple random sample (SRS). In an SRS, every individual in the population has an equal chance of being chosen for the sample. However, in this case, the sample design is a stratified random sample. It involves dividing the population into groups (in this case, male and female engineers) and then selecting a sample from each group using a different sampling rate.
To determine if a sample design is an SRS, we need to ensure that each individual in the population has an equal chance of being chosen, and this probability is the same for every individual. In the case of the given example, the sample design does not meet these criteria, making it not an SRS.
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If a triangle has an angle greater than 90 degrees, then it is not a right triangle true or false?
PLEASE HELP!!!
Taylor bought 200 shares of stock for $18.12 per share last year. He paid his broker a flat fee of $30. He sold the stock this morning for $21 per share, and paid his broker 0.5% commission.
a. What were Taylor's net proceeds?
b. What was his capital gain?
Taylor's net proceeds from the sale of his stock are $4179, after paying his broker's 0.5% commission. His capital gain on the investment is $525, calculated by subtracting the total purchase cost, including the initial broker's flat fee, from the net proceeds of the sale.
a. What were Taylor's net proceeds?
To calculate Taylor's net proceeds, we need to consider the initial cost of buying the shares, any fees paid to the broker, and the amount earned from selling the shares. Here's the breakdown:
Determine initial cost: 200 shares x $18.12 per share = $3624
Add broker's fee when buying: $3624 + $30 = $3654
Subtract broker's fee when selling: $42 (0.5% of $8400 from selling 200 shares at $21 per share)
Calculate net proceeds: $8400 - $42 = $8358
b. What was his capital gain?
To find Taylor's capital gain, subtract the total cost (initial purchase cost + fees) from the total earnings after selling the shares. The calculation is as follows:
what is the sale tax on item that costs $33.50. if the sale tax is 0.06 on every $1?
Average precipitation for the first 7 months of the year, the average precipitation in toledo, ohio, is 19.32 inches. if the average precipitation is normally distributed with a standard deviation of 2.44 inches, find these probabilities.
Using the normal distribution and the central limit theorem, it is found that there is a:
a) 0.7054 = 70.54% probability that a randomly selected year will have precipitation greater than 18 inches for the first 7 months.b) 0.8869 = 88.69% probability that five randomly selected years will have an average precipitation greater than 18 inches for the first 7 months.Normal Probability Distribution
In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
It measures how many standard deviations the measure is from the mean. After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].In this problem:
The mean is of 19.32 inches, hence [tex]\mu = 19.32[/tex].The standard deviation is of 2.44 inches, hence [tex]\sigma = 2.44[/tex].Item a:
The probability that a randomly selected year will have precipitation greater than 18 inches for the first 7 months is 1 subtracted by the p-value of Z when X = 18, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{18 - 19.32}{2.44}[/tex]
[tex]Z = -0.54[/tex]
[tex]Z = -0.54[/tex] has a p-value of 0.2946.
1 - 0.2946 = 0.7054.
0.7054 = 70.54% probability that a randomly selected year will have precipitation greater than 18 inches for the first 7 months.
Item b:
Now, we want the probability that five randomly selected years will have an average precipitation greater than 18 inches for the first 7 months, hence:
[tex]n = 5, s = \frac{2.44}{\sqrt{5}}[/tex]
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{18 - 19.32}{\frac{2.44}{\sqrt{5}}}[/tex]
[tex]Z = -1.21[/tex]
[tex]Z = -1.21[/tex] has a p-value of 0.1131.
1 - 0.1131 = 0.8869.
0.8869 = 88.69% probability that five randomly selected years will have an average precipitation greater than 18 inches for the first 7 months.
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#2
Point C has a _____ abscissa and a _____ ordinate.
positive, negative
negative, positive
negative, negative
positive, positive
What does negative 2 over 3 > −1 indicate about the positions of negative 2 over 3 and −1 on the number line?
Answer:
Step-by-step explanation:
Divide 42 in a ratio of 1:2:3?
(60 POINTS! I NEED HELP NOW PLEASE!)
Given a polynomial function f(x), describe the effects on the y-intercept, regions where the graph is increasing and decreasing, and the end behavior when the following changes are made. Make sure to account for even and odd functions.
-When f(x) becomes f(x) -3
-When f(x) becomes -2 * f(x)
4x-3y+6z=18,-x+5y+4z=48,6x-2y+5z=0 what are x,y,and z?
Peggy needs to pay car insurance 2 times every year. She has divided the payment by 6 and saves that amount each month. Which account below is the best account in which to save this money? a. 401-K retirement fund with matching deposits from Peggy’s employer. b. A checking account, or a savings account for other expenses (predictable), and emergencies. c. A savings account for college tuition. d. Coin jar.
Answer:
b. A checking account, or a savings account for other expenses (predictable), and emergencies.
Step-by-step explanation:
Peggy needs to pay car insurance 2 times every year.
She has divided the payment by 6 and saves that amount each month.
So, the best account in which she can save this money is - A checking account, or a savings account.
Other options are irrelevant as a 401-K retirement fund is some other type of account. It is not used for daily or monthly expenses.
Options C and D are also not correct.
A culture started with 4,000 bacteria. After 5 hours, it grew to 4,800 bacteria. Predict how many bacteria will be present after 15 Hours.
Round your answer to the nearest whole number
Answer:
6900 rounded answer -
Step-by-step explanation:
Plato users, correct.
What is the procedure for multiplying a two digit number by a one digit number?
At a certain college, there are 600 freshman, 400 sophomores, 300 juniors, and 200 seniors. if one student is selected at random, what is the probability that the student is a sophomore?
a video store sold 2 vhs tapes for every 4 dvds if 564 dvds were sold how many vhs tapes were sold?
To find the number of VHS tapes sold when 564 DVDs were sold, divide the number of DVDs by 2.
Explanation:To solve this problem, we can set up a ratio comparing the number of VHS tapes sold to the number of DVDs sold. Since the ratio is given as 2 VHS tapes to every 4 DVDs, we can simplify it to 1 VHS tape to every 2 DVDs. If 564 DVDs were sold, we can divide this number by 2 to find the number of VHS tapes sold:
564 DVDs ÷ 2 = 282 VHS tapes
Therefore, 282 VHS tapes were sold.
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How can u tell
Whether 1080 is cube number
The solution is, 1080 = 2*2*2*3*3*3*5, is not a cube number.
What is multiplication?In mathematics, multiplication is a method of finding the product of two or more numbers. It is one of the basic arithmetic operations, that we use in everyday life.
here, we have,
given that,
the number is 1080
Split into prime factors
1080 = 2*2*2*3*3*3*5
so, we have,
Its not a cube
because although there are 2 triplicates there is also a 5.
as, we know that,
Note 2*2*2*3*3*3 is a cube.
so, 1080 = 2*2*2*3*3*3*5, is not a cube number.
Hence, The solution is, 1080 = 2*2*2*3*3*3*5, is not a cube number.
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How to solve 2(2d+4)=-3(d+2) ?
Find the product: 5(−3)(−2)
PLEASE HELP ASAP!
a physicist working in a large labortory has found out that the light particles traveling in a particle accelerator increase velocity in a manner that can be described by linear function -4x+3y=15, where x is time and y is velocity in kilometers per hour. use this function to determine when a certain particle will reach 30 km/hr
Find the side of a square whose diagonal is of the given measure.
Given = 8 miles
To find the side of a square with a given diagonal length, apply the Pythagorean theorem. With a diagonal of 8 miles, the side length is calculated to be 4√2 miles.
Explanation:To find the side of a square whose diagonal is given, we can use the Pythagorean theorem. In a square, the diagonal creates two right-angled triangles, each with the square's sides as the legs.
Let the side of the square be s, and the given diagonal is 8 miles. According to the Pythagorean theorem, the diagonal (d) is calculated as d = √(s² + s²). Simplifying, this becomes d = √(2 * s²) or s = d / √2. Substituting the value of the diagonal, we get s = 8 / √2. To get rid of the radical in the denominator, we can multiply both the numerator and denominator by √2, giving us s = (8√2) / 2, which simplifies to s = 4√2 miles. Therefore, the side of the square is 4√2 miles.
The regular price of a jacket is $42.75. during a sale, the jacket was marked 12% off. what was the price of the jacket during the sale? (1 point) $5.13 $30.75 $37.62 $42.63
Please help!!!!!
The function graphed shows the total cost for a taxi cab ride for x miles.
Select from the drop-down menus to correctly identify the taxi cab ride information provided by the graph.
The slope is ________
a. 5
b. 3
c. 2.5
d. 0.2
The slope represents _______
a. the total cost of the taxi ride
b. total number of miles traveled
c. cost per mile traveled
d. the initial cost of the taxi ride
The slope of the line between cost of taxi and distance is 2.5 and it represents the cost of taxi per mile of distance travelled.
What is slope of line ?
Slope of line is the angle made by the line from positive x-axis in anticlockwise direction, it also denoted the steepness of the line.
The function graphed shows the total cost for a taxi cab ride for x miles the cost in taxi is shown in y axis and the distance covered by taxi in x axis. Now, to find put the slope we must two coordinates which can be found out from the graph easy by observation as (0,3) and (2,8).
How using slope formula to find the slope of line :
[tex]\begin{aligned}\frac{y_{2}-y_{1}}{x_{2}-x_{1}}&=\frac{8-3}{2-0}\\&=\frac{5}{2}\\&=2.5\end{aligned}[/tex]
It represents the cost of taxi per mile of distance travelled.
Therefore, the slope of the line between cost of taxi and distance is 2.5 and it represents the cost of taxi per mile of distance travelled.
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Jake has $11 left in his pocket. he spent $3 on a bus ticket, $5 on lunch, and $5 on a movie. how much money did he have at the beginning of the day?
Answer:
Jake had twenty-four dollars ($24) at the beginning of the day.
Step-by-step explanation:
Jake has $11 dollars left in his pocket, after spending 3 dollars on a bus ticket, 5 dollars on lunch, and 5 more dollars on a movie. 3+5+5 is equal to 13, therefore if after spending 13 dollars Jake still has 11 dollars, which means that at the beginning of the day Jake had 24 dollars (11+13).
Which transformation is not isometric?
A stack of two hundred fifty cards is placed next to a ruler, and the height of stack is measured to be 5 8 58 inches.
Harry has 1/4 of apple pie that he wants to cut into 3 equal slices.what fraction of the whole pie is each slice?
If Harry has 1/4 of apple pie that he wants to cut into 3 equal slices. The fraction of the whole pie in each slice will be 1/12.
What is a fraction?Fraction number consists of two parts, one is the top of the fraction number which is called the numerator and the second is the bottom of the fraction number which is called the denominator.
It is given that, Harry has 1/4 of apple pie that he wants to cut into 3 equal slices.
We have to apply the ratio which is the comparison of two quantities to determine how many times one obtains the other.
Suppose the fraction of the whole pie is each slice will be x,
x = 1/4 / 3
x = 1 /12
Thus, if Harry has 1/4 of apple pie that he wants to cut into 3 equal slices. The fraction of the whole pie in each slice will be 1/12.
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Maddie has a retangular garden in her backyard. The perimeter of the garden is eighty-two feet. The width is five shorter than the length. What are the length and width of the garden?
This question is based on the following paragraph
Write your answer in standard form with integer coefficients.
2x + 3y = 30 ; (2,-5)