f(x) is discontinuous.
Since LHS ≠ RHS ≠ f(x).
What is a function?A function is a relationship between inputs where each input is related to exactly one output.
Example:
f(x) = 2x + 1
f(1) = 2 + 1 = 3
f(2) = 2 x 2 + 1 = 4 + 1 = 5
The outputs of the functions are 3 and 5
The inputs of the function are 1 and 2.
We have,
f(x) = x + 1, for x ≤ 2
f(x) = 2x - 1, for 1 < x < 2
f(x) = x - 1, for x < 1
Now,
A x = 1
LHS of f(x) = [tex]\lim_{x\to1^-}[/tex] f((h - 1)) = [tex]\lim_{h \to 0}[/tex] (h - 1 - 1) = 0 -2 = -2
RHS of f(x) = [tex]\lim_{x\to1^+}[/tex] f(h + 1) = [tex]\lim_{h \to 0}[/tex] (2(h + 1) - 1) = 2h + 2 - 1 = 0 + 1
f(1) = x + 1 = 1 + 1 = 2
We see that,
LHS ≠ RHS ≠ f(x)
So,
f(x) is not continuous, it is discontinuous.
Thus,
f(x) is discontinuous.
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The complete question.
Show that the following functions are continuous or discontinuous at x = 1.
f(x) = x + 1, for x ≤ 2
f(x) = 2x - 1, for 1 < x < 2
f(x) = x - 1, for x < 1
what is the slope of (-11,12) and (5,4)
The slope of a line passing through two points (x1, y1) and (x2, y2) can be found using the following formula:
m = (y2 - y1) / (x2 - x1)
Plugging in the points (-11, 12) and (5, 4), we get:
m = (4 - 12) / (5 - (-11)) = -8 / 16 = -1/2
So the slope of the line passing through (-11, 12) and (5, 4) is -1/2.
Ten times the sum of a number and eight is -10. What is the number?
Answer:
Step-by-step explanation:
Answer:
The number is -10.4
Explanation:
Assume that the number we are looking for is x.
Now, we want to get 10 times te sum of half the number and 6 and equate this expression to 8. Then we can solve for x
Let's take this step by step:
half the number is 0.5x
the sum of half the number and 6 is 0.5x + 6
10 times the sum of half the number and 6 is 10(0.5x +6)
10 times the sum of half the number and 6 equals 8 is 10(0.5x+6) = 8
Now we can solve for x as follows:
10(0.5x + 6) = 8
5x + 60 = 8
5x = 8 - 60
5x = -52
x = -52/5
x = -10.4
Hope this helps :)
Data for car production per shift at a car manufacturing facility are shown below: 688 711 625 701 688 667 694 630 547 703 688 697 703 656 677 700 702 688 691 664 688 679 708 699 667 703 i.
The mean of the data is given by the equation M = 679.3846 and the standard deviation σ = 34.1535
What is Standard Deviation?The standard deviation refers to a measurement of the data dispersion from the mean. A low standard deviation implies that the data are grouped around the mean, whereas a large standard deviation shows that the data are more dispersed.
Standard Deviation is calculated by
Each number's deviation from the mean must be determined, and then squared.
Next, add up all of these values.
Divide the result by the quantity of numbers.
Then calculate its square root.
σ = √ ( 1/N ) ( ∑₁ⁿ ( x₁ - μ )²
Given data ,
Let the data be represented as Set A
Now , the value of set A is
Set A = { 688 , 711 , 625 ,701 , 688 , 667 , 694, 630, 547, 703, 688, 697, 703, 656, 677, 700, 702, 688, 691, 664, 688, 679, 708, 699, 667, 703 }
The mean of the sample data = Sum of all the values / number of values
The mean M = 17664 / 26
The mean M = 679.3846
Now , the standard deviation is calculated by
σ = √ ( 1/N ) ( ∑₁ⁿ ( x₁ - μ )²
Substituting the values in the equation , we get
The standard deviation σ = √ ( 30328.153846154 / 26 )
The standard deviation σ = √ ( 1166.4674556213 )
The standard deviation σ = 34.1535
Hence , the standard deviation is 34.1535
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is 5.2481 a ratinal number
Answer:
Step-by-step explanation:
Rational numbers are numbers that can be written in the form pq, where p and q are integers and q≠0. The difference between rational numbers and fractions lies in the fact that fractions cannot have negative numerator or denominator.
so, yes.
Find the mean of the data summarized in the given frequency distribution. Compare the computed mean to the actual mean of 51.8 degrees.
Low Temperature
(circle◦F)
40minus−44
45minus−49
50minus−54
55minus−59
60minus−64
Frequency
22
55
1212
77
11
The mean of the frequency distribution is____ degrees. (Round to the nearest tenth as needed.)
Which of the following best describes the relationship between the computed mean and the actual mean?
A. The computed mean is close to the actual mean because the difference between the means is less than 5%.
B. The computed mean is not close to the actual mean because the difference between the means is less than 5%.
C. The computed mean is not close to the actual mean because the difference between the means is more than 5%.
D. The computed mean is close to the actual mean because the difference between the means is more than 5%.
The mean of the frequency distribution is 52 degrees. and option A is carrect . A) The computed mean is close to the actual mean because the difference between the means is less than 5%
First, we calculate the class mid-points (x); This is the mean of the class intervals.
For 40 - 44, x = (40 + 44)/2 = 42. This is applied to all other classes.
40 −44 = 42
45 −49 = 47
50 −54 = 52
55 −59 =57
64 −64 = 62
So:
x 42 47 52 57 62
f 22 55 1212 77 11
mean = ∑f(x)/∑f
= (42*22+47*55+52*1212+57*77+62*11)/(22+55+1212+77+11)
= 71604/1377
=52
actual mean = 51.8
and now mean = 52
percentage difference = ((52-51.8)/51.8)*100
= 0.3861 percentage
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PLEASE I NEED HELP ASAP
A circle has a circumference of 21.98 inches. What is the diameter of the circle? (Use
3.14 for )
Answer:
Step-by-step explanation:
The formula for the circumference of a circle is C = 2 * π * r, where r is the radius of the circle. We can find the radius of the circle by rearranging the formula to r = C / (2 * π).
Substituting the given values into the formula, we get:
r = 21.98 inches / (2 * 3.14) = 21.98 inches / 6.28 = 3.50 inches
The diameter of the circle is equal to twice the radius, so we can find the diameter by multiplying the radius by 2:
d = 2 * r = 2 * 3.50 inches = 7.00 inches.
To find the diameter of a circle with a given circumference, you can use the following steps:
Define the formula for the circumference of a circle: The circumference of a circle is given by the formula C = 2πr, where π (pi) is a mathematical constant approximately equal to 3.14, and r is the radius of the circle.
Substitute the given circumference into the formula: If the circumference of the circle is 21.98 inches, we can substitute that into the formula to get 21.98 = 2πr.
Solve for the radius: To solve for the radius, we can divide both sides of the equation by 2π: 21.98 / 2π = r. Using 3.14 for π, this gives us 21.98 / (2 * 3.14) = r, which can be simplified to r = 2.01 inches.
Calculate the diameter: The diameter of a circle is twice the radius, so the diameter can be calculated as 2 * r = 2 * 2.01 = 4.02 inches.
So, the diameter of the circle with a circumference of 21.98 inches is 4.02 inches.
< Question 4, 1.2.23 >
points
O Points: 0 of 1
Use the method of successive differences to determine the next number in the given sequence.
7; 35; 173; 771; 2529; 6667; 15,095
Assuming that the pattern continues, the eighth term would be
The pattern does not have a consistency to get the differences
How to find the patternThe method of successive differences involves finding the difference between each consecutive term in a sequence, and then using those differences to find the next term.
Here's how the method would work for the sequence given:
7; 35; 173; 771; 2529; 6667; 15,095
Difference between terms:
28; 138; 598; 1758; 4138; 8428
Second difference between terms:
110; 460; 1160; 2380; 4290
Third difference between terms:
350; 700; 1220; 1910
Fourth difference between terms:
350; 520; 690
There is no consistency between the patterns.
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In a survey, 32 people were asked how much they spent on their child's last birthday gift. The results were roughly bell-shaped with a mean of $45 and standard deviation of $19. Construct a confidence interval at a 80% confidence level.
Give your answers to one decimal place.
Answer:
{40.7,49.3}
Step-by-step explanation:
[tex]\displaystyle CI=\bar{x}\pm z\frac{s}{\sqrt{n}}\\\\CI_{80\%}=45\pm1.282\biggr(\frac{19}{\sqrt{32}}\biggr)\\ \\CI_{80\%}\approx45\pm 4.3\\\\CI_{80\%}=\{40.7,49.3\}[/tex]
Thus, we are 80% confident that the true sample mean amount spent is between $40.70 and $49.30
A function has the following properties:
• It is increasing and linear when the value of x is between -5 and -3.
• It remains constant when the value of x is between -3 and 0.
• It is decreasing and linear when the value of x is between 0 and 3.
• It is increasing and nonlinear when the value of x is between 3 and 5.
Which graph best represents this function?
Answer:
c
Step-by-step explanation:
trust me
2 Δ Δ Δ Δ Δ
Δ.
57. If t is any real number, which of the following
statements must be true?
A. -t <0
B. -t
C. -t=t
D. ts|-t
E. (-t)2 = -12
58. Let a and h be positive real num
When t is any real number, the true statements will be:
A. -t < 0:
B. -t = 0
E. (-t)^2 = -t^2
How to get the true statementsIf t is any real number, then the following statements are true:
A. -t < 0: This statement is true if t > 0, and false if t <= 0.
B. -t = 0: This statement is true if t = 0, and false if t ≠ 0.
C. -t = t: This statement is false for all real numbers t, since the negative of any non-zero real number is not equal to the original number.
D. t|-t: This statement is false for all real numbers t, since the absolute value of any real number is non-negative.
E. (-t)^2 = -t^2: This statement is true for all real numbers t, since squaring a real number gives a non-negative result. Note that the negative sign in front of the square is distributive and can be moved inside the square.
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A positive integer is twice another. The difference of the reciprocals of the two positive integers is frac(1,10). Find the two integers.
The two integers are 5 and 10.
What do you mean by Integers?
An integer is a number with no decimal or fractional part and it includes negative and positive numbers, including zero.
Integer is a Latin word which means 'whole' or 'intact'. This means integers do not include fractions or decimals.
A number is positive if it is greater than zero, so it said to be positive integers
A number is negative if it is less than zero, so it said to be negative integers.
A number line is a visual representation of numbers on a straight line. This line is used for the comparison of numbers that are placed at equal intervals on an infinite line that extends on both sides, horizontally.
Given:
Let x be the positive number
2x be the other positive number
Find the first integer
[tex]\frac{1}{x} - \frac{1}{2x} = \frac{1}{10}[/tex]
[tex]\frac{2x - x}{2x^2} = \frac{1}{10}[/tex]
[tex]\frac{x}{2x^2} = \frac{1}{10}[/tex]
[tex]\frac{1}{2x} = \frac{1}{10}[/tex]
2x = 10
x = 5
Therefore, the first positive integer is 5.
Find the other integer
2x = 2(5) = 10
Therefore, the other integer is 10.
To check:
[tex]\frac{1}{x} - \frac{1}{2x} = \frac{1}{10}[/tex]
[tex]\frac{1}{5} - \frac{1}{10} = \frac{1}{10}[/tex]
[tex]\frac{2}{10} - \frac{1}{10} = \frac{1}{10}[/tex]
[tex]\frac{1}{10} = \frac{1}{10}[/tex]
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Please help!! Due tomorrow morning!!
Directions: if QS represents an angle bisect or, solve for x.
(Ignore my answer)
The solution for x in the triangle is x = 52.5
How to solve for x in the triangle?
The bisector of an angle in a triangle divides the opposite side in the ratio of the sides containing the angle. That is:
PQ/QR = PS/SR
PQ = x, QR = 35, PS = 42 and SR = 70 - 42 = 28.
Therefore, we have:
PQ/QR = PS/SR
x/35 = 42/28
28*x = 42*35
28x = 1470
x = 1470/28
x = 52.5
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Solve for x with the given measures
The measure of angle the value of x =-96.
What is parallelogram?
A parallelogram is a special kind of quadrilateral composed of parallel lines. Any angle between the adjacent sides of a parallelogram is allowed as long as the opposite sides are parallel. A quadrilateral is a parallelogram if its two opposite sides are parallel and congruent. A parallelogram is a quadrilateral that has two pairs of opposite sides that are parallel and equal.
With the use of alternate interior angles, we can quickly resolve this. Right, the interior of a parallelogram is always 360 degrees.
(-x-6)° = 90
x = -96
Hence The measure of angle the value of x =-96.
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there are 877 students taking College Algebra or Calculus. There are 478 students taking College Algebra, 428 students taking Calculus, and 29 students taking both College Algebra and Calculus. How many students are taking College Algebra, but not Calculus?
The entry ticket to a local museum costs $12 for an adult and $8 for a child. A group of 100 people paid an entry fee of $1,056. How many adults and how many children were there in this group?.
The number of adults and children in the group were 64 and 36 respectively by using an algebraic equation.
It is given that the entry ticket to a local museum costs $12 for an adult and $8 for a child. A group of 100 people paid an entry fee of $1,056.
Let the number of adults in the group = x
number of adults in the group = y
Total number of people paid an entry fee = 100
So the equation becomes : x + y = 100
or x = 100 - y
Cost for adult ticket = $12
Cost for child ticket = $8
Total entry fee paid = $1,056
Second equation = 12x + 8y = 1056
putting value of x from equation 1
12(100 - y) + 8y = 1056
1200 - 12y + 8y = 1056
4y = 144
y = 36
x = 100 - 36 = 64
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A well of diameter 5 m is dug 24 m deep. The earth taken out of it has been spread evenly all around it in the shape of a circular ring of width 3 m to form an embankment. Find the height of the embankment.
Answer: To find the height of the embankment, we need to first find the volume of the well and then the volume of the embankment.
Volume of the well = πr^2h = π * (d/2)^2 * h = π * (5/2)^2 * 24 = 157.08 cubic meters
Volume of the embankment = π * (R^2 - r^2) * h, where R is the outer radius of the embankment and r is the inner radius.
Outer radius of the embankment = (5 + 3)/2 = 4 m
Inner radius of the embankment = (5 - 3)/2 = 1 m
Volume of the embankment = π * (4^2 - 1^2) * h = π * 15 * h
We know that the volume of the well is equal to the volume of the embankment, so we can write:
157.08 = π * 15 * h
Solving for h, we get:
h = 157.08 / (π * 15) = 157.08 / 47.12 ≈ 3.32 m
So, the height of the embankment is approximately 3.32 meters.
Step-by-step explanation:
A school is having a movie day. They are selling popcorn at the school for $0.50 per bag. If a student purchases popcorn with $5, how much popcorn would the student receive?
This student would receive 10 bag of popcorns.
How to determine the quantity of popcorn?Generally speaking, a unit price can be defined as the price that is being charged by a seller for the sale of a single unit of product.
In order to determine the quantity of popcorn, we would make use of the information provided about the unit price for one bag of popcorn by using the following mathematical expression;
Number of bag of popcorns = Cost of one bag of popcorn/Unit price
Substituting the given parameters into the mathematical expression above, we have the following;
Number of bag of popcorns = 5/0.50
Number of bag of popcorns = 10 bag of popcorns.
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What is the slope between the two points
(-10,50) and (-4,20)
Answer:
Below
Step-by-step explanation:
Slope = rise / run = change in y / change in x = -30 / 6 = -5
Answer: -5
Step-by-step explanation:
Start with the slope formula
m = (y2 - y1) / (x2 - x1)
Substitute point values in the formula
m = (20 - 50) / (-4 - -10)
Simplify the fraction
m = (20 - 50) / (-4 - -10) = -30 / 6
Solve for slope (m)
m = -5
Find the amount of money (Future Value) in an account where $5,100 is deposited (Present Value) at an interest rate of 8% per year compounded continuously and the money is left in the account for 13 years.
round to 2 decimal places.
Answer:
Step-by-step explanation:
The formula for calculating the future value of a continuously compounded investment is:
FV = PV * e^(rt)
where PV is the present value, r is the annual interest rate as a decimal, t is the time in years, and e is the mathematical constant approximately equal to 2.718.
Plugging in the given values, we get:
FV = 5100 * e^(0.08 * 13) = 5100 * e^1.04 = 5100 * 2.856116786 = 14536.86
So, the future value of the investment after 13 years at an 8% interest rate compounded continuously would be approximately $14,536.86. Rounded to 2 decimal places, this is $14,536.86.
which answer is the right choice
7200+80x = 3200+120x
Reason:
Canton starts off with 7200 people. Add on 80x to represent the fact the town grows 80 people per year. That's how we end up with the expression of 7200+80x
Similarly, the expression 3200+120x represents the idea Holly Springs started off with 3200 people and grows at 120 people per year.
The sum of two numbers is 160. One number is 40 more than the other. Find the numbers.
The larger number is
The smaller number is
Answer:
The larger number is 100
The smaller number is 60
Step-by-step explanation:
x + (40 + x) = 160
2x + 40 = 160
x + 20 = 80
x = 60
60 + (40 + 60) = 160
160 = 160
If AB = 12 and BC = 19,
then what does AC equal?
The value of length AC is,
⇒ AC = 31
What is Addition?The process of combining two or more numbers is called the Addition. The 4 main properties of addition are commutative, associative, distributive, and additive identity.
Given that;
The lengths are,
AB = 12 and BC = 19
Now, If the points A, B and C are colinear.
Then, The value of length AC is,
⇒ AC = AB + BC
⇒ AC = 12 + 19
⇒ AC = 31
Thus, We get;
⇒ AC = 31
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19. For the period 2009-2013, if T is the number (in millions) of
Comcast video subscribers, then 7 10.3 is approximately the
number (in millions) of Time Warner Cable video subscrib-
ers (Source: Comcast, Time Warner Cable). There were about
21.7 million Comcast video subscribers in 2013. Estimate the
number of Time Warner Cable video subscribers in 2013.
The estimated number of Time Warner Cable video subscribers in 2013 is approximately 14.8 million.
Estimation by ProportionsWe are given that for the period 2009-2013, 7/10.3 is approximately the number of Time Warner Cable video subscribers for every Comcast video subscriber. We are also given that there were 21.7 million Comcast video subscribers in 2013.
To estimate the number of Time Warner Cable video subscribers in 2013, we can use the following proportion:
7/10.3 = x/21.7
where x is the number of Time Warner Cable video subscribers in 2013.
Solving for x, we get:
x = (7/10.3) x 21.7
x ≈ 14.8 million
Therefore, the estimated number of Time Warner Cable video subscribers in 2013 is approximately 14.8 million.
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Please help with these questions
Answer: number 2 is A
Step-by-step explanation:
What is the y-intercept of the line perpendicular to the line y = 3/4x + 3 that includes the point (3, 1)?
Answer:
Step-by-step explanation:
perp.: -4/3
y - 1 = -4/3(x - 3)
y - 1 = -4/3x + 4
y = -4/3x + 5
y-int: (0, 5)
[tex]\frac{x+8}{3} =7\\[/tex]
Answer:
The solution is 13
Step-by-step explanation:
x+8/3 = 7
x+8 = 7×3
x+8 = 21
x = 21-8
x = 13
Number story that the answer will give me 1/4
Answer:
he division story will be that there are 16 boys who want to share 4 oranges. What fraction will each person get?
Number of boys = 16
Number of oranges = 4
Fraction that each person will get will be:
= Number of oranges / Number of boys
= 4/16
= 1/4
Each boy gets 1/4.
Step-by-step explanation:
Faith needs 300 milliliters of a 57% acid solution for a chemistry experiment. She has two acid solutions, A and B, that can be mixed together to form the solution. Solution A is 50% acid, and Solution B is 60% acid. How much of each solution should she use to create the mixture?
She should use milliliters of Solution A.
She should use milliliters of Solution B.
The needed amount of each solution is given as follows:
90 milliliters of Solution B.210 milliliters of Solution B.How to obtain the amounts?The amounts are obtained using a system of equations, for which the variables are given as follows:
Variable x: amount of Solution A.Variable y: amount of Solution B.Faith needs 300 milliliters of the solution, hence:
x + y = 300.
x = 300 - y.
The percentage is of 57%, hence:
0.5x + 0.6y = 0.57 x 300
0.5x + 0.6y = 171.
Replacing the first equation into the second, the value of y is obtained as follows:
0.5(300 - y) + 0.6y = 171
0.1y = 21
y = 21/0.1
y = 210.
The value of x is then given as follows:
x = 300 - 210
x = 90.
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Serina wants to solve the following system of equations in the most efficient way. 2 x + 3 y = 18. X + 7 y = 31. She plans to solve for x in the first equation as her first step since both 2 and 3 can be divided into 18. Why is serina mistaken?serina should have solved for x in the second equation because it has a coefficient of 1. Serina should have solved for y in the first equation because dividing by 3 instead of by 2 would give a smaller number in the solution. Serina should have solved for y in the second equation because it has the largest coefficient. Serina should have solved for y in the first equation because the division step will be easier since 18 is divisible by 3.
Serina is mistaken because Serina should have solved for x in the second equation because it has a coefficient of 1.
Serina is mistaken because solving for x in the first equation as her first step does not necessarily lead to the most efficient solution. While it is true that both 2 and 3 can be divided into 18, this approach may require additional steps to solve for the other variable, y, in the second equation.
In this case, it is actually more efficient to solve for x in the second equation because it has a coefficient of 1. This means that once we solve for x, we can substitute that value into the first equation and solve for y directly, without needing to perform any additional calculations.
Here's how we can solve the system of equations using this approach:
X + 7 y = 31 (solve for x)
x = 31 - 7y
Substitute x = 31 - 7y into the first equation:
2 x + 3 y = 18
2 (31 - 7y) + 3 y = 18
62 - 14 y + 3 y = 18
-11 y = -44
y = 4
Substitute y = 4 into the equation we used to solve for x:
x = 31 - 7y
x = 31 - 7(4)
x = 3
Therefore, the solution to the system of equations is x = 3, y = 4.
In summary, while it is important to consider which variable to solve for first when solving a system of equations, the most efficient approach may vary depending on the specific equations and coefficients involved. In this case, solving for x in the second equation was the most efficient approach.
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someone help me please
Step-by-step explanation:
This symbol is signma, which is the sub of all the X's after plugging them into the equation.
Just to show the pattern, the first 3 numbers on your table are 15, 13, and 13... doing those 3 with Sigma, and simplifying
[tex]( {1 5 - 2)}^{2} + ({13 - 2)}^{2} + {(13 - 2)}^{2} [/tex]
[tex]169 + 121 + 121[/tex]
You would pretty much do that all the way down the list.
After doing so, we wind up with
15, 13, 13, 4, 19, 12, 6, 5, 14
[tex]169 + 121 + 121 + 4 + 289 + 100 + 16 + 9 + 144[/tex]
Adding these together gets your answer of:
[tex]973[/tex]
