How to tell if a function is continuous or discontinuous?

Answers

Answer 1

A functions is discontinuous if it has stuff like holes, jumps, or vertical asymptotic. In addition, it's continuous if it stays in a line the whole time (it doesn't have to be straight or anything!)

Answer 2

Final answer:

A function is continuous at a point if it is defined at that point, the limit exists as it approaches the point, and the limit is equal to the function's value at that point. If these conditions are not met, the function is discontinuous. The first derivative must also be continuous unless given specific exceptions.

Explanation:

To determine if a function is continuous or discontinuous, you can apply several tests. A function y(x) is said to be continuous at a point if it meets three conditions:

The function must be defined at that point.The limit of the function as it approaches that point exists.The limit of the function as it approaches the point is equal to the function's value at that point.

If any of these conditions fail, the function is discontinuous at that point. For example, a function with a hole, jump, or asymptote at a point would be discontinuous there. Also, if the first derivative of y(x) is not continuous, unless the potential function V(x) is infinite, this indicates a discontinuity. In continuous probability functions, such as probability density functions, discontinuities are not present, and the function's graph does not have breaks or holes. Instead, the entire graph is a single unbroken curve.

Related Questions

Do you know the answer to this question

Answers

Answer: [C]: " f(x) = log₃ (x − 3) " .
_______________________________________________________

Steven investigates the amount of damage to the head gaskets on the trucks in his fleet and finds that the damage index depends on the ambient temperature. He develops the equation y = - 2 3 x + 14 to model the relationship. If the temperature increases 1 degree, what happens to the damage index for the head gasket?

Answers

It's D because it decreases by 2/3 of a point

Answer:

Step-by-step explanation:

Question 1 (3 points) study the Proof Given.

List the reasons for Items 2,3, & 4.

Right the number and then the answer.
Question 2 (3 points
)Study the Proof: Give the reasons for the missing reason. List the number and then state the reason.

Answers

Question 1
2. Definition of midpoint
3. Given
4. Transitive Property

Question 2
1. Given
2. Addition Property of Equality
3. Segment Postulate Property.

What is the area of a circle with a radius of 14 feet? use 22/7 for π?

Answers

So the formula is pi*r^2 and if you plug that in it is 22/7* 14^2 and you get 616

Patty is, right now, 3 times as her brother Terry. in 10 years, the sum of their age will be 36. how old are they right now? show your work.

Answers

Patty at year x will be 3 times the age of his brother

In 10 years (year z) the sum of their age will be 36

36-20=16
(Twenty being ten years gained for each of them)

16 is now the sum of their age in year x

The following are options for the ages they could possibly be
1-15 (1*3 does not equal 15)
2-14(2*3 does not equal 14)
3-13(3*3 does not equal 13)
4-12(4*3 equals 12)!!!!
5-11(5*3 does not equal 11)
6-10(6*3 does not equal 10)
7-9 (7*3 does not equal 9)
8-8(8*3 does not equal 8)

Therefore the answer would be:
Patty is 22 and his brother is 14

Patty is 12 years old, and Terry is 4 years old. After setting up an equation based on their ages now and in 10 years, we determine that Terry is 4, and since Patty is three times as old, she is 12.

The problem presented is a typical age-related algebraic question. To solve it, we can set up equations based on the information given. Let's define Patty's current age as 3T, where T represents Terry's current age. In 10 years, Patty will be 3T + 10 and Terry will be T + 10. According to the problem, the sum of their ages in 10 years will be 36. So, we can write the following equation:

3T + 10 + T + 10 = 36

Solving this equation:

Combine like terms: 4T + 20 = 36Subtract 20 from both sides of the equation: 4T = 16Divide both sides by 4 to find Terry's age: T = 4

With Terry's age being 4, Patty's current age is 3 times Terry's age, which means Patty is 3 × 4 = 12 years old. So, Patty is 12 and Terry is 4 years old right now.

What is the 100th term in the following arithmetic number pattern: 2,7,12

Answers

So the change in between terms is 5 so you would put 5x, then for the first term, which is 2, 5*1(x)(because it's first term) then the difference between 5-2(because it's the 2nd term) So you would plug 100 in for x so 5(100)-3=497. Hope this helped!

six times the reciprocal of a number equals 2 times the reciprocal of 6. find the number.

Answers

Equation:
6(1/x) = 2(1/7)
-----
6/x = 2/7
-----
2x = 42
x = 21

6(1/x) = 2(1/6)

6/x = 1/3

6/x*1/6 = 1/3*1/6

x= 1/18

the number is 18

The quotient of a number and five increased by four is sixteen answer

Answers

The problem would be written out like this: n/5 + 4 = 16. We would have to subtract 4 from both sides to get n/5 = 12. Then, we multiply both sides by 5 and after that step, you get 60. n = 60.

Answer: [tex]\dfrac{x}{5}+4=16\\\\and\\\\x=60[/tex]

Step-by-step explanation:

Since we have given that

The quotient of a number and five increased by four is sixteen a

Let the number be 'x'.

The quotient of a number and five is written as

[tex]\dfrac{x}{5}[/tex]

And it is increased by 4 is written as

[tex]\dfrac{x}{5}+4[/tex]

And the equation is equal to 16.

[tex]\dfrac{x}{5}+4=16[/tex]

And the solution would be

[tex]\dfrac{x}{5}+4=16\\\\\dfrac{x}{5}=16-4=12\\\\x=12\times 5\\\\x=60[/tex]

Hence, [tex]\dfrac{x}{5}+4=16\\\\and\\\\x=60[/tex]

NEED HELP BADLY PLEASEEEE WILL GIVE BRAINLY

List the next three terms in the following sequence:
1, 2, 4, 8, 16, 32,...

a. 48, 80, 128
b. 64, 96, 160
c. 64, 128, 256
d. 46, 92, 184

Answers

Its C because it multiplies by 2

The first number multiplied by two is 2. Two multiplied by two is four. Four multiplied by two is eight. Eight multiplied by two is 16....and so on. So in this case, the answer would be C because 32*2=64, 64*2=128, 128*2=256.

Explain why the following statement are equivalent 4(x+y)=4x+4y

Answers

They are equivalent because they are showing the Distributive Property....The Distributive Property shows that you can Distribute one number into another one, and still get the same answer.....So, you would have to do 4x+4y!!! I hope this helps!!!

The equation 4(x+y)=4x+4y represents the same value after solving.

What is Distributive Property?

This property states that multiplying the total of two or more addends by a number will produce the same outcome as multiplying each addend by the number separately and then adding the results together.

Given:

We have 4(x+y)=4x+4y.

We have the Distributive Property

a x ( b+ c )= a x b + a x c

let x= 2 and y = 5

So, 4 (x+ y) = 4 (2+ 5)= 4(7) = 28

and, 4x+ 4y= 4(2)+ 4(5) = 8+ 20 = 28

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A given cube has a volume of 1728 cubic inches. Find the length of an edge, area of one side, surface area, and surface area in square feet.

Answers

for a cube
v=s³
SA=6s²
area of 1 side=s²
s=length of 1 edge

so

1728=s³
cube root both sides
12=s

length of an edge is 12in

the area of one side is 12² or 144 in²

surface area is 6s² or 6(144) or 864 in²

A given cube with a volume of 1728 cubic inches:

Find the length of an edge:
The edge length (s) can be found using the formula for the volume of a cube: s³ = 1728. Therefore, s = 12 inches.Find the area of one side:
Each face of the cube is a square, so area of one face = s² = 12² = 144 square inches.Find the surface area:
Total surface area = 6 * (area of one face) = 6 * 144 = 864 square inches.Find the surface area in square feet:
To convert square inches to square feet, divide by 144 (since 1 square foot = 144 square inches). Therefore, surface area in square feet = 864 / 144 = 6 square feet.

Y = -27x + 577, where x represents the number of weeks and y represents the amount of money left in her account. After how many weeks will Alma have $145 left in her account?

Answers

To solve this problem, plugin 145 where the y is located and then solve for x
y = -27x +577
145 = -27x +577
145 -577 = -27x
-432 = -27x
-432/-27 = -27x/-27
16 = x
x = 16
After 16 weeks she will have $145 dollars in her account.

It snowed 4 times during October. The snowfall amounts were 0.75 inches, 1.93 inches, 4.73 inches, and 2.33 inches. What was the total snowfall for October?

Answers

9.74 inches i did this problem yesterday

The total snowfall for October was 9.74 inches. Adding the snowfall amounts of 0.75, 1.93, 4.73, and 2.33 inches resulted in this sum.

Let's break down the calculation step-by-step:

Identify the given snowfall amounts for October:

The snowfall amounts are 0.75 inches, 1.93 inches, 4.73 inches, and 2.33 inches.

Add up the snowfall amounts to find the total:

Total snowfall = 0.75 inches + 1.93 inches + 4.73 inches + 2.33 inches.

Perform the addition

Total snowfall = 9.74 inches.

The total snowfall for October was 9.74 inches. This result is obtained by adding all four individual snowfall amounts together.

So, the calculation in step-by-step form is as follows:

0.75 + 1.93 + 4.73 + 2.33 = 9.74 inches.

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The circumference of a circular field is 295.16 yards what is the radius of the field use 3.14 for pi and do not round your answer

Answers

Circumference = 2 x 3.14 x radius

295.16 = 6.28 x radius

295.16 / 6.28 = radius

47 = radius

The radius of circle is 47 yards.

What is Circumference of circle?

The measurement of the circle's perimeter, also known as its circumference, is called the circle's boundary. The region a circle occupies is determined by its area. The length of a straight line drawn from the centre of a circle equals the diameter of that circle. It is typically expressed in units like cm or unit m.

The radius of the circle is considered when applying the formula to determine the circumference of the circle. Therefore, in order to calculate the circle's perimeter, we need to know the radius or diameter value.

Given:

Circumference of circle= 295.16 yards

Now,

circumference of circle= 2πr

295.16= 2 x 3.14 x r

295.16 = 6.28 x r

r = 295.16/ 6.28

r = 47 yards.

Hence, the radius of circle is 47 yards.

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The quotient of 4 and the difference of an integer and 3 is the same as two-thirds. Find the integer.

Answers

To find the integer, you must cross multiply, group the same terms, then divide both side by 2:

Solution:
4/(x-3)= 2/3
12 = 2x-6
18=2x
x= 9

16 pound turkey cost $14.24 what is the price per pound

Answers

$.89 per pound is the answer

The answer would be $0.89,
$14.24 divided by 16

Solve the following problem for the roots by completing the square

3x^2 -12x = 9

Answers

3(x^2+3) is the answer you are looking for. :)

3x^2 - 12x = 9
Divide both sides by 3
x^2 - 4x = 3
To make a square with "x^2 - 4x", you need to have a "+ 4" after it so it can be simplified as (x-2)^2. So add 4 to both sides.
x^2 - 4x + 4 = 7
(x-2)^2 = 7
x - 2 = (square root) of 7
x = 2 (+/-) (square root) of 7.
x = {-0.65, 4.65}

The solutions are -0.65 and 4.65.

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Express the volume v of the box as a function of the length x of the sidfe of the square cut from each corner

Answers

i dont know what the word sidfe

Add the following military times: 0645 + 0745

Answers

14:30 is the answer.
(Aka 2:30 PM)

40 students are going on a field trip to the aquarium. Every student needs to be in a group, and all groups must have the same number of students. If there is 1 group, there will be students in the group. If there are groups, there will be 20 students in each group. If there are 4 groups, there will be students in each group.

Answers

10 is the answer because 40/1 = 40, 40/2= 20 and 40/= 10

If there is 1 group there will be 40 students in the group. There will be 20 students in each group if there are 2 groups. If there are 4 groups, there will be 10 students in each group.

What is Division?

Division is one of the four basic operations of mathematics.

On performing the operation division, a number is divided into some number of equal parts.

The number which is being divided is called the dividend, the number which divides the dividend is called divisor and the number got when division is performed is called the quotient and the remaining number which has not divided is called the remainder.

Total number of students = 40

If there is 1 group, number of students in the group = 40/1 = 40

If there are 20 students in each group, then the number of groups = 40/20 = 2.

If there are 4 groups, then number of students in each group = 40/4 = 10.

Hence there will be 40 students in the group if there is 1 group. There are 2 groups if there will be 20 students in each group. There will be 10 students in each group if there are 4 groups.

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what property is illustrated by this problem 2(3 + 4) = 2x 3 +2 x 4

Answers

Distributive Property

Can someone help me with these maths questions? I totally don't get them.

Do the following coordinates lie on the following graphs?

1) y=2x+4 (7,20)
2) y=4x-14 (-1,-18)
3) y=5x-1 (0.5,2.5)
4) y=-2x+6 (0.8,5.4)
[tex] 5) y=\frac{1}{2}x+2 (0.7,1.6) [/tex]

Answers

1. 2(7) +4 = 18, does not = 20, does not lie on graph

2. 4(-1) -14 = -18, equals -18, does lie on the graph

3. 5(.5)-1 = 1.5, does not = 2.5, does not lie on the graph

4. -2(0.8)+6 = 4.4, does not = 5.4 does not lie on graph

5. 1/2(0.7) +2 = 2.35, does not = 1.6, does not lie on graph

What value of c makes the trinomial below a perfect square x2 + 10x + c?

Answers

hello :
x² + 10x + c
calculate the discriminant Δ = b²-4ac a = 1 b = 10 c = c
Δ = 10²-4c
the trinomial below a perfect square when : Δ = 0
10² -4c = 0
4c = 10²
c= 25
x²+10x+25 = x² +2(5)x+5² = (x+5)²

The sum of the first and third of three consecutive even integers is 128. find the three even integers.

Answers

62, 64, and 66. 62+66=128

Final answer:

The three even integers are 62, 64, and 66.

Explanation:

The sum of the first and third of three consecutive even integers is 128. To find the three even integers, we start by letting the first integer be x. Because the integers are consecutive even numbers, the second integer will be x + 2 and the third will be x + 4. The sum of the first and third integers gives us the equation:

x + (x + 4) = 128

Solving this equation:

Combine like terms: 2x + 4 = 128.

Subtract 4 from both sides: 2x = 124.

Divide both sides by 2: x = 62.

So, the first even integer is 62. Consequently, the second is 62 + 2 = 64, and the third is 62 + 4 = 66. Therefore, the three consecutive even integers are 62, 64, and 66.

Graph the image of the given triangle after the transformation with the rule (x, y)→(y, x) .

Answers

See the attached figure. The original triangle is shown in red (triangle ABC). The line of reflection is the blue dashed line. This is the equation y = x. Reflecting triangle ABC over the line y = x results in the purple triangle A'B'C'.

Note how the x and y values swap places.
We have...
point A = (5,1) turn into point A' = (1,5)
point B = (8,1) turn into point B' = (1,8)
point C = (8,6) turn into point B' = (6,8)
The swap occurs because the rule (x,y) --> (y,x) tells this to happen.

The purple triangle is the final answer.

The perimeter of a rectangle is 44 inches, and its area is 105 square inches. find the length and width of the rectangle.

Answers

Perimeter = (Length + Width)x2
Area = LengthxWidth

P=(L+W)x2
P/2=L+W
L=P/2 - W

Area=(P/2 - W)(W)
A=WP/2-W^2
105=W(44)/2 - W^2
105=22W - W^2
W^2-22W+105=0
(W-7)(W-15)=0
Width= 7 or 15

A=LxW
105=Lx7
105/7=L
L=15

Length is 15 inches, Width is 7 inches.

Two thirds of a number is no more than -10

Answers

Let x be the number in the problem. When it states that x is "no more than -10", it means than it is less than or equal to -10. So, we have

[tex] \frac{2}{3} [/tex]x≤-10

In order to cancel out the [tex] \frac{2}{3} [/tex] on the left and isolate x, we multiply both sides by [tex] \frac{3}{2} [/tex] (since [tex] \frac{3}{2}[/tex]·[tex] \frac{2}{3} [/tex]=1). Thus, we have

x≤-10·[tex] \frac{3}{2} [/tex]=-15

Therefore, x≤-15

With regard to mathematics, what is an accepted statement of fact that is used to prove other statements

Answers

In mathematics, a proof is a deductive agruement for a mathematical statement.In the argument,other prevoiusly establised statements, such as theroms, an be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference.

Is 0 prime odd or even prime or composite?

Answers

0 is a unique case. It is neither prime nor composite, and is neither odd nor even. Zero is an absence of value case which is not made from other numbers...and didn't appear in numbering systems until the past couple of thousand years...it's a weird concept!

Answer:

0 is nor prime or composite

Step-by-step explanation:

The definition of a prime number is a positive integer that has exactly two positive divisors. Therefore, the simplest reason why 0 is not prime is that 0 is not a positive integer, and the axiom says that all whole numbers greater than 1 are either prime or composite.

A kennel owner needs 4 1/2 feet of wall space for each crate. How many crates can she fit along a 27 foot wall?

Answers

6 /////////////////////////////////////

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