The function f(x) = [tex]\left \{ {{x+1} ,for x \leq 2 \atop {2x-1}, for 1 < x < 2} \atop {x-1} , for x < 1 \right.[/tex] is continuous to all the values.
In mathematics, a function is a rule that assigns each element from one set (called the domain) to exactly one element of another set (called the range). The continuity of a function describes how smoothly it behaves as its input values change. In this context, we will analyze the continuity of the given function.
The given function is a piecewise function, which means it has different rules for different intervals of its domain. Let's analyze each interval separately to determine if the function is continuous or discontinuous.
For x less than or equal to 2, the function is f(x) = x + 1. This is a linear function, and we know that all linear functions are continuous everywhere. Therefore, the function is continuous for x less than or equal to 2.
For 1 < x < 2, the function is f(x) = 2x - 1. This is also a linear function, and as we said before, all linear functions are continuous. Therefore, the function is continuous for 1 < x < 2.
For x less than 1, the function is f(x) = x - 1. This is also a linear function and is continuous for all values of x less than 1.
Therefore, we can conclude that the given function is continuous for all values of x.
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Write the number for three million twenty thousand
Answer: 3,020,000
Step-by-step explanation:
three million twenty thousand
Answer: 3,020,000
Step-by-step explanation: 3,000,000 + 20,000 = 3020000
5. A Set of Speakers costs $35 and Songs cost $195 each. The sales tax
rate is 8.25%. The expression 0.0825(35+ 195s) can be used to determine
the amount of sales tax due on the entire purchase of a set of Speakers
and S.Songs. Expand the expression and round to the nearest hundredth.
The amount of sales tax due on the entire purchase of a set of speakers and s songs is $2.89 plus $16.09 for each song purchased, rounded to the nearest hundredth.
What is an expression?
In mathematics, an expression is a combination of numbers, variables, and/or operators that represents a mathematical quantity or relationship. It can be a single number or variable, or a combination of multiple numbers, variables, and operators.
The given expression 0.0825(35+195s) represents the amount of sales tax due on the entire purchase of a set of speakers and s songs, where s is the number of songs purchased.
Expanding the expression, we get:
0.0825(35+195s) = 0.0825(35) + 0.0825(195s)
= 2.8875 + 16.0875s
Rounding this expression to the nearest hundredth, we get:
2.89 + 16.09s
Therefore, the amount of sales tax due on the entire purchase of a set of speakers and s songs is $2.89 plus $16.09 for each song purchased, rounded to the nearest hundredth.
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what ratios are equivalent to 6 : 3?
Answer: 12: 6 and 24: 12 and 3: 1.5
Step-by-step explanation: A ratio can be equivalent through multiplication such as 6: 3 to 12: 6
Answer:
2 : 1,
12/
Step-by-step explanation:
all you have to do is cross out the numbers, basically simplify the
the answer is 2:1 because in 3's multiplication table,
3×2=
and
3×1=
similarly , for 12/6, divide instead of multiplication, there area few other ratios equivalent to 6/
If a²=b+c,b² = c+a and c² = a +b, prove that: (1/a+1)+(1/b+1)+(1/c+1)=1
The proof is as shown below
How to prove that 1/(a+1) + 1/(b+1) + 1/(c+1) = 1?Trigonometry is a branch of mathematics dealing with the relationship between the ratios of the sides of a right-angled triangle with its angles.
If a² = b + c , b² = c + a and c² = a + b
We want to prove that 1/(a+1) + 1/(b+1) + 1/(c+1) = 1
Since a² = b + c
a²+a = a + b + c
a (a+1) = a + b + c
a+1 = (a + b + c)/a
Repeat the same process for b² = c + a and c² = a +b, we have:
1 + b= (a + b + c)/b
1 + c =(a + b + c)/c
Substitute the values 1+a, 1+b and 1+c into 1/(a+1) + 1/(b+1) + 1/(c+1):
1/(a+1) + 1/(b+1) + 1/(c+1) = a/(a+b+c) + b/(a+b+c) + c/(a+b+c) (Taking LCM)
= (a+b+c )/ (a+b+c ) = 1
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the expression n^2 - n - 30 can be written in Factored Form as (n + 5) (n + k), where k represents a number
Answer:
k = -6
Step-by-step explanation:
Expand by substituting -6:
(n+5) (n-6) = n^2 + 5n - 6n (-6×5)
Thus, (n+5)(n-6) = n^2 -1n -30
(In a facotrised form for quadratics, the numbers need to add to 'bx', in this case -1, and multiply to form 'c' (-30). So -6×5 = -30 and k= -6)
a fitness club leases space at a major shopping center. the club has 10,500 square feet which represents 16% of the total leasable space at the shopping center. calculate the total leasable space at the center?
The shopping mall has 65,625 square feet of total leasable area. As stated in the problem, this indicates that the fitness club takes up 16% of the total leasable space.
According to the information provided, a fitness club occupies 10,500 square feet or 16% of the total leasable area at a sizable shopping mall. We can use the proportional and percentage approaches to determine the total leasable area.
Let's assume that the variable "x" stands for the whole amount of leasable space. Since we are aware that the area leased by the gym is 16% of the total leasable area, we can create the following equation:
10,500 = 0.16x
To solve for "x", we can divide both sides of the equation by 0.16:
10,500 / 0.16 = x
Simplifying this expression gives us:
x = 65,625
Therefore, The shopping mall has 65,625 square feet of total leasable area. As stated in the problem, this indicates that the fitness club takes up 16% of the total leasable space.
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Malik just got hired for a new job and will make $96, 000 in his first year. Malik was
told that he can expect to get raises of $5,000 every year going forward. How much
money in salary would Malik make in his 11th year working at this job?
The amount of money Jamar will make in his 18th year working at this job is = $107,500
What is salary outcome?A salary is a form of periodic payment from an employer to an employee, which may be specified in an employment contract. It is contrasted with piece wages, where each job, hour or other unit is paid separately, rather than on a periodic basis.
here, we have,
Calculation of yearly salary outcome
The amount of money Jamar made in his first year= $48,000
The salary rise per year = $3,500
The amount of money he will make for the extra 17 years is
= 17 × $3,500
= $59,500
Therefore the amount he will receive as his salary in the 18th year working at this job = $48,000 + $59,500
= $107,500
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calculate the area of a rectangle in cm2 with a width of 7.6 cm and height of 3.3 cm. be sure to include the proper number of significant figures.
The width and height are given to one decimal place, and we have multiplied them together, the answer should be rounded to one decimal place. Therefore, the area of the rectangle is 25.1 cm²
To calculate the area of a rectangle, we need to multiply its width by its height.
The size of a patch on a surface is determined by its area. Surface area refers to the area of an open surface or the boundary of a three-dimensional object, whereas the area of a plane region or plane area refers to the area of a form or planar lamina. Area can be interpreted as the quantity of material with a specific thickness required to create a model of the shape or as the quantity of paint required to completely cover the surface in a single coat.
Area = width x height
Substituting the given values, we get:
Area = 7.6 cm x 3.3 cm
Area = 25.08 cm²
Since the width and height are given to one decimal place, and we have multiplied them together, the answer should be rounded to one decimal place. Therefore, the area of the rectangle is 25.1 cm² (rounded to one decimal place).
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PLEASE HELP ME WITH THESE!!!!!!!!!!!!!!!!!!!!
If I have an answer please ignore my current answer and just solve for the correct answer. Thanksssss!
The volume of the rectangular prism will be 1008 cubic inches and the volume of the pyramid is 504 cubic inches.
What is volume?The volume of the rectangular prism is defined as the space in three-dimension covered by the rectangular prism having the sides Length, Width, and Height.
The volume of the rectangular prism will be the multiplication of the three parameters which are Length, Width, and Height of the prism.
The volume of the pyramid is the product of half of the area of the base and the height of the pyramid,
The volume of prism = 14 x 12 x 6
The volume of the prism = 1008 cubic inches
The volume of the pyramid,
Volume = 1/2 x Base area x H
Volume = 1/2 x 72 x 14
Volume = 504 cubic inches.
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true or false: the rule of thirds is the process of dividing an image into thirds, using two horizontal lines and two vertical lines, and positioning the most important elements inside the squares the lines create.
True. The rule of thirds is a photography and visual arts guideline that involves dividing an image into thirds using two horizontal lines and two vertical lines, and then placing the image's most important elements at the intersection points or along the lines.
This aids in the creation of a balanced and visually appealing composition. This results in nine rectangular areas of equal size, with four intersection points where the lines cross.
The rule of thirds principle suggests that the points of intersection or the lines themselves are ideal places to position the composition's most important elements. The resulting image is more visually balanced and pleasing to the eye as important elements are placed in these areas.
It is important that, however, that the rule of thirds is not a hard and fast rule, and that there were times when breaking it can result in more dynamic and interesting compositions. The rule of thirds is merely a tool for artists and photographers to use in order to create more effective and engaging images.
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Helppppppppppp meeeeee guyssssssss
Answer:
2. $9.99
3. 45 tickets for $27
4. 20 cents
Step-by-step explanation:
2. 79.92/8 = 9.99
3. 27/45 0.6 and 15/20 = 0.75 and 60 cents is less than 75 cents
4. 6/30 = 20
Answer:
Step-by-step explanation:
79.92/8= 9.12 per cd
15/20= .75
27/45=.6 better deal because it is .60 cents per ticket
6/30= .20 per can
I NEED HELP PLEASEEEE!!!!
Answer:
Step-by-step explanation:
the answer is 420 cuase the manufatcure of math is hard i wont give an explanation.
The price of a gallon of unleaded gas was $2.81 yesterday. Today, the price rose to $2.88. Find the percentage increase. Round your answer to the nearest tenth of a percent.
The percentage increase of the gas would be 2.5%.
How to calculate a percentage increase?We can calculate the value of the percentage increase by the following formula,
percentage increase = (final value - initial value) / initial value × 100%
As per the given question, we have
initial value = $2.81 , and final value = $2.88
Substitute the values in the above formula, and we get
percentage increase= (2.88 - 2.81 ) / 2.81 × 100%
percentage increase = (0.07) / 2.81× 100%
percentage increase = 0.0249 × 100%
Apply the multiplication operation, and we get
percentage increase = 2.49%
Thus, the percentage increase of an item would be 2.5%.
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200$ marked down 20% now increased by 20% is it equal to 200 again?
Answer: nope
Step-by-step explanation:
No, the 200$ marked down 20% would be 200 - 0.2 * 200 = 160$. Then, if it is increased by 20%, it would be 160 + 0.2 * 160 = 192$. So, 200$ is not equal to 192$.
Select the correct answer.
Which graph is the graph of f(x) = 3 log2 (x +1)?
The graph of the equation of function f (x) = 3 log₂ (x + 1) is shown in figure.
What is mean by Function?A relation between a set of inputs having one output each is called a function. and an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).
Given that;
The equation of function is,
⇒ f (x) = 3 log₂ (x + 1)
Now, By using tool we can draw the graph of the equation of function
f (x) = 3 log₂ (x + 1) as shown in figure.
Thus, The graph of the equation of function f (x) = 3 log₂ (x + 1) is shown in figure.
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Mark is buying 4 computers for his business. The table shows the prices and the advertised sales for the same type of computer at 4 computer stores. Based on the advertised sales, at which store will mark get the lowest price on 4 computers?
At a total price of $2720 for 4 Computers, retailer 2 delivers the best deal based on the listed specials.
Based on the stated sales, we need to figure out the entire cost of 4 computers at each retailer, accounting for any discounts or promotions, in order to determine which store will provide the lowest pricing for 4 Computers.
Let's use the following variables to symbolize the costs and special offers at each retailer:
P1: the price of a single computer in a store 1
P2: the price of a single computer in a store 2
P3: the price of a single computer in a store 3
P4: the price of a single computer in a store 4
D1: discount on one computer at store 1
D2: discount on one computer at store 2
D3: discount on one computer at store 3
D4: discount on one computer at store 4
Then, we may figure the total cost of four computers at each store using the following formulas:
Total cost at store 1 = 4(P1 - D1)
Total cost at store 2 = 4(P2 - D2)
Total cost at store 3 = 4(P3 - D3)
Total cost at store 4 = 4(P4 - D4)
The total cost at each store can then be calculated by plugging in the values from the table as follows:
Store 1: Total cost = 4(950 - 0.2*950) = 3040
Store 2: Total cost = 4(800 - 0.15*800) = 2720
Store 3: Total cost = 4(1100 - 0.25*1100) = 3300
Store 4: Total cost = 4(850 - 0.1*850) = 3060
Therefore, At a total price of $2720 for 4 Computers, retailer 2 delivers the best deal based on the listed specials.
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mr hall is doggie-paddling in the school pool at a speed of 1 m/sec when he spots his favorite rubber ducky. He increases his speed to 3m/sec over a period of 3 seconds. What is Mr. Halls acceleration?
Answer:
1m/sec
Step-by-step explanation:
Mr. Hall spent 3 seconds to increase his speed to 3m/sec increasing 1m/sec, don't over think it.
1. what is the last name of whom we honor for discovering ways to express and minimize logic though a new but similar form of algebra? 2. what does the k in k-map stand for? 3,4,5,
Boole. K-map stands for Karnaugh Map and is used to simplify and minimize the expression of logic in a similar form to algebra.
George Boole is the mathematician we honor for discovering ways to express and minimize logic through a new but similar form of algebra. The technique used to simplify and minimize these expressions is known as a Karnaugh Map, or K-map for short. A K-map is a visual representation of a Boolean expression using squares and circles to represent true or false variables in a table. By arranging the variables in the table, the same logic can be expressed in a simpler form with fewer variables. This allows for a simpler and more efficient form of logic expression. K-maps are used in many fields, such as digital logic design, where the efficiency of expression is important.
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Find the indicated IQ score. The graph to the right depicts IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15.
The indicated IQ score, x, is enter your response here. (Round to one decimal place as needed.)
The indicated IQ score is given as 117.
How to solve for the IQ scoreThe area under the curve is given as 0.1431
the area to the left of x = 1 - 0.1431
= 0.8569
for 0.8569 the z stat is 1.07
u = 100
sd = 15
x = u + z * sd
= 100 + 0.8569 * 15
= 116.05
approximately 117
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Find the indicated IQ score. The graph to the right depicts IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15.
xx
0.1431
A symmetric bell-shaped curve is plotted over a horizontal scale. A vertical line runs from the scale to the curve at labeled coordinate x, which is to the right of the curve’s center and peak. The area under the curve to the right of the vertical line is shaded and labeled 0.1431.
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What quantity must be added to 3-7x+4 to obtain - +4x-2?
A-4x2+3x+2
B.-412 + 11x + 2
C.-4x² + 11x-6
D. 2x²-3x+2
E. 2x²-3x+6
Answer:
Step-by-step explanation: (10)(A) Add and subtract polynomials of degree one and degree two. (10)(B) Multiply polynomials of degree one and degree two. (10)(C) Determine the quotient
Hope it helps =}
Answer:
b
Step-by-step explanation:
An athletic track is given in the figure.
The two straight parts are 80 m long.
The total length of the track is 400 m.
Find the radius of the curved part of the track
on both sides.
Subtract the length of the two straight parts
from the entire length of the track.
The radius of the curved part of the track will be 37.3m.
Solution:
Given that
The length of straight parts = 80m each
The total length of the track = is 400m
So, the total length of curved parts = 400 - (80*2) = 240m
Therefore, the radius of the curved parts =
[tex]2\pi r = 240\\r = 240/2\pi \\[/tex]
Hence, r = 37.3m.
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question 2
help me plsssss
Answer:
∠ CAB = 144°
Step-by-step explanation:
the central angle CAB is twice the measure of ∠ BDC , that is
∠ CAB = 2 × 72° = 144°
Someone help, please!
The appropriate response will be (A). The first equation of motion gives the link between "velocity and time."
What are the names of equations?Equation: A statement that two variable or integer expressions are equal. In essence, equations are questions, and the motivation for the development of mathematics has been the systematic search for the answers to these questions.
What are the uses of equations?An equation is a mathematical representation of two equal objects, one on each side of a "equals" sign. We can solve challenges in our daily lives with the aid of equations. Most of the time, we look for help with pre algebra to resolve problems in real life. Pre-algebra concepts contain the fundamentals of mathematics.
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the foreign language club is showing a four-movie marathon of subtitled movies. how many ways can they choose from the available?
The number of ways the foreign language club can choose 4 movies from the available is [tex]nC4 = n! / 4!(n-4)![/tex].
To calculate the number of ways the foreign language club can choose four movies from the available subtitled movies, we need to use the combination formula, which is:
[tex]nCr = n! / r!(n-r)![/tex]
Where n is the total number of available subtitled movies and r is the number of movies to choose (in this case, r=4).
Assuming there are at least four subtitled movies available, the number of ways to choose four movies would be:
[tex]nC4 = n! / 4!(n-4)![/tex]
Simplifying the equation further, we have:
[tex]nC4 = (n * (n-1) * (n-2) * (n-3)) / 4 * 3 * 2 *1[/tex]
Therefore, the number of ways the foreign language club can choose four movies from the available subtitled movies is given by the formula nC4.
The formula calculates the number of possible combinations of four movies that can be selected from the total number of available subtitled movies, without regard to the order in which they are shown.
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how do i do these types of problems?
Answer:
1/24.
Step-by-step explanation:
There are 2 events here and they are independent, so the single probabilities are multiplied.
P(4 when die is rolled) = 1/6
P(A on the spinner) = 1/4
So, P(4 and A)
= 1/6 * 1/4
= 1/24.
if m+1 is an even integer, which one of the flollowing must be add integer?
If m + 1 is an even integer, then 2m + 1 is an odd integer.
What do you understand by even and odd numbers?Even number: An even number is a number that is a multiple of 2. It can be said that any number that is completely divisible by 2 is an even number.
Odd number: Odd numbers are those numbers that are not completely divisible by 2.
let us understand by addition on different combinations of odd numbers and even numbers.
Odd number + Odd number = Even number
Even number + Even number = Even number
Even number + Odd number = Odd number
Odd number + Even number = Odd number
In subtraction
Even number - Even number = Even number
Even number - Odd number = Odd number
Odd number - Even number = Odd number
Odd number - Odd number = Even number
In multiplication
Even number × Even number = Even number
Even number × Odd number = Even number
Odd number × Odd number = Odd number
Given:
m + 1 is an even integer.
since alternate numbers are even.
Therefore, m must be an odd number.
If m is odd, then m - 1 is even.
2m is divisible by 2, so it is always an even number.
So,
2m - 2 is even.
2m + 1 is odd.
2m + 2 is even.
Since 2m + 1 is an odd number.
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Find the slope of the line through each pair of points. ) (11, 8), (- 13, 11)
The slope of the line through each pair of points (11, 8), (- 13, 11) is -1/8.
Slope of a line:The slope of a line is defined as the change in y- coordinate with respect to x - coordinate. The slope of a line represents the direction and steepness of a line. The formula for the slope of a line that passes through the points (x₁, y₁) and (x₂, y₂) is given by
Slope (m) = [ y₂ - y₁]/ [x₂ - x₁]Here we have
The pair of points (11, 8), (- 13, 11)
By the given formula,
Slope (m) = [ y₂ - y₁]/ [x₂ - x₁]
The slope from (11, 8) to (- 13, 11) can be calculated as follows
Slope = [ 11 - 8 ]/ [ -13 -11 ] = 3/-24 = -1/8
Therefore,
The slope of the line through each pair of points (11, 8), (- 13, 11) is -1/8.
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when running Chao has a stride length of 70cm,correct to the nearet 5cm.
Chao runs a distance of 11.2 km, correct to the nearest 0.1km
work out the minumum number of stride that Chao could take to complete this distance
The minimum number of strides that Chao could take to complete a distance of 11.2 km is approximately 16,000.
What is simplification?The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.
Here,
To calculate the minimum number of strides that Chao could take to complete a distance of 11.2 km, we need to first convert the distance to meters since the stride length is given in centimeters.
11.2 km = 11,200 meters (1 km = 1000 meters)
Stride length in meters = 70 cm / 100 cm/m = 0.7 m
Now, we can calculate the minimum number of strides as follows,
Minimum number of strides = (Distance covered in meters) / (Stride length in meters)
Minimum number of strides = 11,200 m / 0.7 m ≈ 16,000
Therefore, the minimum number of strides that Chao could take to complete a distance of 11.2 km is approximately 16,000.
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Solve the system of equations:
Answer:
The solution is (x = 6, y = -3)
Step-by-step explanation:
[tex]\sf \frac{x}{y + 1} = - 3 \: \: ........(1) \\ \\ \sf \frac{y}{x - 3} = - 1 \: \: ........(2) \\ \\ - - - - - - - - - \\ \\ \sf x + 3y = - 3 \: ........(1) \\ \\ \sf x + y = 3 \: \: \: \: \: \: ........(2) \\ \\ - - - - - - - - - \\ \\ \: \: \: \sf x + 3y = - 3 \: \: \: \: \: | \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \sf x + y = 3 \: \: \: \:\: \: \: \: | \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: [/tex]
1) Write the equation of the line passing through point P with slope m, if
a) P(1,4) and m = 3
c) P (3,0) and m = −1
b) P(-1,-2) and m = 4
d) P(0,0) and m = -5
Answer:
a) y - 4 =3(x - 1)
b) y + 2 = 4(x+ 1)
c) y = -(x - 3)
d) y = -5x
Step-by-step explanation:
You did not say which form, so I put them in the point-slope form of a line that follows the pattern:
y - [tex]y_{1}[/tex] = m( x - [tex]x_{1}[/tex])
Answer:
see explanation
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
(a)
given m = 3 , then
y = 3x + c ← is the partial equation
to find c substitute (1, 4 ) into the partial equation
4 = 3(1) + c = 3 + c ( subtract 3 from both sides )
1 = c
y = 3x + 1 ← equation of line
(b)
given m = 4 , then
y = 4x + c ← is the partial equation
to find c substitute (- 1, - 2 ) into the partial equation
- 2 = 4(- 1) + c = - 4 + c ( add 4 to both sides )
2 = c
y = 4x + 2 ← equation of line
(c)
given m = - 1 , then
y = - x + c ← is the partial equation
to find c substitute (3, 0 ) into the partial equation
0 = - 3 + c ( add 3 to both sides )
3 = c
y = - x + 3 ← equation of line
(d)
given m = - 5 , then
y = - 5x + c ← is the partial equation
to find c substitute (0, 0 ) into the partial equation
0 = - 5(0) + c = 0 + c ⇒ c = 0
y = - 5x ← equation of line
