Let f be a function defined on all of that satisfies the additive condition f(x+y)=f(x)+f(y) for all a.) Show that f(0)=0 and that f(-x)=-f(x) for all x in .
b.) Show that is f is continuous at x=0, then f is continuous at every point in c.) Let k=f(1). Show that f(n)=kn for all n in the natural numbers, and then prove that f(z)=kz for all z in . Now, prove that f(r)=kr for any rational number r.
d.) Use (b) and (c) to conclude that f(x)=kx for all x in . Thus, any additive function that is continuous at x=0 must necessarily be a linear function through the origin.

The area of trapezoid is 16 m².

Given in the figure,

Height of trapezoid = 4 m

Length of 1st parallel base = 2 m

Length of 2nd parallel base = 6 m

We know that,

Area of trapezoid = 1/2 * sum of bases * height

∴

Area of trapezoid = 1/2 * (2+6) * 4

                             = 8 * 2 = 16 m²

Therefore, the area of trapezoid is 16 m².

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Answer:

x = 5/2 + (i sqrt(3))/2 or x = 5/2 - (i sqrt(3))/2

Step-by-step explanation:

Solve for x:

x^2 - 5 x + 7 = 0

Subtract 7 from both sides:

x^2 - 5 x = -7

Add 25/4 to both sides:

x^2 - 5 x + 25/4 = -3/4

Write the left hand side as a square:

(x - 5/2)^2 = -3/4

Take the square root of both sides:

x - 5/2 = (i sqrt(3))/2 or x - 5/2 = -(i sqrt(3))/2

Add 5/2 to both sides:

x = 5/2 + (i sqrt(3))/2 or x - 5/2 = -(i sqrt(3))/2

Add 5/2 to both sides:

Answer: x = 5/2 + (i sqrt(3))/2 or x = 5/2 - (i sqrt(3))/2

Answer:

x = - 2

Step-by-step explanation:

the magnitude of a vector < a, b, c > is

[tex]\sqrt{a^2+b^2+c^2}[/tex]

given vector < 1, 2, x > has magnitude of 3 , then

[tex]\sqrt{1^2+2^2+x^2}[/tex] = 3

[tex]\sqrt{1+4+x^2}[/tex] = 3

[tex]\sqrt{5+x^2}[/tex] = 3 ( square both sides to clear the radical )

5 + x² = 3² = 9 ( subtract 5 from both sides )

x² = 4 ( take square root of both sides )

x = ± [tex]\sqrt{4}[/tex] = ± 2

since x is negative then x = - 2

The center of the ellipse is (2, -8), The endpoints of the major axis are 10 units from the center, The endpoints of the minor axis are 9 units from the center, To graph the ellipse, connects (2, 2), (11, -8), (2, -18) and (-7, -8) with  smooth curve.

An ellipse in mathematics is a plane curve that has two focal points and is such that the sum of the two distances from any point on the curve to the focal points is a constant. It generalizes a circle, a special kind of ellipse in which the two focal points coincide.

Glven:

The equation of ellipse is,

[tex]\frac{(x-2)^2}{81}+\frac{(x+8)^2}{100}=1[/tex]

SInce, the general form of ellipse is,

[tex]\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1[/tex]

where,

(h, k) is the center of ellipse

a is the minor axis

b is the major axis.

Here

(h, k) = (2, -8) is the center of the ellipse.

9 is the minor axis and 10 is the major axis of the ellipse.

Also the graph of ellipse connects (2, 2), (11, -8), (2, -18) and (-7, -8) with  smooth curve.

Hence,  the center of the ellipse is (2, -8).

The endpoints of the major axis are 10 units from the center.

The endpoints of the minor axis are 9 units from the center.

To graph the ellipse, connects (2, 2), (11, -8), (2, -18) and (-7, -8) with  smooth curve.

The graph of the ellipse visit.

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Answer:

f^-1(x) = [tex]\sqrt[3]{(x + 4)/3}[/tex]

Step-by-step explanation:

y = 3x^3 - 4  ==> solve for x

y + 4 = 3x^3  ==> isolate x by adding 4 on both sides

(y + 4)/3 = x^3  ==> divide 3 into both sides

x = [tex]\sqrt[3]{(y + 4)/3}[/tex] ==> isolate x by taking the cube root on both sides

y = [tex]\sqrt[3]{(x + 4)/3}[/tex] ==> switch both x and y

f^-1(x) = [tex]\sqrt[3]{(x + 4)/3}[/tex]

Answer:

12

Step-by-step explanation:

2w + 4 = 28

2w = 24

w = 12

w = 12.

[tex]2w+4=28[/tex]

[tex]2w+4-4=28-4\\ \\2w=24[/tex]

[tex]\frac{2w}{2} =\frac{24}{2} \\ \\w=12[/tex]

To verify the answer, write the original expresion and substitute variable "w" by the calculated value (12). If the calculations equal the same number of both sides of the equal sign (=) then the answer is correct.

[tex]2(12)+4=28\\ \\24+4=28\\ \\28=28[/tex]

The same number appears on both sides of the equal sign. Hence, the answer is correct!

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Answer:

$20,400

Step-by-step explanation:

You just have to multiply 680,000 by 0.03 (since 3% is equivalent to 0.03)

The intercepts of the graph are

The polynomial is given as

G(x) = -2(3x+4)(x-1)(x-3)^2

Rewrite as

G(x) = -2(3x+4)(x-1)(x-3)²

The x-intercepts of a polynomial are the points at which the polynomial crosses the x-axis.

To find the x-intercepts, we need to set y = 0 and solve for x.

-2(3x+4)(x-1)(x-3)² = 0

Divide through by -1

(3x+4)(x-1)(x-3)² = 0

Now we can solve for each factor equal to zero:

3x+4 = 0, x = -1.33

x-1 = 0, x = 1

x-3 = 0, x = 3

So the x-intercepts are -4/3, 1, 3

The y-intercept of a polynomial is the point at which the polynomial crosses the y-axis.

For this polynomial, we have

G(0) = -2(30+4)(0-1)(0-3)² = -2(4)(-1)(9) = 72

So the y-intercept is (0, 72)

See attachment for the graph

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The person must leave the money for  18.6 .

         [tex]A= P (1 +\frac{r}{n} )^{nt}[/tex]

        A =Amount = 2900

        P = Principal = 1000

        r = ratio =  5.75%

        2900 = 1000[tex](1 +\frac{5.75}{12} )^{12x[/tex]

        2900 = 1000 [tex](1 +\frac{575}{120000} )^{12x[/tex]

        2900 = 1000[tex](1 +\frac{23}{4800} )^{12x\\\\[/tex]

        2900 = 1000[tex](1 +\frac{4823}{4800} )^{12x\\\\[/tex]

       2900/1000 = 1000[tex](\frac{4823}{4800} )^{12x\\\\[/tex]

               29/ 10 = [tex](\frac{4823}{4800} )^{12x\\\\[/tex]

         12x = [tex]log _{\frac{4823}{4800} }\frac{29}{10}[/tex]

            [tex]x = \frac{log _{\frac{4823}{4800} }\frac{29}{10}}{12}[/tex]

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If the streetlights are spaced in the same way, there will be 48 streetlights on the 4-mile road.

First, we know that there are 18 streetlights evenly spaced on a 1.5 mile road.

Then the space between consecutive streetlights is:

1.5mi/18 = (3/36) mi = (1/12) mi

Now, the number of streetlights that we will have on a 4-mile road is given by the quotient between the length of the road and the distance between consecutive streetlights, it is:

N = 4mi/(1/12 mi) = 12*4 = 48

There will be 48 streetlights on a 4 mile road.

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The exact value of cos (-pi/3) can be found using the unit circle or the angle addition formula.

Using the unit circle, we know that the x-coordinate of the point on the unit circle that corresponds to an angle of -pi/3 is cos(-pi/3) and y-coordinate is sin(-pi/3).

On the unit circle, the angle of -pi/3 is the same as the angle of 2pi/3 (since the unit circle has a period of 2pi), therefore the x-coordinate of the point on the unit circle that corresponds to an angle of -pi/3 is -1/2.

Alternatively, using the angle addition formula:

cos (-pi/3) = cos(pi - pi/3) = cos(pi) * cos (pi/3) - sin(pi) * sin(pi/3) = -1/2

So, the exact value of cos (-pi/3) is -1/2.

Answer:

[tex]x=8.5[/tex]

Step-by-step explanation:

By the definition of a midpoint, [tex]TU=UV[/tex].

[tex]6=2x-11 \\ \\ 2x=17 \\ \\ x=8.5[/tex]

The domain and the range of this function in interval notation include the following;

Domain = [-6, ∞].

Range = [-∞, 0].

In Mathematics, the horizontal extent of a graph represents all domain values and they are read and written from smaller to larger numerical values, and from the left of a graph to the right.

Furthermore, the vertical extent of a graph represents all range values and they are always read and written from smaller to larger numerical values, and from the bottom of the graph to the top.

By critically observing the graph shown in the image attached below, we can reasonably and logically deduce the following:

Domain = -6 < x < ∞ or [-6, ∞].

Range = -∞ < y < 0 or [-∞, 0].

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When a fair coin is tossed nine times, there is a 0.090 (rounded off to the nearest thousandth) chance that at most one will land on heads.

The probability that an event will occur or a claim will be true is measured by probability theory, a branch of mathematics. The probability of an occurrence is a number between 0 and 1, where about 0 represents how likely the event is to occur and 1 represents certainty. A probability is a numerical illustration of the possibility that a specific occurrence will take place. Probabilities can also be expressed as percentages ranging from 0% to 100% or from 0 to 1. the ratio of the total number of outcomes to the proportion of occurrences in a whole set of equally likely options that lead to a specific occurrence.

given

binomial probability ⁿCₓ[(p)ˣ(q)ⁿ⁻ˣ]

where p= probability of tail = 1/2 (favorable outcome)

q = probability of head = 1/2: (1 - p)

n = total outcome = 9

x = 4

ⁿCₓ[(p)ⁿ(q)ⁿ⁻ˣ] = ⁹C₄[(1/2)⁴(1/2)⁵]

=  [1/16 x 1/32]

= (512)

126/512 = 0.24609

≈ 0.090 nearest thousandth rounded off .

When a fair coin is tossed nine times, there is a 0.090 (rounded off to the nearest thousandth) chance that at most one will land on heads.

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Answer:

23.2wx - 13.2w

Step-by-step explanation:

4w (5.8x - 3.3) =

4w * 5.8x - 4w * 3.3 = distribute 4w to 5.8x and 3.3

4 * 5.8 *w * x - 4w * 3.3 =

23.2 * w * x - 13.2w  ==> simplify

23.2wx - 13.2w

These approximate percentages are defined by the Empirical Rule, for a normally distributed variable:

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The solutions are -

S3 = (1/3) - (1/4) + (1/4) - (1/5) + (1/5) - (1/6)

S4 = (1/3) - (1/4) + (1/4) - (1/5) + (1/5) - (1/6) + (1/6) - (1/7)

S5 = (1/3) - (1/4) + (1/4) - (1/5) + (1/5) - (1/6) + (1/6) - (1/7) + (1/7) - (1/8)

The sum of the series is 1/3.

To find S3, we need to find the sum of the first 3 terms of the series:

S3 = (1/3) - (1/4) + (1/4) - (1/5) + (1/5) - (1/6)

To find S4, we need to find the sum of the first 4 terms of the series:

S4 = (1/3) - (1/4) + (1/4) - (1/5) + (1/5) - (1/6) + (1/6) - (1/7)

To find S5, we need to find the sum of the first 5 terms of the series:

S5 = (1/3) - (1/4) + (1/4) - (1/5) + (1/5) - (1/6) + (1/6) - (1/7) + (1/7) - (1/8)

To find the sum of the telescoping series, we can use the property of the telescoping series where the terms will cancel out. In this case, we can see that the first term of each group of two terms cancels out with the second term of the next group of two terms.

S = (1/3) - (1/8)

The series starts at n=2 and ends at infinity, so the sum of the series is the difference between 1/3 and the limit of 1/n as n approaches infinity, which is zero.

S= (1/3) -0 = 1/3

So the sum of the series is 1/3.

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The complete question is -

Calculate S3, S4, and S5 and then find the sum for the telescoping series: S=∑n=2∞(1/n+1)−(1/n+2). where Sk is the partial sum using the first k values of n.

Answer:

B. There are not 10 cubic decimeters in a cubic meter because a cubic meter is a meter cubed. This means that there are 10³ cubic decimeters in a cubic meter.

Step-by-step explanation:

When we cube a number, we multiply the number by itself twice.

Therefore a cubic meter is:

To calculate the number of cubic decimeters in a cubic meter, we need to cube the number of decimeters in a meter.

Given there are 10 decimeters in a meter, then there are 10³ cubic decimeters in a cubic meter:

Therefore , the solution to the given problem of expression comes out to be -8 * 10³.

A number must be substituted for each variable and arithmetic operations must be carried out in order to verify an algebraic expression. The response variable is equivalent to 6 in the illustration above because 6 + 6 equals 12. The variables can be replaced with their values if we know what they are, and the expression can then be evaluated.

Here,

Given : - 8,000

To write -8000 with a power of 10

Thus,

-8000 written into

=> -8 * 10³

So, we get -8 * 10³ as the scientific notation for the given expression

Therefore , the solution to the given problem of expression comes out to be -8 * 10³.

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Answer:

6

Step-by-step explanation:

20 divided by 3 without any remainders, fractions, or decimals is 6.

Hope I helped!

Answer:

Step-by-step explanation:

Since 5 miles is approximately 8 kilometers, and Sawyer lives 12.7 kilometers from the soccer field, Sawyer lives farther than Jabar from the soccer field.

So Sawyer lives about 4.7 kilometers farther from the soccer fields.

The value of  [tex]3\frac{3}{5}[/tex] ×[tex](-8\frac{1}{3})[/tex] is after simplification is C)-30

What is mixed fraction?

A mixed number combines an integer (whole number) with a fraction. It is also sometimes referred to as a mixed fraction (part of a whole number). When a number contains both an integer (whole number) and a correct fraction, it is known as a mixed number or mixed fraction (a fraction whose numerator is less than its denominator).

Here the given expression is ,

=> [tex]3\frac{3}{5}[/tex] ×[tex](-8\frac{1}{3})[/tex]

Now we need to convert them into improper fraction then,

=> [tex]\frac{15+3}{5}[/tex] × [tex]\frac{-(24+1)}{3}[/tex]

=> [tex]\frac{18}{5}[/tex] × [tex]\frac{-25}{3}[/tex]

=> 6 × -5 = -30

Hence the correct option is C)-30.

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Answer is B

y=16.9x+0.7

To Calculate line of best fit you can simply input the values into the stats function on your calculator and using the "Reg" function to get your values of the standard equation form of your line of best fit which is y=bx+a

b=16.9

a=0.7

Answer:

Below

Step-by-step explanation:

3 (x-12) = 15     divide both sides of the equation b y 3 to get :

x-12 = 5           now add 12 to both sides

x = 17       Done.

[tex]\huge\text{Hey there!}[/tex]

[tex]\huge\textbf{Solving for x or simplify this equation}[/tex]

[tex]\mathsf{3(x - 12) = 15}[/tex]

[tex]\huge\textbf{DISTRIBUTE 3 within PARENTHESES}[/tex]

[tex]\mathsf{3(x) + 3(-12) = 15}[/tex]

[tex]\mathsf{3x - 36 = 15}[/tex]

[tex]\huge\textbf{ADD 36 to BOTH SIDES}[/tex]

[tex]\mathsf{3x - 36 + 36 = 15 + 36}[/tex]

[tex]\huge\textbf{SIMPLIFY IT}[/tex]

[tex]\mathsf{3x = 15 + 36}[/tex]

[tex]\mathsf{3x = 51}[/tex]

[tex]\huge\textbf{DIVIDE 3 to BOTH SIDES}[/tex]

[tex]\mathsf{\dfrac{3x}{3} = \dfrac{51}{3}}[/tex]

[tex]\huge\textbf{SIMPLIFY IT}[/tex]

[tex]\mathsf{x = \dfrac{51}{3}}[/tex]

[tex]\mathsf{x = 17}[/tex]

[tex]\huge\text{Therefore, your answer should be:}[/tex]

[tex]\huge\boxed{\mathsf{x = 17}}\huge\checkmark[/tex]

[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]

A table which has a constant of proportionality between y and x of 0.9 include the following: B. table B.

In order to have a proportional relationship and equivalent ratios that is constant, the variables x and y must have the same constant of proportionality.

Next, we would determine the constant of proportionality (k) based on the given parameters in each table:

Constant of proportionality (k) = y/x

Constant of proportionality (k) = 17/18

Constant of proportionality (k) = 0.94 (False).

For the variables x and y in table B, the constant of proportionality (k) is given by;

Constant of proportionality (k) = y/x

Constant of proportionality (k) = 3.6/4 = 5.4/6 = 10.8/12

Constant of proportionality (k) = 0.9 (True).

For the variables x and y in table C, the constant of proportionality (k) is given by;

Constant of proportionality (k) = y/x

Constant of proportionality (k) = 3.3/3

Constant of proportionality (k) = 1.1 (False).

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Answer:

7 students.

Step-by-step explanation:

A trick for this is to find 10% of the students in the Environmental Club and then divide by 2.

10% of 140 is simple; just remove the 0 and you got 14.

Next, divide 14/2 to get 5%

14/2 = 7

7 students are apart of the Environmental Club

An IV is infusing at 77mcgtt/min. rate of flow equal in mL/h is 77 mL/h

1gtt/min equals 3.00 mL/h

If you simply need to figure out the infusion rate, or the mL per hour to infuse, take the total volume in mL, divided by the total time in hours that the medication is ordered to be infused over, to equal the rate in mL per hour.

1 drop per minute to milliliters per hour = 3.00 mL/h

2 drops per minute to milliliters per hour = 6.00 mL/h

Change hours to minutes. Change milliliters to drops.

1 ml : 60mcgtt

1hr :60min

77mcgtt/min =77*60/60 ml/h

                  =77ml/h

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The Z-Score associated on moving one standard devations to the right is equals to the 1.0, i.e., Z = 1.00. So, correct answer is option (c).

We have, The number of classes, n = 5

Mean of size of five classes,µ = 44

Standard deviations for five classes,σ = 8

The dynamic equation for conversion to z-scores in real-time is, Z = (Xᵢ - µ )/standard deviations (s)

Z-Score : A Z-Score is a metric that represents that how closely a value relates to mean of set of values. If Z score value is zero that means data point's scores are same as mean score.

Formula for Z- Score is Z = (X - µ)/σ

where, X ---> data values

µ --> mean of values

σ --> standard deviation

For 32, Z = (X - µ)/σ = (32 - 44)/8 = -1.50

For 42 , Z= (X- µ)/σ = (42 - 44)/8 = -0.25

For 48, Z = ( 48 - 44)/8 = 0.5

For 52, Z = ( 52 - 44)/8 = 1.0

For 54, Z = (54 - 44)/8 = 1.25

When we move One standard deviations to right , that means X = 44 + 8 = 52 , So the Z-Score for 52 is Z = 1.0. Hence, required value is 1.0.

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Complete question:

The dot plot below shows the sizes for five classes with a mean of 44 and standard deviation of 8. The dynamic equation shows the mathematical conversion to z-scores in real-time. Drag the vertical line to see the corresponding z-score for each possible class size,

Z =( Xᵢ - X)/s Created by Gary M. McClelland, Professor Emeritus | University of Colorado Boulder Cengage Learning. All Rights Reserved. 1. Make sure the green line is on the mean value of x= 44. Next, move the green line one standard deviation to the right. What Z-score is associated with this value?

a. ) Z-0.75

b) Z= 0.80

c) Z = 1.00

d) Z = 0.5

Answer: b

Step-by-step explanation: