Sherry paid 6.5% interest on a one-year loan. if she paid a total of 149.50 in interest, what was the original amount of the loan? A. 2,300 B. 971.75 C. 9717.50 D. 230.00

(a) The probability that a 0 is received is 66.7%

(b) By using the Bayes theorem, the probability that a 0 was transmitted, given that a 0 was received is 90%

Probability:

In statistics, probability deals with finding out the likelihood of the occurrence of an event.

Given,

Here the space probe near Neptune communicates with earth using bit strings. suppose that in its transmissions it sends a 1 one-third of the time and a 0 two-thirds of the time. when a 0 is sent, the probability that it is received correctly is 0.6 and the probability that it is received incorrectly (as a 1) is 0.4. when a 1 is sent the probability that it is received correctly is 0.8 and the probability that it is received incorrectly (as a 0) is 0.2.

Now, we have to find the probability of the following.

Here the probability that a 0 is received is calculated as,

=> 0.9 of 2/3(0 received when a 0 is sent) and 0.2 of 1/3(0 received when a 1 is sent). So

Now, the probability is

=> p = (0.9 x 2)/3 + (0.2 x 1)/3

=> p = 0.6667

When we convert this into percentage then we get, 66.67% probability that a 0 is received.

Similarly, by using the Bayes theorem, the probability that a 0 was transmitted, given that a 0 was received is calculated as,

Here we know that, Event A: 0 received and Event B: 0 transmitted.

Then 0.6667 = 66.67% probability that a 0 is received, which means that

P(A) = 66.7%

And the zero is transmitted two-thirds of time, which means that

P(B) = 66.7%

Where as if 0 is sent, then the probability that it is received correctly is 0.9, which means that

P(A|B) =0.9

Therefore, the probability is

=> 66.7 x 0.9/66.7

=> 0.9

We know that 0.9 = 90% probability that a 0 was transmitted, given that a 0 was received.

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The cost of the flooring task is $35

Consider the image I attached:

Actual area of floor = Area of  rectangle A + Area of  rectangle B

Actual area of floor = (6×3) + (2×1) = 18 + 2 = 20 ft²

With Estimate of 5% more than the actual area in case of mistakes:

Area = 20 + (5/100 × 20)

Area = 20 + 1 = 21 ft²

Since Cost: $20 for 12 square feet of flooring. Thus:  

Cost of task = 21/12 × 20 = $35

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There are 12 chaperones in total.

In molecular biology, molecular chaperones are proteins that help big proteins or macromolecular protein complexes fold or unfold conformationally. There are different groups of molecular chaperones, all of which have the same purpose: to help big proteins fold properly during or after synthesis as well as following partial denaturation. Protein translocation for proteolysis involves chaperones as well.

The first molecular chaperones to be identified are an assembly-type of chaperones that help nucleosomes form from folded histones and DNA.

Since heat stress increases the propensity for protein aggregation, one of the main roles of molecular chaperones is to prevent the accumulation of misfolded proteins. As a result, many chaperone proteins are categorized as heat shock proteins.

Chaperones and students combined: 3 + 20 = 23

by the number of individuals in total:

92 / 23 = 4

Divide that sum by the proportional number of chaperones:

4 x 3 = 12

There are 12 chaperones in total.

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The probability that a seventh-grader chosen will randomly play the keyboard as a fraction in simplest form is P=0.68.

Probability is considering a numerical description of how likely an event is to occur or how likely it is true. In many situations, it is necessary to foresee how something will turn out in the actual world. We may be aware of the result of an occurrence or not. When this occurs, we state that there is a possibility that the event will occur. In general, probability has many excellent applications in gaming, commerce, and this newly growing area of artificial intelligence.

Seventh-grade students guitar= 8

All total students' guitar = 8+8+7+2

= 25

A instruments other than guitar = 25-8

= 17

So, p = 17/25

= 0.68

Here we use simple random sampling n/n. Thus, the answer is 0.68.

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for the given normally distributed random variable,

(a) P(16 < X < 22) = 0.7333

(b) P(X < k1) = 0.2946; such that the value of k1 = 16.625

(c) P(X > k2) = 0.8186; such that the value of k2 = 15.725

(d) The two cut-off values that contain the middle 90% of the normal curve area are -1.64 and +1.64.

The formula for finding the z-score is

z = (X - μ)/σ

Here the mean (μ) and the standard deviation (σ) are applied.

It is given that, a normally distributed random variable X has a mean μ = 18 and the variance σ² = 6.25.

So, its standard deviation is σ = √6.25 = 2.5

(a) P(16 < X < 22)

The z-score for X = 16 is

z = (X - μ)/σ ⇒ z = (16 - 18)/2.5 = -0.8

The z-score for X = 22 is

z = (X - μ)/σ ⇒ z = (22 - 18)/2.5 = 1.6

So, the probability becomes

P(16 < X < 22) = P(-0.8 < z < 1.6)

                       = P(z < 1.6) - P(z < -0.8)

From the distribution table, we have P(z < 1.6) = 0.9452 and P(z < -0.8) = 0.2119.

So, we get

P(16 < X < 22) = 0.9452 - 0.2119 = 0.7333

(b) The value of k1 such that P(X < k1) = 0.2946:

we have P(X < k1) = 0.2946

⇒ P(X < k1) = P(z < (k1 - μ)/σ) = 0.2946

From the table, we know that,

P(z < -0.55) = 0.2946

So,

(k1 - μ)/σ = -0.55

⇒ (k1 - 18)/2.5 = -0.55

⇒ k1 - 18 = -0.55 × 2.5 = -1.375

∴ k1 = -1.375 + 18 = 16.625

(c) The value of k2 such that P(X > k2) = 0.8186:

we have P(X > k2) = 0.8186

⇒ P(z > (k2 - μ)/σ) = 0.8186

⇒ 1 - P(z < (k2 - μ)/σ) = 0.8186

⇒ P(z < (k2 - μ)/σ) = 1 - 0.8186 = 0.1814

From the table, we know that P(z < -0.91) = 0.1814

So,

(k2 - μ)/σ = -0.91

⇒ (k2 - 18)/2.5 = -0.91

⇒ k2 - 18 = -0.91 × 2.5 = -2.275

∴ k2 = -2.275 + 18 = 15.725

(d) The two cut-off values that contain the middle 90% of the normal curve area are -1.64 and +1.64 since the z-score for a 90% confidence interval is 1.64.

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Rational Function: any function that is found by a rational fraction.

Its form: [tex]\frac{1}{x}[/tex]

Finding Zeros of  a Rational Function:

[tex]f(x)=\frac{(x-3)(x+2)}{(x-5)(x-3)}[/tex]

   2. Find restrictions by equating the denominator to 0

[tex]x-5=0[/tex]                [tex]x-3=0[/tex]      when x=-5 and 3, the function is undefined

[tex]x=-5[/tex]                   [tex]x=3[/tex]

  3. Set the numerator equal to zero

[tex]x-3=0[/tex]              [tex]x+2=0[/tex]  

[tex]x=3[/tex]                    [tex]x=-2[/tex]

Be careful - since we have x-3 on both the numerator and denominator the graph will not intersect the x-axis at that point. This is what is called a removable discontinuity.

Therefore, the only zero occurs at (-2,0)

Answer:

----------------------------------------------

Given is the parallelogram.

Properties:

Solve the following:

Answer:

w = 72°

x = 36°

2x = 72°

3x = 108°

y = 72°

z = 108°

Step-by-step explanation:

The arrows on the sides of figure ABCD indicate that opposite sides are parallel.  Therefore, ABCD is a parallelogram.

In a parallelogram:

As adjacent angles sum to 180°:

⇒ m∠A + m∠B = 180°

⇒ 3x + 2x = 180°

⇒ 5x = 180°

⇒ x = 36°

Therefore:

⇒ m∠A = 3(36) = 108°

⇒ m∠B = 2(36) = 72°

As opposite angles are equal:

⇒ y = m∠D = m∠B = 72°

⇒ z = m∠BCD = m∠A = 108°

Angles on a straight line sum to 180°:

⇒ z + w = 180°

⇒ 108° + w = 180°

⇒ w = 72°

The first four terms of the sequence given the sequence definition are 17, 15, 13 and 11

From the question, we have the definition of the function that can be used in our computation:

f(n) such that

f(1) = 17

f(n) = f(n - 1) - 2 if n > 1

The above definitions imply that we simply subtract 2 from the previous term to get the current term

Using the above as a guide,

So, we have the following representation

f(2) = 17 - 2

f(2) = 15

Also, we have

f(3) = 15 - 2

f(3) = 13

Lastly, we have

f(4) = 13 - 2

f(4) = 11

Hence, the the first four terms are 17, 15, 13 and 11

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The line that would appear parallel to HG and would go through point D, is DC.

When describing parallel lines, we describe two lines in the same plane that are equally spaced apart and never cross each other. Both horizontal and vertical shapes are possible.

The only line that would go through point D and would be parallel to line HG, would therefore be Line DC. This is because this line is equally spaced from line HG and would never cross it.

In conclusion, line DC would be parallel to line HG.

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To find the range of a function: consider a function y = f(x). The spread of all the y values from minimum to maximum is the range of the function. In the given expression of y, replace all the values of x to investigate whether it is positive, negative or equal to other values.

The range of a function is the complete set of all possible resulting or output values of the dependent variable (usually labeled by y), after we have replaced the domain. Typically, this will be represented by the letter y or f(x). The range is any number that you can get when you plug in any number for x.

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A space whose constituents are all single point sets is said to be completely isolated. Related to this property, the space X is such that the two distinct open sets of  x and y are disjoint. In some cases, they are said to be completely isolated. We use U and x and V and y, X is the becomes a join. Of course, all perfectly isolated spaces are completely isolated, but not vice versa. For example, take two copies of the rational number Q and identify them at all points except zero. The resulting quotient topology space is completely separated and subset. In fact, it is not Hausdorff, and strictly speaking, the condition of complete separation is stronger than the condition of being Hausdorff.

In topology and related mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint, non-empty, open subsets. Connectivity is one of the most important topological properties used to distinguish topological spaces.

A subset of a topological space X is a connected space when viewed as a subspace of X.

Some related but stronger conditions are path connections, single connections, and n connections. Another related term is locally attached. It does not imply or derive connectivity.

At a given point x in the topological space X, X is an arbitrary collection of connected subsets each containing x is the union of The connected component of a point x in X is the uniquely largest connected subset of X containing  is the union (the subset of X associated with containing x . The maximally connected subset of the non-empty topological space (containing) are called connected components of the space. The components of any topological space X form a partition of X. , is not empty, and their union is the entire space: each component is a closed subset of the original space. If that number is finite, each component is also an open subset. However, that may not be the case if the number of them is infinite. For example, the connected component of the set of rational numbers is an open singleton.

Let Гₓ be the topological space X and Г'ₓ be the intersection of all Clepen sets containing x (called subcomponents of x).

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Answer:

$5.25 C

Step-by-step explanation:

17 student = $89.25

per student(1) = x

cross multiply

17x = $89.25

x = $89.25/17

x = $5.25

RATE AS BRAINLIEST

Answer:

5.25

Step-by-step explanation:

the person above is right

Answer:

No, she would need to print out 4536 pages

Step-by-step explanation:

2 x 28 x 81 = 4536

Answer:

No, that does not sound right.

Ms. Stewart needs to print a total of 4,536 pages in this time!

Step-by-step explanation:

This situation is an example of ratios and proportions.

⭐What is a ratio?

⭐What is a proportion?

Let's set up the 1st ratio:

"Every school day, Ms. Stwewart prints out 2 pages of worksheets for each [student]"

1 student/2 pages

Let's set up the 2nd ratio:

"Ms. Stewart is a teacher with 28 students"

28 students/x pages

Now, we want to solve how many pages Ms. Stewart prints in a day for 28 students.

Thus, we need to set up a proportion with these 2 ratios.

Make sure the same units are on the numerator and denominator.

[tex]\frac{1}{2} = \frac{28}{x}[/tex]

Multiply LHS and RHS by [tex]\frac{1}{28}[/tex] to solve for x.

[tex]\frac{1}{2}(\frac{1}{28}) = \frac{28}{x}(\frac{1}{28})[/tex]

[tex]\frac{1}{56} = \frac{1}{x}[/tex]

∴ Ms. Stewart needs to print 56 worksheets for 1 day.

However, this isn't what the problem is asking us. We want to know how many worksheets Ms. Stewart needs to print for the coming school days.

To find out, we need to make another proportion. We only have 1 ratio so far: 56 worksheets/1 day, so we need to make another one for the coming school days.

Let's set up the 2nd ratio:

"There are 81 school days left in this school year."

y worksheets/ 81 days

Thus, we need to set up a proportion with these 2 ratios.

Make sure the same units are on the numerator and denominator.

[tex]\frac{56}{1} = \frac{y}{81}[/tex]

Multiply both sides by 81 to isolate y.

[tex]56(81) = \frac{y}{81}(81)[/tex]

[tex]y = 4,536[/tex]

∴ Ms. Stewart needs to print 4,536 worksheets for the next 81 days!

The value of x is 5.

What is a angle of a triangle?

In a triangle, there are three angles. These angles are created by the triangle's two sides coming together at the triangle's vertex. Three inner angles added together equal 180 degrees. The side length generates an outside angle if we stretch it outside.

Given that, ∠KLM = 81°,  ∠LKM = 4x°, and ∠KMN = (13x+36)°.

Theorem of exterior angle:

If a triangle side is created, the outside angle that results is equal to the product of the two opposite internal angles.

According to theorem of exterior angle,

∠KMN = ∠KLM + ∠LKM

(13x+36)° = 81° + 4x°

Subtract 4x from both sides:

9x + 36 = 81

Subtract 36 from both sides:

9x = 45

Divide both sides by 9:

x = 5

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69 Passengers (20 plus additional 49) will result in maximum revenue for the owner of the yacht. Maximum Revenue will be $38,088.

current: 20 passengers at $944 each

let the number of additional passengers be x

cost per passenger = 944 - 8x

revenue (R) = (20+x)(944-8x)

= 18880 - 160x + 944x -8 x^2

dR/dx = -160 + 944 - 16x = 0 for a max of R

16x = 784

x = 49

There should be an additional 49 or a total of 69 passengers

The cost per passenger would be 944-8(49) or $552 per day

Maximum revenue for the yacht:

revenue (R) = (20+x)(944-8x)

revenue (R) = (20+49)(944-8(49))

revenue (R) = (69)(944-392)

revenue (R) = $38,088

Hence , 69 Passengers (20 plus additional 49) will result in maximum revenue for the owner of the yacht. Maximum Revenue will be $38,088.

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Probability can be used to make predictions or decisions in a variety of situations, such as in gambling, finance, and science. In these situations, probabilities can be calculated based on statistical data or by using mathematical models.

To find the probability of making all 3 free throws, we can use the formula for the probability of an independent event occurring multiple times:

P(A) * P(A) * P(A) = P(A^3).

To find the probability of missing all 3 free throws, we can use the same formula with the probability of missing a single free throw, which is 1 - 0.91 = 0.09.

Plugging this into the formula gives us 0.09 * 0.09 * 0.09 = 0.0029.

To find the probability of making at least one free throw, we can subtract the probability of missing all 3 free throws from 1. 1 - 0.0029 = 0.9710.In this case, P(A) is the probability of making a single free throw, which is 0.91.

Plugging this into the formula gives us 0.91 * 0.91 * 0.91 = 0.8281.

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The Euler, explicit Trapezoid, and fourth-order Runge-Kutta methods were applied to a given initial value problem on a grid/mesh of step-size h=0.1 in [0,1]. The results were printed in a table, showing the t values, approximations, and global error at each step.

Euler Method

t Approximation Global Error

0 1 0

0.1 1.011 -0.011

0.2 1.0454 -0.0454

0.3 1.1044 -0.1044

0.4 1.1921 -0.1921

0.5 1.3125 -0.3125

0.6 1.4713 -0.4713

0.7 1.6749 -0.6749

0.8 1.9301 -0.9301

0.9 2.2447 -1.2447

1 2.63 -1.63

Explicit Trapezoid Method

t Approximation Global Error

0 1 0

0.1 1.0055 -0.0055

0.2 1.0246 -0.0246

0.3 1.0592 -0.0592

0.4 1.1021 -0.1021

0.5 1.1564

The Euler method is a numerical technique for solving initial-value problems, which involves approximating the solution of a differential equation by the combination of a series of tangent lines. The explicit trapezoid method is an improved version of the Euler method, which uses the average of two slopes instead of the slope at the previous point whencalculating the next approximation. The fourth-order Runge-Kutta method is a technique that uses a weighted average of four different slopes at different points to approximate the solution of a differential equation.

For the given initial value problem, we applied the Euler method, the explicit Trapezoid method, and the fourth-order Runge-Kutta method on a grid/mesh of step-size h=0.1 in [0,1]. We printed a table of the t values, approximations, and global error at each step.

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a) The hypotheses are given as follows:

b) The test statistic is given as follows: [tex]\chi^2 = 37.79[/tex]

c) The p-value is given as follows: 0.61

d) The conclusion is given as follows: There is not enough evidence to conclude that the pulse rates of men have a standard deviation different of 10 bears per minute.

At the null hypothesis, it is tested if there is not evidence to conclude that the pulse rates of men have a standard deviation different of 10 bears per minute, that is:

[tex]H_0: \sigma = 10[/tex]

At the alternative hypotheses, it is tested if there is enough evidence to conclude that the standard deviation is different of 10, that is:

[tex]H_1: \sigma \neq 10[/tex]

The equation for the test statistic is given as follows:

[tex]\chi^2 = \frac{(n - 1)s^2}{\sigma^2}[/tex]

In which:

The parameters for this problem are given as follows:

[tex]n = 42, s = 9.6, \sigma = 10[/tex]

Hence the test statistic is calculated as follows:

[tex]\chi^2 = \frac{(n - 1)s^2}{\sigma^2}[/tex]

[tex]\chi^2 = \frac{41 \times 9.6^2}{10^2}[/tex]

[tex]\chi^2 = 37.79[/tex]

Using a chi-square distribution calculator, with a two-tailed test, as we are testing if the standard deviation is different of a value, with 42 - 1 = 41 df and [tex]\chi^2 = 37.79[/tex], the p-value is given as follows:

0.61.

As the p-value is greater than the significance level of 0.10, the conclusion is given as follows: There is not enough evidence to conclude that the pulse rates of men have a standard deviation different of 10 bears per minute.

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The total income produced by a continuous income stream in the first 15 years is $45,000.

What is continuous income stream?

We can see the money flowing into the account continuously rather than in a lot of discrete payments if the deposits are made often enough and the intervals between deposits are short compared to the account's overall lifespan. This scenario will be referred to as a constant revenue stream.

Given function is

f(t) = 3000

Now integrate both side of the function and the lower limit is t = 0 and upper limit is t = 15.

[tex]= \int^{15}_{0}3000 dt[/tex]

[tex]=3000[t]_0^{15}[/tex]

= 3000 [ 15 - 0]

=45000

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_______________________________________________________

1. Convert element to fraction: [tex]3=\frac{3}{1}[/tex]

[tex]\frac{3}{1} * \frac{3}{5}[/tex]

2. Apply the fraction rule: [tex]\frac{a}{b}* \frac{c}{d}=\frac{a*c}{b*d}[/tex]

[tex]\frac{3*3}{1*5}[/tex]

3. Multiply the numbers: [tex]3 * 3=9[/tex]

[tex]\frac{9}{1*5}[/tex]

4. Multiply the numbers: [tex]1*5=5[/tex]

[tex]\frac{9}{5}[/tex]

^Hence, our solution.

_________________________________

Hope this helps!

a) The area under the standard normal curve to the right of z=2.25 is 0.0122

b) The area under the standard normal curve between z=-2.48 and z=-1.02.

The possibility of something is how probable it is to happen. When we don't know how an event will turn out, we can talk about the probabilities of many outcomes—how likely they are. Statistics is the study of events with a probability component.

Probability theory, which is extensively used in fields of study like statistics, mathematics, science, finance, gambling, artificial intelligence, machine learning, computer science, game theory, and philosophy, has given these ideas an axiomatic mathematical formalization. For example, it can be used to infer information about the expected frequency of events. The mechanics and regularities that underlie complex systems are also described using probability theory.

a) P(z>2.25)

=1-P(z≤2.25)

=1-0.9878

=0.0122

b) P(-2.48<z<-1.02)

=P(z<-1.02)-P(z<-2.48)

=0.1539-0.0066

=0.1473

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The origin is the point in the rectangle coordinate system which is located on the point of intersection of the x-axis and y-axis.

The four parts of the cartesian plane divided by the x-axis and the y-axis are,

Quadrant 1, Quadrant 2, Quadrant 3 and Quadrant 4.

Given, a cartesian plane divided by x-axis and y-axis.

we have to mention the four regions created by intersection of the x-axis and the y-axis.

Also, we have to mention the intersection of the x-axis and y-axis in the coordinate.

The intersection point of the x-axis and the y-axis is the point on cartesian plane and known as Origin.

The four parts of the cartesian plane divided by the x-axis and the y-axis are,

Quadrant 1, Quadrant 2, Quadrant 3 and Quadrant 4.

Also, The origin is the point in the rectangle coordinate system which is located on the point of intersection of the x-axis and y-axis.

Hence, the origin is the point in the rectangle coordinate system which is located on the point of intersection of the x-axis and y-axis.

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From the data we conclude that; (c) The sampling minimum appears to be biased estimator since it consistently over estimated the population minimum.

The sampling distribution of the sample minimum is given as 51 points

The number of tests in the sample = 6 tests

The results of the tests are summarized in the dot plot

In order to be an unbiased estimator the sampling distribution for the statistic has to be evenly distributed about the true parameter for the population .

and In this question , the true parameter for the population is that the sample minimum of the population is 99 points .

So , From sampling distribution of sample Minimum it is clear that distribution of sample maximum is unevenly distributed.

Hence , it is not an unbiased estimator that means it is an biased estimator which is overestimating the population minimum .

Therefore , the correct option is (c) .

-- The given question is incomplete , the complete question is

"Silas was curious if sample minimum was an unbiased estimator of population minimum. He started with a large normally distributed population of test scores whose minimum was 51 points. He then took a random sample of 6 tests and calculated the minimum of the sample. He replaced those tests and repeated this process for a total of 40 trials. His results are summarized in the dot plot , where each dot represents the minimum score from a sample of 6 tests.

50 , 70 , 75 , 55 , 60 , 65

(a) Sample minimum appears to be an unbiased estimator for the population minimum.

(b) Sample minimum appears to be an biased estimator since it consistently under estimates the population minimum.

(c) Sample minimum appears to be an biased estimator since it consistently over estimates the population minimum." --

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Answer:

circulation was 2,250. In 2020, the circulation was 2,350.

Step-by-step explanation:

To Bisect the line segment CD, the next step is option B. Swing an arc on both sides of line segment CD.

Bisection is a process of dividing a given line, or angle into two equal parts or measures respectively. This could either be a line bisector or an angle bisector as the case may be.

Thus to bisect the line CD, the following steps are required:

The next step for Ian is to swing an arc on both sides of line segment CD i.e option B.

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Answer:

b

Step-by-step explanation:

took test

The value of piecewise function at x = -1 is f(-1) = 11, and the value of piecewise function at x = 9 is f(9) = 1.

what is piecewise function?

A piecewise-defined function in mathematics is one that is composed of several smaller functions, each of which has a specific interval of the domain it applies to. Instead of being a property of the function itself, piecewise definition is a way to express the function.

We have to evaluate the piecewise function for x = -1 and x = 9.

Let, function has value for x = -1 in the domain x ≤ 5 which is 11.

And function has value for x = 9  in the domain 6 < x which is 1.

⇒  f(-1) = 11 and f(9) = 1

Hence, the value of piecewise function at x = -1 is f(-1) = 11, and the value of piecewise function at x = 9 is f(9) = 1.

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Answer:

Given:

Sum of edge lengths = TSA of cube

there are 12 edges in a cube

[tex]12n = 6 {n}^{2} \\ 6 {n}^{2} - 12n = 0 \\ 6n(n - 2) = 0 \\ 6n = 0 \\ n = 0 \\ n - 2 = 0 \\ n = 2[/tex]

n ≠ 0 (since length can't be 0)

therefore n=2

Answer:

width ≈ 70.7 m

Step-by-step explanation:

let the width be w , then length = 2w

the area (A) of a rectangle is calculated as

A = lw ( l is length and w is width )

given A = 10,000 , then

2w × w = 10,000

2w² = 10,000 ( divide both sides by 2 )

w² = 5,000 ( take square root of both sides )

w = [tex]\sqrt{5,000}[/tex] ≈ 70.71 ( to 2 decimal places )

approximate width is 70.7 m ( to 1 decimal place )

Here are some properties of the exponential growth functions

What are exponential functions ?

An exponential function can be written in forms

f(x)=abˣ

where

a is the initial value because f(0)=a. In the growth and decay models that we examine in this finite math textbook, a>0.

b is often called the growth factor. We restrict b to be positive (b>0) because even roots of negative numbers are undefined. We want the function to be defined for all values of x, but bx would be undefined for some values of x if b<0.

Properties of Exponential Growth Functions

The function y=f(x)=ab× represents growth if b>1 and a>0.

The growth rate r is positive when b>1. Because b=1+r>1, then r=b−1>0

The function y=f(x)=ae× represents growth if k>0 and a>0.

The function is an increasing function; y increases as x increases.

Domain: { all real numbers} ; all real numbers can be input to an exponential function

Range: If a>0, the range is {positive real numbers} The graph is always above the x axis.

Horizontal Asymptote: when b>1, the horizontal asymptote is the negative x axis, as x becomes large negative. Using mathematical notation: as x → −∞, then y → 0.

The vertical intercept is the point (0,a) on the y-axis. There is no horizontal intercept because the function does not cross the x-axis.

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Its is true that C ⊆ D means Every element of C is present in D

According to he question,

Let C = {n ∈ Z | n = 6r – 5 for some integer r}

D = {m ∈ Z | m = 3s + 1 for some integer s}

We have to prove : C ⊆ D

Proof : Let n ∈ C

Then there exists an integer r such that:

n = 6r - 5

Since -5 = -6 + 1

=> n = 6r - 6 + 1

Using distributive property,

=> n = 3(2r - 2) +1

Since , 2 and r are the integers , their product 2r is also an integer and the difference 2r - 2 is also an integer then

Let s = 2r - 2

Then, m = 3r + 1 with r some integer and thus m ∈ D

Since , every element of C is also an element of D

Hence ,  C ⊆ D proved !

Similarly, you have to prove D ⊆ C

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