Given that f(x) = x² - 3 and g(x) = 3x + 5, find (g- f)(9), if it exists.

Answer:

[tex]{ \sf{ \frac{dy}{dx} = y( {x}^{2} + 1)}} \\ \\ { \sf{ \frac{dy}{y} = ( {x}^{2} + 1)dx}} \\ \\ { \sf{ \int \frac{1}{y} dy = \int ( {x}^{2} + 1) \: dx}} \\ \\ { \sf{ ln(y) = \frac{ {x}^{3} }{3} + x + c }}[/tex]

Answer:

[tex]\large\text{$y& = e^{\frac{1}{3}x^3+x+\text{C}}$}[/tex]

Step-by-step explanation:

Given differential equation:

[tex]\dfrac{\text{d}x}{\text{d}y}=\dfrac{1}{y(x^2+1)}[/tex]

Rearrange the equation so that all the terms containing y are on the left side, and all the terms containing x are on the right side:

[tex]\begin{aligned}\implies \dfrac{\text{d}x}{\text{d}y}&=\dfrac{1}{y(x^2+1)}\\\\(x^2+1)\;\dfrac{\text{d}x}{\text{d}y}&=\dfrac{1}{y}\\\\(x^2+1)\;\text{d}x}&=\dfrac{1}{y}\;\text{d}y\\\\\dfrac{1}{y}\;\text{d}y&=(x^2+1)\;\text{d}x}\end{aligned}[/tex]

Integrate both sides:

[tex]\begin{aligned}\implies \displaystyle \int \dfrac{1}{y}\;\text{d}y &= \int (x^2+1)\;\text{d}x}\\\\\int \dfrac{1}{y}\;\text{d}y &= \int x^2\;\text{d}x}+\int 1\;\text{d}x}\\\\\ln y & = \dfrac{1}{3}x^3+x+\text{C}\\\\e^{\ln y} & = e^{\frac{1}{3}x^3+x+\text{C}}\\\\y& = e^{\frac{1}{3}x^3+x+\text{C}}\\\\\end{aligned}[/tex]

Therefore, the solution to the given differential equation is:

[tex]\large\text{$y& = e^{\frac{1}{3}x^3+x+\text{C}}$}[/tex]

Integration rules used:

[tex]\boxed{\begin{minipage}{3.5 cm}\underline{Integrating $\frac{1}{x}$}\\\\$\displaystyle \int \dfrac{1}{x}\:\text{d}x=\ln |x|+\text{C}$\end{minipage}}[/tex]

[tex]\boxed{\begin{minipage}{3.5 cm}\underline{Integrating $x^n$}\\\\$\displaystyle \int x^n\:\text{d}x=\dfrac{x^{n+1}}{n+1}+\text{C}$\end{minipage}}[/tex]

[tex]\boxed{\begin{minipage}{5 cm}\underline{Integrating a constant}\\\\$\displaystyle \int n\:\text{d}x=nx+\text{C}$\\(where $n$ is any constant value)\end{minipage}}[/tex]

(a) The area of the triangle = 6 cm squares

(b) The area of the whole shape = 76 cm squares

The shape given is formed by a triangle and a rectangle.

First we need to find the area of the triangle.

Height of the triangle = 3 cm

Base of the triangle = 4 cm

Area of the triangle = 1/2 x base x height

                                 = 1/2 x 4 x 3

                                 = 6 cm squares

Area of the whole shape = Area of the triangle + Area of the rectangle

Length of the rectangle = 10 cm

Width of the rectangle = 7 cm

Area of the rectangle = length x width

                                    = 10 x 7

                                    = 70 cm squares

Therefore, area of the whole shape = 6 + 70

                                                            = 76 cm squares

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Given:

Find the property.

Sol:

The additive identity property is:

So use the Additive Identity property.

Step-by-step explanation:

cook time = 65 min + 11 min + 60 seconds + 30 seconds

= 65 min + 11 min + 1.5 minute

cook time = 77.5 minutes or: 1 hour, 17 minutes and 30 seconds.

For food to be ready by 4PM:

4PM - 1 hour, 17 minutes and 30 seconds = when to put food in

put food in = 2:42:30 PM

Answer:

$14540

Step-by-step explanation:

you are earning 4 dollars per 400 dollars per month. there are 12 months in a year. 400 * 12*30 = 144000. 144000 divided by 400 is 360.

360 times 4 is 1440.

Answer:

Explanation:

Let x represent the number of defective cars.

Assuming that number of defective cars are uniformly distributed in the total number of cars, we can go ahead and solve for x by setting up proportion as shown below;

Answer:

p(2) = 2

f(3) = 7

Step-by-step explanation:

Given p(x) = -3(x - 4)² + 14 find p(2)

Simply substitute x with the value 4 in the above expression

p(2) = -3(2-4)² + 14

= -3(-2)² + 14

= -3(4) -+ 14

= -12 + 14

= 2

f(3) can directly read from the table

for x = 3, f(x) = 7

so f(3) = 7

Answer: y = −6

Step-by-step explanation:

Answer:

5y−3=6y+3

Step 1: Subtract 6y from both sides.

5y−3−6y=6y+3−6y

−y−3=3

Step 2: Add 3 to both sides.

−y−3+3=3+3

−y=6

Step 3: Divide both sides by -1.

−y

−1

=

6

−1

y=−6

Step-by-step explanation:

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The value of the ratio of skateboards wheel in its simplest form is 1:4.

The ratio refers to the numerical relationship between two variables.

Ratios show the quantity of one variable contained in another.

Ratios are expressed in fractions, decimals, or percentages.  They can also be written using the ratio symbol (:).

The total number of skateboards = 12

The total number of wheels = 48

The ratio value of skateboards wheels = 48:12

= 4:1 or 1:4

Thus, we can conclude that for every skateboard, there must be 4 wheels, given their ratio values.

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Answer:

  (d) Domain: {x|x∈ℝ} and range: {y|y∈ℝ}

Step-by-step explanation:

You want the domain and range of the relation shown in the graph.

The graph shows a line. The arrows on the end of it mean that it extends to infinity in both directions (up and left; down and right).

The domain is the horizontal extent. It includes all real numbers (from -∞ to +∞).

The range is the vertical extent. It includes all real numbers (from -∞ to +∞).

Domain: {x|x∈ℝ} and range: {y|y∈ℝ}

If Isaiah is measuring 3 1/2 inches using a ruler marked in 1/8 inch increments. The total number of eighths that Isaiah will need to count out to make his measurement is B.28.

We would be making use of the given hint to determine the total number  of eight.

The hint is to use  3 1/2 into an improper fraction with a denominator of 8.​

Hence,

Converting 3 1/2 to improper fraction

3 1/2 = 7/ 2 inches

Now let determine the total number with a denominator of 8.​

= 7 × 4 / 2 ×4

= 28 /8

Therefore the correct option is B.

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Stocks and bonds both have costs, risks, and returns associated with them. Bonds offer an interest payment through a maturity date while stocks may offer dividends that will grow until it is sold.

Over the long term, stocks have the highest potential for growth for investors. Investors who have chosen to hold onto stocks for an extended length of time have typically been rewarded with robust, profitable returns. However, stock prices fluctuate both up and down.

An IOU-like debt security called a bond. Bonds are issued by borrowers to attract capital from investors ready to extend a loan to them for a specific period of time. When you purchase a bond, you are making a loan to the issuer, which could be a corporate, government, or municipality.

A dividend is a payment made by a corporation to its shareholders that is decided by the board of directors. Dividend payments are frequently made quarterly and might take the form of cash payments or stock reinvestments.

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The table which represents the transformation of function f given by g(x) = 3f(x) is:

x                   - 3   - 1     1    3

f(x)                - 5   - 1     3    7

g(x) = 3f(x)    - 15  - 3    9    21

When a function is transformed, the graph's curve of the parent function either "moves to the left/right/up/down," "expands or compresses," or "reflects."

A table for every value of x and its corresponding value of f(x) is given as:

x      - 3   - 1     1    3

f(x)   - 5   - 1     3    7

Now the transformed function g is given as:

g(x) = 3f(x)

Therefore, the table that represents the transformation of f given by g will be:

x                   - 3   - 1     1    3

f(x)                - 5   - 1     3    7

g(x) = 3f(x)    - 15  - 3    9    21

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We construct  an equation and calculate that number as 7 .

The equals sign is a symbol used in mathematical formulas to denote the equality of two expressions. The word equation and its cognates can have slightly different meanings in different languages. For instance, in English, an equation is any well-formed formula that consists of two expressions linked by the equals sign, whereas in French, an equation is defined as containing one or more variables.

The equation's solutions are the values of the unknown variables that satisfy the equality, which are often referred to as the variables for which the equation must be solved.

let the number be x .

9 times the number is = 9x

5 times the number added to 3 is

(x + 3) × 5

Now let us set up an equation and find that number :

(x + 3) × 5 = 9x - 13

Simplifying the given equation we get :

5 (x + 3) = 9x - 13

5x + 15 = 9x - 13

5x - 9x = - 13 - 15

- 4x = - 28

x = - 28/-4

x = 7

Hence the required number calculated from the equation is 7 .

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Disclaimer: The complete question is :

If the sum of a number and 3 is multiplied by 5 , the result is equal to 9 times the number decreased by 13. Find the number

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Answer:

13

Step-by-step explanation:

3X4=12

To know if a number is a perfect sqare of a perfect cube you:

perfect sqare

determine if its sqaure root is an integer. If is not an interger, then the number is not a perfect sqare

perfect cube

determine if its Cube root is an interger. If is not an interger then the number is not a perfect cube

Then so, 121 is a perfect cube, because is the result of (11*11)

121 is not a perfect cube

The length and width of the rectangular box are 13 ft and 6 ft respectively.

The length of the batter's box on a softball field is 1 ft more than twice the width.

The area of the batter's box is 78 ft².

The length and width of the rectangular batter's box can be found as follows:

Therefore,

area of a rectangle = lw

where

Therefore,

area of the rectangular box = 78 ft²

l = 1 + 2w

Hence,

78 = (1 + 2w)w

78 = w + 2w²

2w² + w - 78 = 0

Hence,

w = 6  and w =  -13  /2

we can only use positive values.

Therefore,

width = 6 ft

length  = 2(6) + 1 = 13 ft

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Take the prime factorization of both numbers to find the GCF. Then list the characteristics that they share. If they have more than one of the same factor (for example, two 2's), write them both down. Then multiply the factors they share.

A positive integer's prime factors are the prime numbers that divide it exactly. Finding these numbers is known as integer factorization or prime factorization. According to the fundamental theorem of arithmetic, every positive integer has a unique prime factorization.

The GCF (Greatest Common Factor) of two or more numbers is the greatest number among the given numbers' common factors. The greatest common factor (GCF) of two natural numbers x and y is the largest possible number that divides both x and y without leaving any remainder. GCF can be calculated in three ways: division, multiplication, and prime factorization.

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1) Let's call Temperature by T

Since the temperature is not more than 82 F, then we exclude any value above 82º F, and we include 82.

T≤82

Answer:

Step-by-step explanation:

d=3n

0.10d+0.05n=1.40

0.10(3n)+0.05=1.40

0.3n+0.05n=1.40

0.35n=1.40

n=4

d=3(4)

d=12

12(0.10)+.05(4)=1.40

1.40=1.40

As a result of the sequences, the numbers for the first equation are 31,37,43, 64,128,256, and 37,42,47 for the second and third equations, respectively.

An ordered group of numbers with a shared difference between each succeeding word is known as an arithmetic sequence. For instance, the common difference in the arithmetic series 3, 9, 15, 21, and 27 is 6. An arithmetic progression is another name for an arithmetic sequence.

A geometric progression, sometimes referred to as a geometric sequence in mathematics, is a series of non-zero numbers where each term following the first is obtained by multiplying the preceding one by a constant, non-zero value known as the common ratio.

so for the first equation:

we see the common difference is constant :

second term/third term - first term/second term =7-1 or 13-7

                        =6

the next three terms in this series would be calculated by adding the common difference to consecutive last terms  =31,37,43

for the third equation

the common difference is a constant:

second term-first term=5

the next three terms in this series would be calculated by adding the common difference to consecutive last terms  =37,42,47

for the second equation we don't see a constant value as a common difference ,then we conclude that it is a Geometric progression

difference in series = second term/first term

                              =4/2

                              =2

so for the three consecutive terms we will multiply the last consecutive terms by 2=64,128,256

Therefore the numbers for First equation are 31,37,43 for the second equation are 64,128,256 and third equation are 37,42,47

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nswer:

cvents A and B are independent because P(A|B) = P(A)

pxplanation:

f aA and B are independent, then:

In this case,

P(A) = 0.4, P(B) = 0.2

If we multiply:

Which verify what the problem tell us.

Next, the definition of two independent events is P(A|B) = P(A)

An exponential equation that passes through the points (2,1.8) and (5,14.4) is y = (0.6708)2ˣ

Exponential equation:

If you have two points, (x1, y1) and (x2, y2), you can define the exponential function that passes through these points by substituting them in the equation

y = abˣ

and solving for a and b.

In general, you have to solve this pair of equations:

y1 = abˣ¹ and y2 = abˣ²

Given,

The points (2,1.8) and (5,14.4).

Here we need to find the exponential equation that passes through the points.

This yields the following pair of equations:

1.8 = ab²

14.4 = ab⁵

If you divide the second equation by the first, you get

b³ = 14.4/1.8

b³ = 8

b = ∛8

b = 2

so b = 2. It's possible for b to also be equal to -2, but in this case, assume it's positive.

Now, we can substitute this value for b in either equation to get a. It's easier to use the first equation, so:

1.8 = a² x (2)²

1.8 = a²  x 4

a²  = 1.8/4

a²  = 0.45

a = √0.45

a = 0.6708

Therefore, the exponential form of the equation is,

y = (0.6708)2ˣ

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Answer: 60

Step-by-step explanation:6x10=60

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The left side of the two-column proof should be filled as follows:

Statements                                             Reasons_______________

32 = 2x + 4y; x = 2          ⇒                       Given

32 = 2(2) + 4y                ⇒                     Substitution Property

32 - 4 = 4 + 4y - 4        ⇒                   Subtraction Property of Equality

28/4 = 4y/4                  ⇒                  Division Property of Equality

7 = y, y = 7                    ⇒                  Symmetric Property of Equality

The division property of equality states that when both sides (left and right) of an algebraic expression or equation are divided by the same (common) real number that isn't equal to zero (0), the quotients would always remain equal.

The subtraction property of equality states that the two (2) sides of an algebraic expression or equation would still remain equal even when the same number has been subtracted from both sides of an equality.

In conclusion, it has been proven that y is equal to 7 by using the two-column proof.

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Answer:

[tex]g( - 16) = - 3[/tex]

Step-by-step explanation:

substitute

[tex]x = - 16 \: by \: g(x) = \frac{1}{4} x + 1 \\ g( - 16) = \frac{1}{4} x( - 16) + 1 \\ g( - 16) = - \frac{1}{4} x16 + 1 \\ g( - 16) = - 4 +1 \\ g( - 16) = - 3[/tex]

SOLUTION:

Answer:

[tex]x = \frac{196}{13} [/tex]