Given f(x) = 2x and g(x) = f(x − 4) + 5, write the new function rule (equation) for function and

describe

The average rate of change of the function on the interval from x = 3 to x = 5 is 40.

In mathematics, the average rate of change is a measure of how much a dependent variable (such as y) changes in proportion to a change in the independent variable (such as x). It is calculated by dividing the change in the dependent variable by the change in the independent variable during a certain time period. If the value of y rises by 5 units when the value of x rises by 2 units, the average rate of change is 5/2 Equals 2.5 per unit change in x. Calculus students frequently use this idea to compute the slope of a line on a graph or to calculate the rate at which a function changes. It is a crucial idea in several domains.

How to solve?

f(x) = 10(2)x,the interval is from x = 3 to x = 5

At x = 3,  10(2)3 = 80

at x = 5, 10(2)5 = 160

total change over the interval from x = 3 to x = 5 is 160 - 80 = 80.

the length of the interval is 5 - 3 = 2,

average rate of change is 80 / 2 = 40.

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Answer:

He saved $121

Step-by-step explanation:

All you have to do is divide $726/6 which gives you $121.

Answer:

1

Step-by-step explanation:

x²-10x+9 = 0

(x-9)(x-1) = 0

x-9 = 0  =>  x=9

x-1 = 0  =>  x=1

The solution for the function R(x) = x - 7 / x + 2 ≤ 0 is −2<x≤7 or x (-2, 7].

Each element of X receives exactly one element of Y when a function from one set to the other is used. The sets X and Y are collectively referred to as the function's domain and co-domain, respectively.

Given:

The function R(x) = x - 7 / x + 2

The given condition x - 7 / x + 2  ≤ 0

Solve the inequality as shown below

Here, the critical points are x = 7 and x = -2

Check the denominator for the critical point, x = -2 will not be suitable for the function as it becomes undefined,

Thus

−2<x≤7 or x (-2, 7]

Therefore, the solution for the function R(x) = x - 7 / x + 2 ≤ 0 is −2<x≤7 or x (-2, 7].

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Answer:

Step-by-step explanation:

30,248.35

The two integers are equal to 1 and 7 respectively.

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

Let x represent the smaller number. Then the given relationships say ...

1/x + 2/(x+6) = 9/7

Multiplying by 7x(x+6), we have ...

7(x+6) +14x = 9x(x+6)

9x² +54x = 21x +42 . . . . .eliminate parentheses, swap sides

9x² +33x = 42 . . . . . . . . ...subtract 21x

3x² +11 -14 = 0 . . . . . . . . . .subtract 42 and divide by 3

(3x +14)(x -1) = 0 . . . . . . . . factor

Values of x that make this true are x = 1 and x = -14/3. Then for the positive integer x=1, the other integer is x+6=7.

Therefore, the two integers are equal to 1 and 7 respectively.

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11) (Level 2):

I believe if you just multiply the shaded area (10x6), then maybe subtract the non-shaded area (20cm and 10 cm). After doing this process, I believe you will get: 200-60 = 140! (But then again I could be wrong, I’ll admit it I’m not a pro but at least I gave effort!)

Fraction: 14/10

Decimal: 1.4

Percent: 140%

12) (Level 3):

I have absolutely no idea about this question and I don’t want to guess! Sorry!

The distances from the point (x,y) are given as follows:

Suppose that we have two points with coordinates given by [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex].

The shortest distance between them is given by the equation presented as follows:

[tex]D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

The coordinates of the focus of the parabola are given as follows:

(-3,2).

Hence the distance of the point (x,y) to the focus of the parabola is given as follows:

√(x+3)²+(y-2)².

The equation that defines the directrix is given as follows:

y = 4.

The y-coordinate of the point is of:

y.

Hence the distance is of:

|y - 4|.

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Answer:

D x = 3 y = 4

(3.4)

Step-by-step explanation:

See below

Answer: f^-1(x)=(3x-2)/4

Step-by-step explanation:

1. Change the x and y

y=(4x+2)/3 → x=(4y+2)/3

2. Swap sides (optional)

x=(4y+2)/3 → (4y+2)/3=x

3. Multiply the equation by 3 on both sides

(4y+2)/3=x → (3)*(4y+2)/3=x*(3)

4. Subtract 2 from both sides

(3)*(4y+2)/3=x*(3) → 4y=3x-2

5. Divide 4 from both sides

4y=3x-2 → y=(3x-2)/4

6 Substitute f^-1(x) for y

y=(3x-2)/4 → f^-1(3x-2)/4

The amount you have after 18 months is $ 1143.96 .

what is compound interest?

Interest that is added to a loan or deposit sum is known as compound interest. In our daily lives, it is the notion that is employed the most frequently. Compound interest is calculated for a sum using the principal and interest accrued over time. Compound interest and simple interest differ primarily in this way.

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

Given:

P = 1000 , r in decimal  = 0.09 ,

n = 12 ,  t = 1.5 ( 18 months = one and a half years).

Substituting the values into the formula:

A = 1000 ( 1 + 0.09/ 12 ) ^[12 ( 1.5 )]          

  ≈ 1000 ( 1.143960 )          

  = 1143.960

Rounding to the nearest cent, we get $ 1143.96 .

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Answer: 47

Step-by-step explanation: Let's call the number of chairs Liam has x. We can set up the following equations to represent the problem:

7r = x - 5

5r = x - 17

Where r is the length of each row.

To solve for x, we can set the two equations equal to each other:

7r = x - 5

5r = x - 17

Then we can solve for x by combining like terms and solving for the value of x:

7r - 5r = x - 5 - x + 17

2r = 12

r = 6

We can substitute the value of r back into either of the original equations to solve for x:

7r = x - 5

7 * 6 = x - 5

x = 47

Therefore, Liam has 47 chairs.

Answer:

  no solution

Step-by-step explanation:

You want the solution to the absolute value equation |6y -7| = -1.

The expression on the left side of the equal sign is the output of the absolute value function. That output can never be negative: all negative input values are transformed to positive output values by the absolute value function.

The expression on the right side is negative. There is no value of the variable that will make the absolute value function give a negative output. There can be no solution.

Answer:

Step-by-step explanation:

10 = 2(x - 2)

How would I solve for X?

10 = 2(x - 2)

10 = 2x - 4

14 = 2x

x = 7

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

check

10 = 2(7 - 2)

10 = 2 x 5

10 = 10

the answer is good

Answer:

x=7

Step-by-step explanation:

Given:

A linear equation is given as:

[tex]10=2(x-2)[/tex]

To Find:

The objective is to solve the equation and find the value of x.

Explanation:

[tex]10=2(x-2)\\[/tex]

⇒ [tex]\frac{10}{2}=x-2[/tex]

⇒[tex]5=x-2[/tex]

⇒ [tex]x=7[/tex]

Final Answer :

The value of x is 7.

Answer:

Part (c):

Part (d):

Part (e):

Step-by-step explanation:

Given equations:

[tex]\textsf{1.\;\;Mr.\;Pace}: \quad y = \dfrac{3}{4}x + 4[/tex]

[tex]\textsf{2.\;\;Mr.\;Glintz}: \quad y = \dfrac{2}{3}x + 7[/tex]

[tex]\textsf{3.\;\;Ms.\;Liskey}: \quad y = 3x[/tex]

[tex]\textsf{4.\;\;Mr.\;Brininger}: \quad y = \dfrac{1}{2} + 10[/tex]

Plot the lines on the given coordinate plane.

(See attachment).

Draw a line at y = 25 to represent the distance at the finish line.

Find the points of intersection of the linear equation of each contestant and the finish line equation y =25.  The point with the smallest x-value is the contestant who is first place.

Mr Pace

[tex]\begin{aligned}\implies 25&= \dfrac{3}{4}x + 4\\\dfrac{3}{4}x&=21\\x&=28\; \sf seconds\end{aligned}[/tex]

Mr Glintz

[tex]\begin{aligned}\implies 25&= \dfrac{2}{3}x + 7\\\dfrac{2}{3}x&=18\\x&=27\; \sf seconds\end{aligned}[/tex]

Mr Liskey

[tex]\begin{aligned}\implies 25&= 3x\\x&=8.33\; \sf seconds\;(2\;d.p.)\end{aligned}[/tex]

Mr Brininger

[tex]\begin{aligned}\implies 25&= \dfrac{1}{2}x + 10\\\dfrac{1}{2}x&=15\\x&=30\; \sf seconds\end{aligned}[/tex]

Therefore, the order in which the contestants reached the finish line is:

From inspection of the attached graph, Ms Liskey (red line) passes all three contestants.  To find the time at which she passes the other contestants, substitute her equation into the contestant's equation and solve for x.

Mr Pace

[tex]\begin{aligned}\implies 3x&= \dfrac{3}{4}x + 4\\\dfrac{9}{4}x&=4\\x&=1.78\; \sf seconds\;(2\;d.p.)\end{aligned}[/tex]

Mr Glintz

[tex]\begin{aligned}\implies 3x&= \dfrac{2}{3}x + 7\\\dfrac{7}{3}x&=7\\x&=3\; \sf seconds\end{aligned}[/tex]

Mr Brininger

[tex]\begin{aligned}\implies 3x&= \dfrac{1}{2}x + 10\\\dfrac{5}{2}x&=10\\x&=4\; \sf seconds\end{aligned}[/tex]

Therefore:

If you want to win the race, either:

The Maclaurin series of sin(3x²) up to n = 3 is

[tex]=3x^2-\dfrac{9}{2}x^6+\frac{81}{40}x^{10}+\ldots[/tex]

The function is given in the question as

g(x) = sin(3x²)

Taylor (Maclaurin) series of sin(3x²) up to n = 3

A Maclaurin series is given by f(x),

[tex]f\left(x\right)=f\left(a\right)+\frac{f^'\left(a\right)}{1!}\left(x-a\right)+\frac{f^{''}\left(a\right)}{2!}\left(x-a\right)^2+\frac{f^{'''}\left(a\right)}{3!}\left(x-a\right)^3+\ldots[/tex]

[tex]f\left(x\right)=f\left(0\right)+\frac{f^'\left(0\right)}{1!}\left(x\right)+\frac{f^{''}\left(0\right)}{2!}\left(x\right)^2+\frac{f^{'''}\left(0\right)}{3!}\left(x\right)^3+\ldots[/tex]

We need to do to get the desired polynomial is to calculate the derivatives, evaluate them at the given point, and plug the results into the given formula.

f⁽⁰⁾(x) = f(x) =sin(3x²)

Evaluate the function at the point: f(0)=0

1st derivative: f(1)(x)=(sin(3x²))′ = 6xcos(3x²)

1st derivative at the given point: (f(0))′=1

2nd derivative: f(2)(x)=(f(1)(x))′=(6xcos(3x²))′=−36x2sin(3x2)+6cos(3x²)  

Evaluate the 2nd derivative at the given point: (f(0))′′=6

3rd derivative: f(3)(x)=(f(2)(x))′=(6xcos(3x²))′=−36x2sin(3x²)+6cos(3x²)  

Evaluate the 3rd derivative at the given point: (f(0))′′′=0

Now, use the calculated values to get a polynomial:

[tex]g(x)=0+\frac{0}{1!}x+\frac{6}{2!}x^2+\frac{0}{3!}x^3+\frac{0}{4!}x^4+\frac{0}{5!}x^5[/tex]

Thus, the Maclaurin series of sin(3x²) up to n = 3 is

[tex]=3x^2-\dfrac{9}{2}x^6+\frac{81}{40}x^{10}+\ldots[/tex]

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The statement that is true about the algebraic expression 450 −15t is B. A student has 450 marbles. They give t marbles to each of their 15 friends. The expression represents the total marbles the student is left with.

In mathematics, an algebraic expression is an expression composed of variables and constants as well as algebraic operations (addition, subtraction, etc.). Terms combine to form expressions.

In this case, when a student has 450 marbles. They give t marbles to each of their 15 friends.

The expression represents the total marbles the student is left with will be:

= 450 - (15 × t)

= 450 - 15t

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The natural logarithms of the given questions are f(e⁴) = 4, f(e ㏑ 5) = 5, and f(e³ ㏑ 5) = -0.5108.

What is the natural logarithm?

A number's natural logarithm is its logarithm to the base of the transcendental and irrational number e, which is roughly equivalent to 2.718281828459.

Given,

f(x) = ln x,

a) f(e⁴)

f(e⁴) = In(e⁴)

= 4In(e)

= 4 * 1

f(e⁴) = 4

b) f(e ㏑ 5)

= f(e⁻⁵)

= [tex]f(e^{\frac{1}{5} })[/tex]

= In(1) - In(e⁵)

= 0 - 5In(e)

= 5*1

f(e ㏑ 5) = 5

c) f(e³ ㏑ 5)

[tex]= f(e^{\frac{3}{5} })[/tex]

=  In(3) - In(e⁵)

= ln(3) - ln(5)

f(e³ ㏑ 5) = -0.5108

Hence, The natural logarithms of the given questions are f(e⁴) = 4, f(e ㏑ 5) = 5, and f(e³ ㏑ 5) = -0.5108.

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Using the given function, the income for day 31 is

the total Income for working 31 days is

Using the given function an = a1 (r)^n-1

where

a1 = first day = 1

r = common ratio = 2

The income on day 31, n = 31

= 1 * 2^(31 - 1)

= 2^30

= 1,073,741,824 cents

= $10,737,418.24

Total Income for working 31 days n = 31

Using Sn= a1 (1-r^n) / 1 - r

Sn = (1 * (1 - 2^31)) / (1 - 2)

Sn = (1 - 2^31)) / - 1

Sn = (1 - 2147483648) /-1

Sn = 2147483647

Sn = 2,147,483,647 cents

Sn = $21 474 836.47

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Answer:

p = - 123/10 = -123/10 = -12.3

Step-by-step explanation:

-3(0.5 + 3p) = -6(p - 5.9)

Use the distributive property to multiply -3 by 0.5 + 3p

-1.5 - 9p = -6 (p - 5.9)

Use the distributive property to multiply -6 by p - 5.9

-1.5 - 9p = -6p + 35.4

Add 6p to both sides

-1.5 - 3p = 35.4

Add 1.5 to both sides

-3p = 35.4 + 1.5

Add 35.4 and 1.5 to get 36.9

-3p = 36.9

Divide both sides by -3

p = 36.9/-3

Expand 36.9/-3 by multiplying both numerator and the denominator by 10.

p = 369/-30

Reduce the fraction 369/-30 to lowest terms by extracting and canceling out 3.

p = -123/10

Answer:

P= 8

Step-by-step explanation:

-1 (-2p = -3p + 8)

2p = 3p - 8

-p = -8

p = 8

to check this...

-2(8) = -3(8) + 8

-16 = -24 + 8

-16 = -16

Answer:

C

Step-by-step explanation:

Answer:

I think the Answer choice = C

Step-by-step explanation:

Answer:

11

Step-by-step explanation:

If it is correct, could you please give me brainliest? I would really appreciate it! Thank you!

The answer you are looking for would be 44.

Here,

The number lies between 41 and 46 are 42, 43, 44, and 45

From the straightforward observation and according to the condition that must be divisible by 2 and 11 there is only one number that is divisible by both 2 and 11 is 44.

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The largest possible area will be 14400 meters², the pen's length should be 120 meters, and the width should be 120 meters.

The perimeter of a rectangle is defined as the addition of the lengths of the rectangle's four sides.

Let's assume that the length of the pen would be x.

We have been given the perimeter as 480 meters.

Let w be the width of the rectangle  and l be the length of the rectangle

The perimeter of a rectangle = 2(l+w)

So, 480= 2(x + w)

240 = x + w

w = 240 - x

Length = x and width = (240 – x)

Area = x(240 - x) = 240x - x²

To get the maximum area, we equate the first derivative of the area to zero.

A = 240x - x²dA/dx = 240 - 2x = 0

2x = 240

x = 120

So, width = 240 – x = 240 – 120 = 120.

Maximum area = length × width

Maximum area = 120 × 120 = 14400 meters²

Thus, the pen should be 120 meters long and 120 meters wide, with a maximum possible area of 14400 meters².

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The set of rational numbers arranged from least to greatest is -3.5, -1/4, 1/3, 2, which is option (b).

Any number that can be represented as the ratio of two integers and where the denominator is not zero is referred to as rational.

The given sets of rational numbers are:

(a) -3.5, -1/4, 2, 1/3

(b) -3.5, -1/4, 1/3, 2

(c) 2, 1/3, -1/4, -3.5

(d) -1/4, 1/3, 2, -3.5

Note that 1/4 = 0.25 and 1/3= 0.33

Therefore, the set of rational numbers arranged from least to greatest is -3.5, -1/4, 1/3, 2, which is option(b).

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Step-by-step explanation:

Firstly we determine how many people clara will reach in 40 phone calls.

6 people=12 calls

X people=40 calls.

make X the subject of the formula.

X=(6×40)÷12=20

So clara will reach 20 people in 40 phone calls.

Now lets determine how many people Toby will reach in 40 phone calls.

7 people=13 calls

Y people=40 calls.

Therefore making Y the subject of the formula we have,

Y=(7×40)÷13=21 people.

So toby will reach more people in 40 calls