F(x)=1-2x part e) find f^-1 (x)

According to the given rate the total profit made by the company in the first year is R240,000.

The problem statement does not provide enough information to determine the rate of return or interest rate. In the absence of this information, it is not possible to calculate the rate of return or interest rate earned on the capital contributions or on the profit made by the company.

According to the agreement, the profit made by the company should be divided among the partners in the ratio of their capital contributions, which in this case is 225,000:150,000:75,000 or 3:2:1.

Let's assume that the total profit made by the company in the first year is P. Dee's share of the profit is R40,000, which is 1/6 of the total profit, since her capital contribution is 1/6 of the total capital.

So, we can set up the equation:

1/6 * P = R40,000

Multiplying both sides by 6, we get:

P = R240,000

Therefore, according to the given rate the total profit made by the company in the first year is R240,000.

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If it's a decimal round to the nearest tenth then the value οf x is 4/3.

In mathematics, a prοpοrtiοn is an equatiοn that relates twο ratiοs. It is expressed in the fοrm οf a/b = c/d, where a, b, c, and d are numbers οr quantities.

In a prοpοrtiοn, the crοss-prοducts are equal, meaning that the prοduct οf the numeratοr οf the first ratiο and the denοminatοr οf the secοnd ratiο is equal tο the prοduct οf the denοminatοr οf the first ratiο and the numeratοr οf the secοnd ratiο.

the given prοblem can be sοlved by using ratiοs οf the cοrrespοnding sides.

we knοw that, the ratiο οf cοrrespοnding sides is equal.

sο, we have :

[tex]$\frac{8}{x+2} = \frac{12}{5}[/tex]

8 × 5 =  12 × (x+2)

40   = 12x + 24

12x  = 40 - 24

= 16

∴ x = 16/12 = 4/3

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f[g(-2)] = 2 as the function f(x) = x - 5 must then be used to evaluate the function f[g(-2)] .

A function is a mathematical construct that transforms a set of inputs (referred to as the domain) into a single output from another set (called the range). A function is, in other words, a rule that allots precisely one output to each input in the domain. The notation f(x) is frequently used to denote functions, where f stands for the function name and x for the input value. As an illustration, the function f(x) = 2x + 1 doubles any input x, adds 1, and returns the result.

given

To begin, we must calculate g(-2) using the formula g(x) = x2 + 3:

[tex]g(-2) = (-2)^2 + 3 = 4 + 3 = 7[/tex]

The function f(x) = x - 5 must then be used to evaluate the function f[g(-2)]:

f[g(-2)] = f(7) = 7 - 5 = 2

Hence, f[g(-2)] = 2.

f[g(-2)] = 2 as the function f(x) = x - 5 must then be used to evaluate the function f[g(-2)] .

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Answer: 80 mph fet stream

Step-by-step explanation:

2576/4 + the jet stream=3216/4- jet stream

simplified, 80 mph

The critical value Zc is ±2.170.

What is a confidence interval?

An estimated range for an unknown parameter is known as a confidence interval. The 95% confidence level is the most popular, however other levels, such as 90% or 99%, are occasionally used when computing confidence intervals.

Here, we have

Given: a confidence interval at the level of confidence shown below.

c=0.97

We have to find the critical value Zc.

c = 0.97

we get a = 1 - c

a = 1 - 0.97

a = 0.03

Critical value Zc = ±2.170

Hence, the critical value Zc is ±2.170.

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Triangle JKL's area is 3 square units as a result.

The vertices' coordinates and the following formula must be used to determine the area of triangle JKL:

Area equals 1/2*base*height

We must first determine the triangle's height and base length. The distance between points J and K, which is the base, is:

base = √[(x2 - x1)² + (y2 - y1)²]

= √[(-4 - (-7))² + (-8 - (-9))²]

= √[3² + 1²]

= √10

The height is calculated as the distance between point L and the base-containment line. To calculate it, we can use the following formula for the separation between a point and a line:

height = |(y2 - y1)x0 - (x2 - x1)y0 + x2y1 - y2x1| / √[(y2 - y1)² + (x2 - x1)²]

= |(-9 - (-8))(-5) - (-7 - (-4))(-6) + (-4)(-9) - (-8)(-7)| / √[(1)² + (3)²]

= 6 / √10

Now, we can enter the height and base values into the triangle's area formula:

Area = 1/2 * √10 * 6 / √10

= 3

Triangle JKL's area is 3 square units as a result.

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The total garden area obtained by adding the new area of the garden is obtained as 36  sq. ft.

Your polygon's four sides must be two congruent pairings in order to have two pairs with parallel sides.

The length of the left and right sides, as well as the base and top, will be equal. A four-sided polygon with two sets of parallel, congruent sides with four internal angles of 90° each is required to be a rectangle.

Given data:

current area of garden = 24 square feet.

Coordinates of other garden:

(−17, 15), (−20, 15), (−17, 11), and (−20, 11)

Length of new garden = -20 + 17 = 3 ft.

Width of new garden = 15 - 11 = 4 ft.

Area of new garden = length * width

Area of new garden = 3ft * 4 ft

Area of new garden = 12 sq. ft.

total area of the garden = current area +  new garden area

total area of the garden = 24 sq. ft. + 12 sq. ft.

total area of the garden = 36  sq. ft.

Thus, the total area of the garden obtained by adding the new area of the garden is obtained as 36  sq. ft.

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Answer:

The answer is (d) 200 m².

Step-by-step explanation:

Let's denote the width of the rectangle as w. Then the length of the rectangle is twice the width, which means the length is 2w.

The perimeter of a rectangle is given by the formula:

P = 2l + 2w

Substituting the values, we get:

60 = 2(2w) + 2w

Simplifying the equation, we get:

60 = 6w

Dividing both sides by 6, we get:

w = 10

So the width of the rectangle is 10 m and the length is twice the width, which is 20 m.

The area of a rectangle is given by the formula:

A = lw

Substituting the values, we get:

A = 10 x 20

Simplifying the equation, we get:

A = 200

Therefore, the area of the rectangle is 200 m².

Answer:

differentiate between tropical and subtropical climate zone on the basis of climate and economic activities in 4 points class 10

The answer of the given question based on series to match the correct summation is the given series matches with option 2: 1 + 0.5 + 0.25 + 0.125 + ...

In mathematics, an infinite series is a sum of infinitely many terms. An infinite series can be thought of as the limit of the sequence of partial sums, which is obtained by adding only the first n terms of the series. If the limit of the sequence of partial sums exists and is finite, then the infinite series is said to be convergent; otherwise, it is said to be divergent.

Infinite series have many applications in mathematics, science, and engineering, and are used to approximate functions, solve differential equations, and model physical phenomena.

The given infinite series is: 3 - 9/4 + 27/16 - 81/64 + ...

This series is a geometric series with first term a = 3 and common ratio r = -3/4.

The formula for the sum of an infinite geometric series with first term a and common ratio r (|r| < 1) is given by:

S = a / (1 - r)

Plugging in the values of a and r, we get:

S = 3 / (1 - (-3/4)) = 3 / (7/4) = 12/7

Therefore, the given series matches with option 2: 1 + 0.5 + 0.25 + 0.125 + ...

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Answer:

the answer is : 2

Answer:

Step-by-step explanation:

Solve from the inside outwards:

         [tex]g(-4)=\frac{-4}{2} =-2[/tex]

         [tex]f(g(-4))=f(-2)=10\times(-2)-4=-24[/tex]

Solution:  [tex]f(g(-4))=-24[/tex]

Answer:

  3 6 4 2 1 5

Step-by-step explanation:

You want to know the sequence of solving the problem of finding the time until the rope hits the ground when thrown at 10 m/s down a 50 m cliff.

The equation describing the height of the rope is of the form ...

  h(t) = -4.9t² +v₀t +h₀

where v₀ is the initial upward velocity (meters per second), and h₀ is the initial height (meters).

Here, the initial velocity is described as 10 m/s downward, so we expect v₀ to have a value of -10. However, the offered equation(s) show v₀ = +10, so we will be solving the problem assuming the rope is initially thrown upward.

For v₀ = 10 and h₀ = 50, the equation for the height of the rope is ...

  h(t) = -4.9t² +10t +50 . . . . . . . step 1

We want to find the time (t) when the height is 0, so we substitute h(t) = 0 into the above equation:

  0 = -4.9t² +10t +50 . . . . . . . . step 2

This can be solved nicely using the quadratic formula. We recognize that a=-4.9, b=10, c=50, so the filled-in formula is ...

  [tex]t=\dfrac{-10\pm\sqrt{10^2-4(-4.9)(50)}}{2(-4.9)}\qquad\textsf{step 3}[/tex]

Simplifying this and taking the square root, we have ...

  [tex]t\approx\dfrac{-10\pm32.86}{-9.8}\qquad\textsf{step 4}[/tex]

Evaluating this expression gives ...

  t ≈ -2.33  or  t ≈ 4.37 . . . . . . . . . step 5

Writing the answer as a sentence is the last step:

  The rope will hit the ground in about 4.37 seconds. . . . . . step 6

The sequence of step numbers is ...

  3 6 4 2 1 5

__

Additional comment

The equation assume ballistic motion, with no air resistance or other impediment to the rope falling only due to gravity. We assume that Krystal plans to use the rope for rappelling, so it is probably coming uncoiled as it falls. That process usually changes the motion from one that is purely defined by the force of gravity.

We also note that if the initial throwing direction is downward, then the solution to the equation h(t) = -4.9t² -10t +50 is actually t ≈ 2.33 s.

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The summary table of the function obtained using differential calculus, and the graph of the function are presented as follows;

[tex]\begin{tabular}{|c|c|c|} \cline{1-3} Interval & Increasing/Decreasing &Concave Up/Down\\ \cline{1-3}(-\infty, 0)&Decreasing&Concave Up \\ \cline{1-3} (0, 3/2) & Decreasing & Concave Down \\ \cline{1-3} (3/2, \infty) & Increasing& Concave Up\\ \cline{1-3}\end{tabular}[/tex]

Please find attached, the graph of the function, created using MS Excel

Differential calculus is the part of calculus that is concerned with the rate of change of quantities.

The steps to determine the intervals where f(x) = x⁴ - 2·x³ is increasing or decreasing, or constant are presented as follows;

The derivative of the function is first found using calculus, differentiation as follows;

f'(x) = 4·x³ - 6·x²

The critical points at which the slope of the graph changes from increasing are found at the points where f'(x) = 0 as follows;

f'(x) = 0 = 4·x³ - 6·x²

4·x³ - 3·x² = 0

2·x²·(2·x - 3) = 0

Therefore, at the critical points, x = 0, and x = 3/2

The first derivative test can be used to determine the interval the function is increasing or decreasing as follows;

f(x) is increasing at x < a, if f'(x) > 0 for x < a and f(x) is decreasing at x > a if f(x) < 0 at x > a, where a is a critical point

f(x) is decreasing at x < a, if f'(x) < 0 for x < a and f(x) is increasing at x > a if f(x) > 0 at x > a,

The concavity of the function at x = a if f'(x) = 0 at x = a can be determined using the second derivative as follows;

f''(x) = 12·x² - 12·x

f''(x) > 0 at x = a if f(x) is concave up

f''(x) < 0 at x = a if f(x) is concave down

When a = 0, we get;

f''(0) = 12 × 0² - 12 × 0 = 0

f''(0.5) = 12 × 0.5² - 12 × 0.5 = -3 (Concave down)

f''(-0.5) = 12 × (-0.5)² - 12 × (-0.5) = 9 (Concave up)

f''(0) = 0 at x = 0, therefore, x = 0 is an inflection point

f'(-1) = 4·(-1)³ - 6·(-1)² = -10

f'(1) = 4·(1)³ - 6·(1)² = -2

Therefore, the graph changes from decreasing and is also decreasing at x > 0

Similarly we get;

f'(2) = 4·(2)³ - 6·(2)² = 8 (Increasing)

Therefore;

f'(1) = 4·(1)³ - 6·(1)² = -2

There

f''(3/2) = 12 × (3/2)² - 12 × (3/2) = 9

The second derivative at x = 3/2 = 9 > 0, therefore;

The function, f(x) is concave up at x = 3/2

Therefore, we get the following table;

Please find attached the graph of the function created using MS Excel

and the following points;

x   [tex]{}[/tex]      f(x)

-0.9    [tex]{}[/tex]2.1141

-0.8 [tex]{}[/tex]   1.4336

-0.1 [tex]{}[/tex]    0.0021

0 [tex]{}[/tex]       0

0.4 [tex]{}[/tex]    -0.1024

0.5 [tex]{}[/tex]    -0.1875

1.5 [tex]{}[/tex]     -1.6875

2 [tex]{}[/tex]        0

2.5 [tex]{}[/tex]     7.8125

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The function oscillates between two horizontal asymptotes, y=3 and y=-3, as x approaches the vertical asymptotes.

How to study graph?

To graph the function y=3cot(x) between 0 and 2, we can start by identifying the key characteristics of the cotangent function. The cotangent function is defined as the ratio of the adjacent side to the opposite side of a right triangle, with respect to a given angle. As such, the cotangent function is periodic with a period of π and has vertical asymptotes at x=nπ, where n is an integer.

The coefficient of 3 in the given function y=3cot(x) vertically stretches the function by a factor of 3, and does not affect the period or the vertical asymptotes.

Between 0 and 2, there are two vertical asymptotes at x=0 and x=π. To graph the function, we can plot a few points along the curve, using the symmetry properties of the cotangent function. Specifically, the cotangent function is odd, meaning that cot(-x)=-cot(x), and thus the graph is symmetric about the origin.

At x=π/4, we have cot(π/4)=1, and thus y=3cot(π/4)=3. Similarly, at x=3π/4, we have cot(3π/4)=-1, and thus y=3cot(3π/4)=-3. Using the symmetry property, we can also plot points at x=-π/4 and x=-3π/4, with y=3cot(-π/4)=-3 and y=3cot(-3π/4)=3, respectively.

Connecting these points with smooth curves, we can see that the graph of y=3cot(x) oscillates between two horizontal asymptotes, y=3 and y=-3, as x approaches the vertical asymptotes at x=0 and x=π.

Overall, the graph of y=3cot(x) between 0 and 2 can be summarized as follows:

There are two vertical asymptotes at x=0 and x=π.

The function oscillates between two horizontal asymptotes, y=3 and y=-3, as x approaches the vertical asymptotes.

The graph is symmetric about the origin.

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The composite function [g . f](x) when evaluated has its value to be 4x² + 20x + 22

Given that, we have the following functions

f(x) = 2x + 5

g(x) = x² - 3

To find [g . f](x) we make use of g(f(x)), where we first need to find f(x) and then plug it into g(x) in place of x.

Given, f(x) = 2x + 5 and g(x) = x^2 - 3.

So,

g(f(x)) = g(2x + 5) [Substitute f(x) in place of x in g(x)]

g(f(x)) = (2x + 5)² - 3 [Substitute 2x + 5 in place of x in g(x)]

g(f(x)) = (4x² + 20x + 25) - 3

g(f(x)) = 4x² + 20x + 22

This means that

[g . f](x) = 4x² + 20x + 22

Therefore, [g . f](x) = 4x² + 20x + 22

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Answer:

[tex]1 \frac{1}{81} \ [/tex]

Solution:

[tex] \frac{a - 1}{a} = \frac{2}{3} [/tex]

1) 2a = 3(a - 1)

2a = 3a - 3

2a - 3a = -3

-a = -3 / * (-1)

a = 3

2) Replace the a with 3 in this:

[tex] \frac{ {a}^{4} + 1}{ {a}^{4} } [/tex]

[tex] \frac{ {3}^{4} + 1}{ {3}^{4} } = \frac{81 + 1}{81} = \frac{82}{81} = 1 \frac{1}{81} [/tex]

I don't know if I got this right, but at least I tried

A. The equation that relates these variables is:

d = s * t. B. The biker would travel 10 miles in 20 minutes. C. The biker would bike for 96 minutes to travel 48 miles. Rounded to the nearest tenth, this is 96.0 minutes.

a. Let's define the variables:

t = time in minutes

d = distance traveled by the biker in miles

s = speed of the biker in miles per minute

The equation that relates these variables is:

d = s * t

b. If the biker travels for 20 minutes, we can use the equation above to find the distance traveled:

d = s * t

d = s * 20 (since t = 20 minutes)

We need to know the speed of the biker to calculate the distance traveled. Let's assume that the biker's speed is 0.5 miles per minute (which is equivalent to 30 miles per hour):

d = 0.5 * 20

d = 10

Therefore, the biker would travel 10 miles in 20 minutes.

c. If the biker traveled 48 miles, we can rearrange the equation above to find the time:

d = s * t

t = d / s

We need to know the speed of the biker to calculate the time. Let's assume that the biker's speed is still 0.5 miles per minute:

t = 48 / 0.5

t = 96

Therefore, the biker would bike for 96 minutes to travel 48 miles. Rounded to the nearest tenth, this is 96.0 minutes.

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The lengths of the edges of the cubes are 45cm and 63cm respectively.

To determine the edge length of a cube with a volume of 91,125 cm³, we can use the formula:

Volume of a cube = edge length³

Plugging in the given value, we get:

91,125 = edge length³

Taking the cube root of both sides, we get edge length

edge length ≈ 45

Therefore, the edge length of the cube is approximately 45 cm.

To determine the side length of a square with an area of 3,969 cm², we can use the formula

Area of a square = side length²

Plugging in the given value, we get:

3,969 = side length²

side length = √3,969

Using a calculator, we can simplify this to

side length ≈ 63

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Answer:

Step-by-step explanation:

Let x be the side length of the square window. Then the area of the square window is x^2, and the area of the rectangular window is 16 * 4 = 64.

We can write the equation:

x^2 = 64

To solve for x, we can take the square root of both sides:

x = √64

x = 8

Therefore, the side length of the square window is 8 inches.

Answer:23200cm^3

Step-by-step explanation:

we are going to watch this figure as 2 different prisms. 1st-parallelepiped

and 2nd-trapezoidal prism;

The volume of the parallelepiped is equal to a*b*c which is the product of its sides(width, height, and length) and in this case, it is equal to V1=a*b*c=10*12*80(because the trapezoidal prisms side is parallelogram and facing sides of parallelogram are equal so c=80cm)=9600cm^3;

V2=Area of base*h

area of base is equal to area of transpazoid which is

A=((a+b)/2)*h=(12+22)/2*10=34/2*10=17*10=170cm^2

and as i said earlier h=80 because its a parallelogram so,

V2=170+80=13600cm^3

V=V1+V2=9600+13600=23200cm^3

The function with the greater initial value in this problem is given as follows:

Function f(x), the graphed function.

The initial value of a function, often denoted as f(0), is the value of the function when the independent variable (usually denoted as x) is equal to 0. In other words, it is the value of the function at the starting point of its domain.

Hence the initial value of each function in this problem is given as follows:

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Answer:

960 pages

Step-by-step explanation:

Answer:

first option (30x4 − 20x3 + 15x2)

By answering the presented question, we may conclude that There are polygon four vertex designated HT , HB, TE and EB in the provided picture.

A polygon is a two-dimensional shape that is formed by joining three or more points in a closed loop. Triangles, tetrahedra, pentagons, hexagons, and other forms are examples of polygons. Polygons are a fundamental geometric concept used to define and analyze a wide range of real-world forms and structures, such as buildings, layouts, and computer graphics. Polygons are examined not just for their perimeter, area, and angles, but also for their symmetry.

The vertex of the polygon in the image are the line segments that link the polygon's opposing vertices and are perpendicular to its base. There are four vertex designated HT , HB, TE and EB   in the provided picture.

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Therefore, the solutions to the quadratic equation x² + 7x - 8 = 0 are -8 and 1. The answers are D and E.

An equation is a mathematical statement that shows that two expressions are equal. It typically consists of variables, constants, and mathematical operations. Equations can be solved by manipulating the expressions to find the value of the variables that satisfy the equation. Equations can be used to model real-world situations, and they are an important tool in many fields, including mathematics, physics, engineering, and economics.

Here,

To check which values are solutions to the quadratic equation, we can substitute each value into the equation and see if it equals zero.

Substituting -1 into the equation:

(-1)² + 7(-1) - 8 = 1 - 7 - 8 = -14, which is not equal to zero.

Substituting 2 into the equation:

2² + 7(2) - 8 = 4 + 14 - 8 = 10, which is not equal to zero.

Substituting -4 into the equation:

(-4)² + 7(-4) - 8 = 16 - 28 - 8 = -20, which is not equal to zero.

Substituting -8 into the equation:

(-8)² + 7(-8) - 8 = 64 - 56 - 8 = 0, which is equal to zero. Therefore, -8 is a solution to the equation.

Substituting 1 into the equation:

1² + 7(1) - 8 = 1 + 7 - 8 = 0, which is equal to zero. Therefore, 1 is a solution to the equation.

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Answer:

The area of the shaded region is 21.

Step-by-step explanation:

Since point E lies halfway between AB and BC, the area of the shaded region (As) consists of two identical triangles with base equal to AB and height equal to half the measure of BC:

[tex]A_s=2\times A_t[/tex]

Where At is the area of each triangle.

[tex]A_t=\dfrac{AB\times BC/2}{2}[/tex]

[tex]A_t=\dfrac{AB\times BC}{4}[/tex]

We know AB=7 and BC=6, thus:

[tex]A_t=\dfrac{7\times6}{4} =\dfrac{42}{4}[/tex]

Simplifying:

[tex]A_t=\dfrac{21}{2}[/tex]

Finally:

[tex]A_s=2\times\dfrac{21}{2}[/tex]

[tex]A_s=21[/tex]

The area of the shaded region is 21.

Note the area of the shaded region is half the area of the rectangle.

Some of the radicals below that are like terms are; √2, 5√2, − 3√2, √32.

The like terms can be combined as 7√2.

The reason why the leftover terms are NOT like terms with the others is that the radicals lack the same root number also  radicand (expression under the root).

Radicals with the same root number AND radicand are similar radicals (expression under the root)

A number's radical and root are the same thing. A cube root, a square root, or an nth root could be the root. Therefore any number or phrase that uses a root is a radical. The word "radical" is derived from the Latin word radix, which means root. The radical can be used to explain a number's square, cube, fourth, and other forms of roots. The number that comes before the radical is known as the index number or degree.

3a. Some of the radicals that are like terms are; √2, 5√2, − 3√2, √32 and the reason why √32 is part of it is that √32 = 4√2

3b. The like terms can be combined as ;

(√2+ 5√2 − 3√2 + √32)

= (√2+ 5√2 − 3√2 + 4√2)

=7√2

3c. The reason why the leftover terms are NOT like terms with the others is that the radicals  do not have the same root number as well as  radicand (expression under the root).

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