What are the domain and range of the logarithmic function?

The correct equation which represent an identity will be;

⇒ 4(5x+2) = 20x+8

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Given that;

All the equations are,

1) 2x + 1 = 3x + 4

2) 4x - 3 = 2(2x + 7)

3) 6x + 3 = 2x + 10

4) 4(5x + 2) = 20x + 8

Now,

Solve each equation and find an identity as;

1) 2x + 1 = 3x + 4

⇒ 1 - 4 = 3x - 2x

⇒ - 3 = x

⇒ x = -3

Thus, It is not represent an identity.

2) 4x - 3 = 2(2x + 7)

⇒ 4x - 3 = 4x + 14

⇒ - 3 ≠ 14

Thus, It shows no solution.

3) 6x + 3 = 2x + 10

⇒ 6x - 2x = 10 - 3

⇒ 4x = 7

⇒ x = 7/4

Thus, It is not represent an identity.

4) 4(5x + 2) = 20x + 8

⇒ 20x + 8 = 20x + 8

⇒ 20x - 20x = 8 - 8

⇒ 0 = 0

Clearly, '0' is an additive identity.

Thus, This equation shows an identity.

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Answer:

2nd choice:  -3x - 5 = 10

Step-by-step explanation:

10 - x = -5

-x = -5 - 10 = -15

-3x - 5 = 10

-3x = 10 + 5

x = 15/-3 = -5

The possible values of x for the given triangle ABC are 0.75<x<8.

The triangle inequality theorem describes the relationship between the three sides of a triangle. According to this theorem, for any triangle, the sum of lengths of two sides is always greater than the third side.

Given that, m∠A < m∠B < m∠C.

a) From triangle ABC,

BC<AC<AB

2x+3<6x<5x+8

Now, 2x+3<6x

Subtract 2x on both the sides of an inequality, we get

3<4x

Divide 4 on both the sides of an inequality, we get

0.75<x

Here, 6x<5x+8

Subtract 5x on both the sides of an inequality, we get

x<8

So, possible values of x are 0.75<x<8

b) From triangle ABC,

BC<AC<AB

3x+1<4x+8<7x+2

Now, 3x+1<4x+8

Subtract 3x on both the sides of an inequality, we get

1<x+8

Subtract 8 on both the sides of an inequality, we get

-7<x

Here, 4x+8<7x+2

Subtract 4x on both the sides of an inequality, we get

8<3x+2

Subtract 2 on both the sides of an inequality, we get

6<3x

Divide 3 on both the sides of an inequality, we get

2<x

So, possible values of x are -7<x<2

Therefore, the possible values of x for the given triangle ABC are 0.75<x<8.

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Rebecca can save $27 in finance charge when she purchases the $1,400 gift for her friend on the laset day of the billing cycle instead of on the first day

Two or more expressions with an Equal sign is called as Equation.

The billing cycle for the month = 28 days

Current beginning balance = x

Annual percentage rate (APR) = 24%

Monthly percentage rate (MPR) = 2% (24% ÷ 12)

Purchase on the last day = $1,400

Total finance charge for the month = $51

Finance charge for the last-day purchase = $1 ($1,400 x 2% x 1/28)

Finance charge for the beginning balance = $50 ($51 - $1)

The current beginning balance, x = $2,500 ($50 ÷ 2%)

Equation for current beginning balance, x = 51 - 1 ÷ 0.02

Total finance charge for the last-day purchase = $28 ($1,400 x 2%)

Savings in finance charge by purchasing on the last day = $27 ($28 - $1)

Hence, Rebecca can save $27 in finance charge when she purchases the $1,400 gift for her friend on the last day of the billing cycle instead of on the first day.

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The partial fraction expansion of (x³ + 4)/ ((x² - 1)(x² + 3x + 2)) is -3/ 4(x + 1) + 5/12(x - 1) + 4/3(x + 2) - 3/ 2(x + 1)². So the coefficients A = -3/4, B = 5/ 12, C = 4/3 and D = -3/2.

A partial fraction expansion is a way of expressing a rational function (a function that can be written as the ratio of two polynomials) as the sum of simpler fractions, each with a numerator that is a constant or a simple polynomial. The process of finding the partial fraction expansion of a rational function is also known as partial fraction decomposition.

The denominator is

(x² - 1)(x² + 3x + 2) = (x² - 1)(x² + x + 2x + 2)

(x² - 1)(x² + 3x + 2) = (x + 1)(x - 1)(x + 1)(x + 2)

So by  partial fraction expansion

(x³ + 4)/ ((x² - 1)(x² + 3x + 2)) = (x³ + 4)/ ((x + 1)(x - 1)(x + 1)(x + 2))

(x³ + 4)/ ((x + 1)(x - 1)(x + 1)(x + 2)) = A/ (x + 1) + B/(x - 1) + C/(x + 2) + D/ (x + 1)²

A(x - 1)(x + 1)(x + 2) + B(x + 1)²(x + 2) + C(x + 1)²(x - 1) + D(x - 1)(x + 2) = x³ + 4

(A + B + C)x³ + (2A + 4B + C + D)x² + (-A + 5B - C + D)x + (-2A + 2B - C - 2D) = x³ + 4

Thus from coefficients of x³, x², x and 1,

A + B + C = 1

2A + 4B + C + D = 0

-A + 5B - C + D = 0

-2A + 2B - C - 2D = 4

Solving

A = -3/4, B = 5/ 12, C = 4/3 and D = -3/2

--The question is not clear, the question is given below--

"Enter (A,B,C,D) in order below if A, B, C, and D are the coefficients of the partial fractions expansion of

[tex]$$12\cdot\frac{x^3 + 4}{(x^2-1)(x^2 +3x+ 2)}[/tex]  "

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Answer:

-1, 0, 4

Step-by-step explanation:

2f - 8 ≤ 6f + 4

-4f ≤ 12

f ≥ 12/-4  **flip the inequality when you divide by a negative number

f≥ -3

Check all the answers that are ≥ -3

The y-intercept of the sine function is -3

We know that the standard form of the sine function is, y = a sin(bx + c) + d

where, amplitude = |a|

period = 2π/b

phase shift = -c/b

and vertical shift = d

Here, a sine function has an amplitude of 3, a period of π, and a phase shift of π/4

a = 3,

So, 2π/b = π

⇒ b = 2

And phase shift -c/b = π/4

⇒ -c/2 = π/4

⇒ -c = π/2

⇒ c = -π/2

So, a sine function would be,

y = 3 sin(2x + -π/2) + 0

y = 3 sin(2x + -π/2)

When x = 0,

y = 3 sin(0+ -π/2)

y = 3 sin(-π/2)

y = 3 × (-1)

y = -3

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The radius of the sphere is 9 centimeters.

The area of the outer surface of a sphere is the surface area of a sphere.

And its formula:

The surface area of the sphere is 4πr².

Where r is the radius of the sphere.

Given:

The surface area of the sphere is 4πr².

Where r is the radius of the sphere.

So,

surface area of the sphere = 4πr²

324π cm² = 4πr²

Simplifying,

r² = 324π/4π

r² = 324/4

r² = 81

r = 9

Therefore, the radius is 9 centimeters.

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In order to use the Normal approximation for the sampling distribution of p

the sample should have at least 10 successes and at least 10 failures.

The probability distribution of a statistic that is acquired by repeated sampling of a particular population is called the sampling distribution. It outlines a variety of potential possibilities for a statistic, such as the mean or mode of a population's mean or the mode of a variable.

The study of statistics focuses on gathering, organizing, organizing, analyzing, interpreting, and presenting data. It is customary to start with a statistical population or a statistical model to be researched when applying statistics to a scientific, industrial, or social problem.

For a sample of size n = 125,

The condition for the sample size to be large enough is that both np and       n(1-p) should be greater than or equal to 10. This means that the sample should have at least 10 successes and at least 10 failures. If this condition is met, we can use the Normal approximation for the sampling distribution of p.

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Answer:

48cm3

Step-by-step explanation:

Volume of a rectangular box = a.b.c
The volume of the solid figure: (2.2.5) + (7.2.2) = 20 + 28 = 48 cm3

Answer:

see explanation

Step-by-step explanation:

if the denominator of a rational expression is zero then the expression will be undefined.

the numerator is the part of the rational expression that makes it zero.

solve the numerators in each to find values of x

1

[tex]\frac{x+6}{x-4}[/tex]

x + 6 = 0 ( subtract 6 from both sides )

x = - 6 ← value that makes expression equal to zero

2

[tex]\frac{(x+4)(x-2)}{x+6}[/tex]

(x + 4)(x - 2) = 0

equate each factor to zero and solve for x

x + 4 = 0 ⇒ x = - 4

x - 2 = 0 ⇒ x = 2

x = - 4 and x = 2 make the expression equal to zero

3

[tex]\frac{2x+10}{3x-12}[/tex]

2x + 10 = 0 ( subtract 10 from both sides )

2x = - 10 ( divide both sides by 2 )

x = - 5 ← value that makes expression equal to zero

Answer:

1)  x = -6

2)  x = -4  and  x = 2

3)  x = -5

Step-by-step explanation:

A rational expression is undefined when the denominator equals zero.

A rational expression equals zero when the numerator equals zero.

Given rational expression:

[tex]\dfrac{x+6}{x-4}[/tex]

Set the numerator to zero and solve for x:

[tex]\implies x+6=0[/tex]

[tex]\implies x=-6[/tex]

Given rational expression:

[tex]\dfrac{(x+4)(x-2)}{x+6}[/tex]

Set the numerator to zero:

[tex]\implies (x+4)(x-2)=0[/tex]

Apply the zero-product property and solve for x:

[tex]\implies x+4=0 \implies x=-4[/tex]

[tex]\implies x-2=0 \implies x=2[/tex]

Given rational expression:

[tex]\dfrac{2x+10}{3x-12}[/tex]

Factor the numerator and denominator:

[tex]\dfrac{2(x+5)}{3(x-4)}[/tex]

Set the numerator to zero and solve for x:

[tex]\implies 2(x+5)=0[/tex]

[tex]\implies x+5=0[/tex]

[tex]\implies x=-5[/tex]

The value should be added will be [tex]y = -4x^{2} +9x+6[/tex]

Now, According to the question:

Let the value y be added to  [tex]7x^2-5x+1[/tex] to give [tex]3x^2+4x+7[/tex]

y =  [tex]3x^2+4x+7[/tex] - ( [tex]7x^2-5x+1[/tex])

y = [tex]3x^{2} +4x+7-7x^{2} +5x-1[/tex]

Combine like terms:

y = [tex]3x^{2} -7x^{2} +4x+5x+7-1[/tex]

Calculate the subtraction and addition.

[tex]y = -4x^{2} +9x+6[/tex]

Hence, The value should be added will be [tex]y = -4x^{2} +9x+6[/tex].

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The complete question is this:

What should be added to  [tex]7x^2-5x+1[/tex] to get [tex]3x^2+4x+7[/tex] ?

Answer:

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

Find the difference in time and the difference in the number of batches.

The two numbers will help to find the rate of change in that time interval.

This is same as finding the slope of a line with points (2.5, 8) and (4, 11):

The smallest distance it has to go to get back to the starting tree is [tex]3\sqrt{5}[/tex]when traveling 6 miles to the south before turning 3 miles to the east.

According to Pythagoras's Theorem, the square of the hypotenuse side in a right-angled triangle is equal to the sum of the squares of the other two sides. Perpendicular, Base, and Hypotenuse are the names of this triangle's three sides. The Pythagorean Theorem demonstrates how to calculate the side lengths of a right triangle by adding the areas of three intersecting squares. As the foundation for more intricate trigonometry theories like the Pythagorean theorem's opposite, this theorem is a very helpful tool.

given

by pythagoras theorem :-

shortest distance = [tex]c^{2} = a^{2} + b^{2} \\[/tex]

[tex]c^{2} = 6^{2} + 3^{2} \\c^{2} = 36 + 9\\c^{2} = 45\\c = 3\sqrt{5}[/tex]

The smallest distance it has to go to get back to the starting tree is [tex]3\sqrt{5}[/tex]when traveling 6 miles to the south before turning 3 miles to the east.

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The equivalent quadratic equation to  (x - 2)(x + 9) = 0 is x² + 7x - 18 = 0.

A quadratic equation is an algebraic expression in the form of variables and constants.

A quadratic equation has two roots as its degree is two.

We know if α and β are the roots of a quadratic equation ax² + bx + c = 0,

Then we can construct the equation by the factors (x - α)(x - β).

Given, We have to construct a quadratic equation with (x - 2)(x + 9) = 0.

Now, Multiply the terms we have,

x² + 9x - 2x - 18 = 0.

x² + 7x - 18 = 0.

So, Our required quadratic equation is x² + 7x - 18 = 0.

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I think it's answer this .

In order for an expression, such as x² + 4x + , to become a perfect square trinomial, it must be in the form of a² + 2ab + b².

Perfect squares are whole numbers that are the result of multiplying a given number by itself. For example, 4 is a perfect square because it is the result of multiplying 2 by itself. Similarly, 9 is a perfect square because it is the result of multiplying 3 by itself. Other examples of perfect squares include 16 (4x4), 25 (5x5), 36 (6x6), and so on.

To do this, you must factor the expression. In this case, the expression can be factored into (x + 2)².

Once the expression has been factored into a perfect square trinomial, it can be written as (x + 2)².

This is a perfect square trinomial because it represents the area of a square, as the length of each side is equal to (x + 2).

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Step-by-step explanation:

a. 5a²c-3a²d + 5b²c3b²d l can't this question but l can calculate a. 5a²c-3a²d + 5b²c-3b²d.

I this your question is a little worng.

Answer:

Step-by-step explanation:

Take the percentages for both 1992 and 2000 and make the average. that is your answer

Plotting method is the graphical representation of mathematical expression.

A plot is a graphical representation of a list of data. Besides, a plot is the graphical analysis of mathematical expressions and equations because plot can show the relationship between two or more variables. There are many plotting methods such as hand plotting method, computer aided plotting method and so on.

At first, we need to choose the coordinate system by which we plot the point.

suppose we choose a XY coordinate plane. At first identify the horizontal or X axis and vertical or Y axis.

besides, we need to determine the required value that will be plotted on the plan.

if we have a list of X and Y value then find the value of X on the x axis and next find the value of Y on the Y axis.

we will notice some points for every pair of x and y value. Finally, we will join the points to achieve a required shaped graph.

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[tex]\begin{array}{|c|c|} \cline{1-2}\text{Class} & \text{Frequency}\\\cline{1-2}20-29 & 5\\\cline{1-2}30-39 & 12\\\cline{1-2}40-49 & 9\\\cline{1-2}50-59 & 6\\\cline{1-2}60-69 & 3\\\cline{1-2}\end{array}[/tex]

=====================================================

Explanation:

The given data set is

{37, 60, 35, 38, 59, 34, 56, 42, 55, 48, 39, 33, 36, 35, 43, 52, 39, 64, 41, 35, 20, 31, 29, 45, 32, 54, 66, 44, 47, 45, 27, 24, 48, 28, 55}

It sorts to

{20, 24, 27, 28, 29, 31, 32, 33, 34, 35, 35, 35, 36, 37, 38, 39, 39, 41, 42, 43, 44, 45, 45, 47, 48, 48, 52, 54, 55, 55, 56, 59, 60, 64, 66}

The five values 20,24,27,28,29 are between 20 and 29. This means the first class or bin will have a frequency of 5.

The next interval spans from 30 to 39. There are 12 items in this interval, and those 12 items are this subset {31, 32, 33, 34, 35, 35, 35, 36, 37, 38, 39, 39}. This means we have a frequency of 12 for the interval from 30 to 39.

Keep this process going until you get to the end of the data set.

83.2 square feet is the area of the figure.

A three dimensional shape can be defined as a solid figure or an object or shape that has three dimensions—length, width, and height.

The given figure is a cone with radius 3 ft and height 5 ft.

We need to find the area of cone=A=πrl+πr²

where l is √r²+h²

A=πr(r+√r²+h²)

=π×3(3+√3²+5²)

=3π(3+√34)

=3×3.14(3+5.83)

=3×3.14(8.83)

=83.2 square feet

Hence, 83.2 square feet is the area of the figure.

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Therefore, the height of the ball after four seconds is 448 feet.

Height refers to the vertical distance between something's top and bottom or its base and anything above it. Height is used to describe everything that may be measured vertically, high or low.

Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria.

We need to plug t = 4 into the equation h = -16t^2 + 176t.

h = -16(4^2) + 176(4)

h = -16(16) + 704

h = -256 + 704

h =  448 feet.

So, 448 feet is the height after four seconds.

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Answer:

C. 94

Step-by-step explanation:

|A| = [tex]\left[\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right][/tex]

det(A) = a * [tex]\left[\begin{array}{ccc}e&f\\h&i\\\end{array}\right][/tex] - b * [tex]\left[\begin{array}{ccc}d&f\\g&i\\\end{array}\right][/tex] + c * [tex]\left[\begin{array}{ccc}d&e\\g&h\\\end{array}\right][/tex]

OR

det(A) = aei - afh - bdi + bfg + cdh - ceg

|A| = [tex]\left[\begin{array}{ccc}-2&-2&-5\\2&7&-3\\8&9&9\end{array}\right][/tex]

det(A) = -2 * [tex]\left[\begin{array}{ccc}7&-3\\9&9\\\end{array}\right][/tex]  - (-2) * [tex]\left[\begin{array}{ccc}2&-3\\8&9\\\end{array}\right][/tex] + (-5) * [tex]\left[\begin{array}{ccc}2&7\\8&9\\\end{array}\right][/tex]

OR

det(A) = (-2)(7)(9) - (-2)(-3)(9) - (-2)(2)(9) + (-2)(-3)(8) + (-5)(2)(9) - (-5)(7)(8)

det(A) = 94

Therefore , the solution to this problem of equation is y = 16.95  cost for 94 minutes.

A mathematical equation is a formula that links two statements by indicating their equivalence with the equal sign (=). A mathematical statement that proves the equality of two mathematical expressions is known as an equation in algebra. For instance, the components 3x + 5 and 14 in the equation 3x + 5 = 14 are separated by an equal sign. The relationship between two sentences on either side of a letter is expressed mathematically. There is frequently only one variable, which doubles as the symbol. say that 2x - 4 Equals 2.

Here,

y = mx + b

(50,13.98)(78,17.06)

slope = (17.06 - 13.98) / (78 - 50) = 3.08 / 28 = 0.11

slope(m) = 0.11

use either of ur points...(50,13.98)...x = 50 and y = 13.98

now sub into the formula and find b, the y int

13.98 = 0.11(50) + b

13.98 = 5.5 + b

13.98 - 5.5 = b

8.48 = b

so our equation is : y = 0.11x + 8.48.......for 94 minutes...sub in 77 for x

y = 0.11(94) + 8.48

y = 8.47 + 8.48

y = 16.95 <=== this is the cost for 94 minutes

Therefore , the solution to this problem of equation is y = 16.95  cost for 94 minutes.

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The solution of the function f(x) = 2(x - 3)² is at x = 3 as shown in the graph.

An equation is used to show the relationship between numbers and variables.

Polynomial can be classified based on degree as linear, quadratic, cubic.

Given the function:

f(x) = 2(x - 3)²

The graph of f(x) is plotted. The solution of the graph can be gotten from its x intercept. Hence the solution is at x = 3

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''By taking log'' on both side Mike solve this equation.

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Given that;

The expression is,

⇒ 37ˣ = 12

Now,

Since, The expression is,

⇒ 37ˣ = 12

Solve this expression as;

⇒ 37ˣ = 12

Taking log both side, we get;

⇒ log 37ˣ = log 12

⇒ x log 37 = log 12

⇒ x = log 12 / log 37

⇒ x = 1.08 / 1.57

⇒ x = 0.69

Thus, The value of x = 0.69

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Answer:

∠ R = 35° , ∠ T = 45°

Step-by-step explanation:

since the triangles are similar then corresponding angles are congruent, so

∠ R = ∠ U = 35°

∠ T = ∠ Q = 45°

Perimeter is the total length of all sides of a shape, also known as the circumference.

Perimeter is a measure of the distance around a two-dimensional shape. It is the sum of the lengths of all the sides of a shape. Perimeter is also known as the circumference. When calculating perimeter, you measure the length of each side of the shape and add them together. For example, the perimeter of a square with sides of 4 inches each would be 16 inches. Perimeter is an important concept in geometry, and it can be used to calculate area, the space within a shape or figure. It can also be used to calculate the distance between two points. Knowing the perimeter of a shape can help you solve many problems in math, design, and science.

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