Need help with the last two questions. I can't solve it with using Scientific Notation or Standard Notation. Also I need to be sure it's three significant digits that are included.

We're asked to find inverse of the function,

[tex]\longrightarrow\rm{f(x)=\dfrac{e^x-e^{-x}}{e^x+e^{-x}}+2}[/tex]

Take [tex]\rm{y=f(x).}[/tex]

[tex]\longrightarrow\rm{y=\dfrac{e^x-e^{-x}}{e^x+e^{-x}}+2}[/tex]

Subtract 2.

[tex]\longrightarrow\rm{y-2=\dfrac{e^x-e^{-x}}{e^x+e^{-x}}}[/tex]

We apply componendo - dividendo rule.

[tex]\rm{\dfrac{a}{b}=\dfrac{c}{d}\quad\iff\quad\dfrac{b+a}{b-a}=\dfrac{d+c}{d-c}}[/tex]

Thus,

[tex]\longrightarrow\rm{\dfrac{y-2}{1}=\dfrac{e^x-e^{-x}}{e^x+e^{-x}}}[/tex]

[tex]\Longrightarrow\rm{\dfrac{1+(y-2)}{1-(y-2)}=\dfrac{(e^x+e^{-x})+(e^x-e^{-x})}{(e^x+e^{-x})-(e^x-e^{-x})}}[/tex]

[tex]\longrightarrow\rm{\dfrac{y-1}{3-y}=\dfrac{2e^x}{2e^{-x}}}[/tex]

[tex]\longrightarrow\rm{\dfrac{e^x}{e^{-x}}=\dfrac{y-1}{3-y}}[/tex]

[tex]\longrightarrow\rm{e^{2x}=\dfrac{y-1}{3-y}}[/tex]

[tex]\longrightarrow\rm{(e^x)^2=\dfrac{y-1}{3-y}}[/tex]

Take square root.

[tex]\longrightarrow\rm{e^x=\sqrt{\dfrac{y-1}{3-y}}}[/tex]

Take natural logarithm.

[tex]\longrightarrow\rm{x=\log_e\sqrt{\dfrac{y-1}{3-y}}}[/tex]

[tex]\longrightarrow\rm{x=\dfrac{1}{2}\log_e\left(\dfrac{y-1}{3-y}\right)}[/tex]

Now take [tex]\rm{x=f^{-1}(y).}[/tex]

[tex]\longrightarrow\rm{f^{-1}(y)=\dfrac{1}{2}\log_e\left(\dfrac{y-1}{3-y}\right)}[/tex]

Replace y by x and finally we get,

[tex]\longrightarrow\rm{\underline{\underline{f^{-1}(x)=\dfrac{1}{2}\log_e\left(\dfrac{x-1}{3-x}\right)}}}[/tex]

This is the inverse of the given function.

To express a mathematical operation symbolically, we employ a mathematical expression.

In mathematics, an expression is a sentence that has at least two terms and an operator in the middle. Expressions may be divided into two categories: numerical expressions, which include simply numbers, and algebraic expressions, which contain both numbers and variables. A mathematical expression consists of two parts: terms and operator, where terms refer to variables and integers. It is referred to as a constant term if an expression just contains numbers, and it is known as a coefficient if it also contains numbers and a variable.

We utilize expressions to symbolically represent any mathematical activity as well as mathematical models. The specifics of our computation, for instance, can be succinctly represented in terms of two or more mathematical terms. We can identify unknown numbers with the equal sign, allowing us to resolve difficult issues. In real life, we utilize expressions to figure out how much we make and spend, how quickly things change, how long it will take to go a certain distance, and other things. When calculating the rate of change, we use the differentiated term dy/dx. When calculating speed, time, and distance, as well as time and speed, we employ equations with variables.

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Answer:

To try and relate or understand our feelings and try and give an example. For example the expression “it’s raining cat’s and dog’s” well that means it’s raining heavily!

Hope I helped

Step-by-step explanation:

The value of a continuous function is that it produces a single output for any given input, no matter how small the change in the input. This makes it reliable and predictable, allowing for precise calculations and predictions.

Continuous functions are a type of mathematical function that produces a single output value for any given input. This means that no matter how small the change in the input is, the output value remains the same. This makes continuous functions reliable and predictable, allowing for precise calculations and predictions. As an example, a continuous function can be used to calculate how far an object will travel over a certain amount of time if provided with the initial velocity and acceleration. This is because the function will not change its output value when the input is altered, no matter how small the change is. Continuous functions are therefore an essential tool for many applications in mathematics and science.

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The value of x when the quadratic expression  t² - 8t + 16 is factorised to (t + a)² is -4.

Quadratic expression is an expression with the variable with the highest power of 2. The quadratic equation in its standard form is ax² + bx + c = 0, where a and b are the coefficients, x is the variable, and c is the constant term.

Therefore, let's factorised the expression t² - 8t + 16 in the form (t + a)². Where a is an integer.

Hence,

t² - 8t + 16

t² - 4t - 4t + 16

t(t - 4) -4(t - 4)

(t - 4)(t - 4)

Hence,

(t + a)² = (t - 4)²

Therefore,

a = -4

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6) The time required by James if he completes the work alone is 18 hours.

Time and work are concerned with the amount of time it takes a person or a group of people to complete a task and the effectiveness of the work performed by each of them. The fundamental idea behind Time and Work is comparable to the idea behind Proportionality, which is present in all arithmetic disciplines.

"How much work one person can complete in one day is what the term "work efficiency" refers to. For instance, someone can complete a task in two days. In other words, he may complete 50% of the work in a single day. His effectiveness will therefore be 50%.

Given that,

Monica is fencing a yard

Monica took 9 hours to fence the yard.

If she takes James help to fence the yard then she estimates that she can do it in 6 hours.

Now, we have to find how much time James requires if he work alone=?

Then,

(9x)/(9+x)=6

9x=54+6x

3x=54

x=18

So, the time required by James if he completes the work alone is 18 hours.

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the placebo effect means in this specific situation that The benefit is due to the fact that the co-workers are given a "treatment" (coffee, although decaffeinated) and not an effect of the "active ingredient" (caffeine).

Therefore, it appears that Janae is serving decapitated coffee to its customers, who have said that it has given them more energy. We want to examine the placebo effect, and when we think of the placebo effect, we imagine that a guy should publish. The best response you have provided is to recognize that they are receiving treatment, even though it is not caffeine, and causing them to believe that they are experiencing an increase in energy and that it is not the result of an active ingredient. As a result, they are receiving treatment and are having an impact by which. They aren't actually being impacted by the caffeine, though. You will typically feel more energized as a result of that.

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The inverse of 7 modulo 11 is 4, which means that 4 multiplied by 7 gives a number congruent to 1 (mod 11). This means that 4 is the number that when multiplied by 7 gives a remainder of 1 when divided by 11.

In modular arithmetic, the inverse of a number is the number that when multiplied by the original number gives a result that is congruent to 1 (modulus). For example, if we take the modulus of 11 as an example, then the inverse of 7 modulo 11 is 4, since 7*4 = 28 ≡ 1 (mod 11). This means that 4 is the number that when multiplied by 7 gives a remainder of 1 when divided by 11. The inverse of a number modulo a given modulus can be calculated using the Extended Euclidean algorithm. This algorithm allows us to find the inverse of a number modulo a given modulus by solving a series of equations and using the result to calculate the inverse. For example, in the case of 7 mod 11, the Extended Euclidean algorithm gives us the result 4, which is the inverse of 7 mod 11.

11 = 7*1 + 4

7 = 4*1 + 3

4 = 3*1 + 1

3 = 1*3 + 0

By reading the equations from right to left, we can see that the inverse of 7 mod 11 is 4, since 4*7 = 28 ≡ 1 (mod 11).

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Answer:

In the set {1, 1, 2, 2, 3, 3, 4, 4}, the value that occurs most frequently is 1, 2, 3, and 4, so the mode of this set is 1, 2, 3, and 4...

The mode of a set of numbers is the value that occurs most frequently in the set. In the set {1, 1, 2, 2, 3, 3, 4, 4}, the value that occurs most frequently is 1, 2, 3, and 4, so the mode of this set is 1, 2, 3, and 4.

It's important to note that a set of numbers can have more than one mode if multiple values occur with the highest frequency. A set of numbers that has multiple modes is called a multimodal set. In the example set {1, 1, 2, 2, 3, 3, 4, 4}, all four values occur with the same frequency, so the set is multimodal.

On the other hand, if no value occurs more frequently than any other value, the set is said to have no mode. For example, the set {1, 2, 3, 4, 5} has no mode because all values occur with the same frequency.

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The resultant answer after solving the given inequality by taking the subject x is x < (12 + 2y)/3.

An inequality in mathematics is a relation that compares two numbers or other mathematical expressions in an unequal way.

The majority of the time, size comparisons between two digits on the number line are made.

So, we have the inequality:

3x - 2y < 12

Now, solve the inequality by taking x as the subject as follows:

3x - 2y < 12

3x < 12 + 2y

3x/3 < 12/3 + 2y/3

x < (12 + 2y)/3

Therefore, the resultant answer after solving the given inequality by taking the subject x is x < (12 + 2y)/3.

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Correct question:

What is the solution to the inequality 3x - 2y < 12?

The maximum value of f(x,y) on the unit of circle is [tex]\frac{48 \sqrt{21}}{2,401}[/tex].

The full question is in the attachment. From the unit circle

Substitute to f(x,y)

f(x,y) = x³y⁴ = x³ (y²)² = x³ (1 - x²)²

f(x,y) = x³ (1 - 2x² + x⁴)

f(x,y) = x³ - 2x⁵ + x⁷

df/dx = 3x² - (2 × 5)x⁴ + 7x⁶

df/dx = 3x² - 10x⁴ + 7x⁶

df/dx = x² (3 - 10x² + 7x⁴)

To find the maximum or minimum value, the derivated equals zero.

0 = x² (7x⁴ - 10x² + 3)

0 = x² (x - 1) (x + 1) (7x² - 3)

7x² - 3 = 0

Using ABC formula

[tex]x_{1,2} \:=\: \frac{-b \pm \sqrt{b^2 \:-\: 4ac}}{2a}[/tex]

[tex]x_{1,2} \:=\: \frac{-0 \pm \sqrt{0^2 \:-\: (4 \times 7 \times - 3)}}{2 \times 7}[/tex]

[tex]x_{1,2} \:=\: \frac{\pm \sqrt{4 \times 21)}}{14}[/tex]

[tex]x_{1,2} \:=\: \frac{\pm 2 \sqrt{21}}{14}[/tex]

[tex]x_{1,2} \:=\: \frac{\pm \sqrt{21}}{7}[/tex]

The maximum value from x₁.

The minimum value from x₂.

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Similar to one-step and two-step equations, the basic objective of multi-step problems is to isolate the unknown variable on one side of the equation while maintaining the constant or number on the other side.

Solving a 3-step problem typically involves the following steps:

It can be helpful to break the problem down into smaller parts and solve each part separately. This can make the problem feel more manageable and help you avoid making mistakes.

If you get stuck at any point, it can be helpful to take a break and come back to the problem later, or to ask someone for help.

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In the XY plane, the circles with equations (x-3) + y = 11 and (x+2) + y = r are externally tangent. the value of r is -6.

Generally, To find the value of r, we can start by solving for the centers and radii of the two circles.

The equation of the first circle is of the form (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius. We can rewrite the equation of the first circle as (x - 3)² + y² = 11², so the center of the circle is (3, 0) and the radius is 11.

The equation of the second circle is also of the form (x - h)₂ + (y - k)₂ = r². We can rewrite the equation of the second circle as (x + 2)² + y² = r², so the center of the circle is (-2, 0) and the radius is r.

Since the two circles are externally tangent, the distance between their centers is equal to the sum of their radii. Therefore, the distance between the centers is equal to 11 + r. We can use the distance formula to find the distance between the centers:

distance = √((h₂ - h₁)^2 + (k₂ - k₁)²)

Plugging in the coordinates of the centers, we get:

distance = √((-2 - 3)^2 + (0 - 0)^2) = √(5^2 + 0^2) = √(25) = 5

Since the distance between the centers is equal to the sum of the radii, we can set up the equation:

5 = 11 + r

Solving for r, we get:

r = 5 - 11 = -6

Therefore, the value of r is -6.

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Answer:

D

Step-by-step explanation:

( g-h)(n).

n+5 - (n^2-5n)

-n^2+5n+n+5

-n^2+6n+5

Answer:

y = 1x+-5

Step-by-step explanation:

The slope is going up hill so there for the slope is positive. The y-intercept is -5 bc the line crosses at -5 on the y-axis.

Answer:

6 1/2 hours     12-hour clock

18 1/2 hours   24- hour clock

Step-by-step explanation:

I attached a picture of how I solved it!

The clock just crossed the 9th hour.

Given is that a clock that strikes only the hours has just finished striking a total of 9 times in the last 24 hours.

Since, the clock strikes only on hours, the clock just crossed the 9th hour.

Therefore, the clock just crossed the 9th hour.

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The length of r, to the nearest 10th of an inch, is 2.4 inches.

The relationship between the sides and angles of non-right (oblique) triangles is defined by the Law of Sines. Simply put, it states that the ratio of the length of a triangle's side to the sine of the angle opposite that side is the same for all triangle sides and angles.

Given that in ΔQRS, s = 5.2 inches, ∠S=129° and ∠Q=30°. The value of side r will be calculated by using the sin angle theorem.

The value of r is,

r / sin21 = s / sin129

r = 5.2 x ( sin21/sin129)

r = 5.2 x 0.46

r = 2.4 inch

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The question uses 6 mathematical phrases. The coefficient is made up of two integers, and the other variables are x, y, z, and A.

A mathematical expression is a sentence that describes the relationship between two or more concepts and one or more symbols. There are many distinct words, including numbers, variables, and variables with coefficients. Different mathematical operators can be used to denote an equation, including the signs for addition and subtraction, plus and minus, multiplication and division, equal and inequal signs, and so on.

Using the formula above, 5xyA2 = 3xyz

There are six words and one equal sign, as we can see.

A, B, C, and There are six words and one equal sign, as we can see.

Five, three, and A are numerals and four variables, respectively. The coefficient of the variables is represented by the numbers.

A, y, and x are all used twice, as well. A2 or the square of A is how you write the product of A and A.

When we multiply 5 by x, y, and A2, we get the formula 3xyz, where 3 is multiplied by x, y, and z.

hence, we determine 6 terms in the question

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The quadratic equation in standard form given the sum and product of roots is

x² - (α + β)x + αβ = 0.

Polynomial is an equation written as the sum of terms of the form kx^n.

where k and n are positive integers.

We have,

A polynomial with degree 2 is called a quadratic equation.

Now,

The standard form of a quadratic equation is ax² + bx + c.

Now,

Let the roots of the quadratic equation be α and β.

α + β = -b/a

αβ = c/a

Thus,

The quadratic equation is x² - (α + β)x + αβ = 0.

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a) The slope of the main portion of the slide is of: 5/6 = 0.8333.

b) If the bottom of the slide was only 12 inches above the ground, then the slope would be of 1/3 = 0.3333.

The slope of an inclined object is obtained as the division of the height of the object by the length of the object.

From the text, the parameters for the slide are given as follows:

Then the slope of the slide is of:

Slope = 10/12 = 5/6 = 0.8333.

After the change in the bottom, the parameters would be of:

Then the new slope of the slide is given as follows:

Slope = 4/12 = 1/3 = 0.3333.

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The exponential function that goes through the points (0, 14) and (3, 896) is y = 14·3ˣ. This function has a base of 3 and a coefficient of 14.

An exponential function in the form y =abˣ is a function

where y is equal to a multiplied by b raised to the power of x.

To find the function that goes through the points (0, 14) and (3, 896),

We can write the following equation to find the base of the exponential function that goes through the points (0, 14) and (3, 896):

b³ = 896/14

We can then solve for b by taking the cube root of both sides of the equation to get:

b = 3

Now that we know the base of the function, we can use the point (0, 14) to find the value of the coefficient a.

Since the x-value of this point is 0, we can plug 0 in for x in the exponential function y=abˣ to get:

y = ab⁰ = a

Since the y-value of the point (0, 14) is 14, we can set this equal to a to solve for the value of a:

a = 14

So, the exponential function is y = 14·3ˣ

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The process to calculate slope of  line is explained below.

The Slope of line passing through two points (a,b) and (c,d) can be calculated by using  formula ;

slope = (d - b)/(c - a) ;

The slope of a line is denoted by letter "m" .

For Example : If the line is passing through the points (0,2) and (3,8)  .

on comparing we get the values as a = 0 , b = 2 , c = 3 and d = 8 .

On substituting values in the slope formula that is mentioned above  ,

we get ;

Slope(m) = (8 - 2)/(3 - 0)

On Simplifying further ,

we get ;

= 6/3

= 2 .

So , the slope of line will be = 2 .

The given question is incomplete , the complete question is

How do you calculate slope of a line quickly ?

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On solving the provided question, we can say that - There exist 333 integers smaller than 1000 that are evenly divisible by 3 if you divide 1000 by 3, which results in 333 with a residual of 1.

The divisibility rule is a concise and practical approach to check, often by looking at the integer's digits, whether a given integer is divisible by a given set divisor without actually executing division. Without actually doing the division procedure, you may quickly discover if a given number can be divided by a defined divisor using the divisibility test. When dividing two numbers exactly, the quotient must be an integer and the remainder must be zero.

here,

1000/3

remainder = 1

divisor = 333

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The number of terms that the expression 5xy 9yz 3zx 5x 4y has is 5 terms.

What is an algebraic expression?

An algebraic expression is a set of numbers and letters that make up an expression that has a meaning, the letters are variables and the numbers are coefficients or independent terms, algebraic expressions can be part of an equation and model mathematical processes.

How to determine terms?

To determine the number of terms in an algebraic expression, we only have to count the spaces (addition and/or subtraction symbols) and add the unit, in this way we will have the number of terms.

5xy 9yz 3zx 5x 4y has 4 spaces

4 + 1 = 5 terms.

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The function H(x) = 3x⁴ + 2x — 5/x + 9x³ - 7 is not a polynomial function

A polynomial function is a function in an equation, such as the quadratic equation, cubic equation, etc., that only uses non-negative integer powers or only positive integer exponents of a variable.

A polynomial with an exponent of 1 is, for instance, 2x+5.

The given polynomial in standard form is written as

H(x) = 3x⁴ + 2x — 5/x + 9x³ - 7

H(x) = 3x⁴ + 9x³ + 2x  - 7 - 5x⁻¹

this shows that on of the term has a negative exponent 5x⁻¹ and hence not a polynomial

the leading coefficient is 3

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The nature of roots of the quadratic equation ax²+bx+c= 0 can be describe in three ways depending on the value of b²-4ac.

For quadratic equation ax²+bx+c=0 if

Case 1) If b²-4ac>0  then the roots of equation will be real and distinct.

Case 2) If b²-4ac=0  then roots will be real and equal.

Case 3) If b²-4ac<0 then roots will be imaginary and complex number pairs.

also a) if b2-4ac >0 and perfect square then the roots are real,distinct, and rational

b) if b2-4ac>0 and not a perfect square then the roots are real, distinct but irrational.

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Answer:

315

Step-by-step explanation:

multiply (3.78x10^6) first to get 3780000 then divide 3,780,000 by 12,000 to get 315

The girl's basketball team won 75 games and the boy's basketball team won 151 games.

Let's assume the number of games the girl's basketball team won "g," and the number of games the boy's basketball team won "b."

We know that the total number of games won by both teams is 226, so we can set up our first equation as follows:

g + b = 226

We also know that the boy's basketball team won one more than twice as many games as the girl's basketball team. We can represent this as follows:

b = 2g + 1

Now we have two equations and two unknown values, so we can solve for g and b using algebra. Let's substitute the second equation into the first equation to eliminate one of the unknowns:

g + (2g + 1) = 226

This simplifies to:

3g + 1 = 226

Subtracting 1 from both sides gives us:

3g = 225

Dividing both sides by 3 gives us:

g = 75

Now that we know the value of g, we can substitute it back into the second equation to solve for b:

b = 2 x 75 + 1 = 151

Therefore, the girl's basketball team won 75 games and the boy's basketball team won 151 games.

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DMAS is four type of arithmetic we deal everyday in our mathematical problem

What are four basic property of arithmetic?

Commutative, associative, distributive and identity are the four basic properties of arithmetic.

Addition and multiplication ate two arithmetic which is commutative.

Division, addition, subtraction and multiplication are the four type of arithmetic

Examples:

1+1=2 is addition

4-2=2 is subtraction

3*2=6 is multiplication

4/2=2 is division

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To find the missing term, we can use the expansion of the square of a binomial, which is given by the formula (a + b)^2 = a^2 + 2ab + b^2. In this case, the binomial is x + 12, so we can use the formula like this:

[tex](x + 12)^2 = x^2 + 2 * x * 12 + 12^2[/tex]

Therefore, the missing term is 144.

You can also find the missing term by expanding the left-hand side of the equation using the same formula:

[tex](x + 12)^2 = (x^2 + 24x) + __[/tex]

When we expand the left-hand side using the formula, we get:

[tex](x + 12)^2 = x^2 + 24x + 144[/tex]

So the missing term is also 144.