inverse function of f(x) = (x^2+1)^2 step by step

The value of the variable x when y = 6 is 2

It is important to note that a proportional relationship between variables or events means that as one increases in its value, the other also increases in the same way.

From the information given, we have that;

The variable y and x is proportional

This is represented as;

y ∝ x

Find the constant value

k = y/x

Substitute the values

k = 12/4

k = 3

Then when y = 6, we get;

x = y/k

x = 6/3

x = 2

Hence, the value is 2

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so a and b are positive integer

so when we put the all options in place of a one by one in equation we will get

a+26=b²

10+26= b²

b=√36

b= 6

so a must be 10.

32 thousands 6 hundreds 8 ones can be rewritten as 32608

According to the BODMAS rule, the brackets have to be solved first followed by powers or roots (i.e. of), then Division, Multiplication, Addition, and at the end Subtraction. Solving any expression is considered correct only if the BODMAS rule or the PEMDAS rule is followed to solve it.

Given here: 32 thousands 6 hundreds 8 ones

Thus we can rewrite as 32×1000+6×100+8×1=32608

Numbers in words are written using the English alphabet. Numbers can be expressed both in words and figures. For example, 100,000 in words is written as One Lakh or One hundred thousand. Numbers in words can be written for all the natural numbers, based on the place value of digits, such as ones, tens, hundreds, thousands, and so on.

Hence, 32 thousands 6 hundreds 8 ones can be rewritten as 32608

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The cosine of the angle  θ  is √5/5  whose terminal side passes through the given point

In mathematics, trigonometric functions are real functions that relate an angle of a right-angled triangle to ratios of two side lengths. The six trigonometric functions are Sine, Cosine, Tangent, Secant, Cosecant, and Cotangent.

Given here: The terminal side goes through the point  (√5/5,2√5/5)

Thus we have cosθ =√5/5 and sin θ =-2√5/5

now tan θ =sinθ /cosθ

        tanθ    =-2√5/5 / √5/5

              θ = 63.434

Hence, cosθ =√5/5

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The graph that best represents the situation, given the person cycling from home to the library, is Graph C.

The graph that would best represent the situation would show that the person cycling, stopped moving after 8 kilometers because they were at the library.

The graph should also show that after 30 minutes, the person left the library which is shown in Graph C as:

= 50 - 20 minutes

= 30 minutes

The best graph is therefore Graph C.

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The given function

g(x)  =  (x√x)/(x - 6)²

is continuous at  x = 36,

In mathematics, The function is called as continuous at the particular value of x,  if right hand limit, left hand limit and the value of function at x all are equal.

RHL   = LHL  = F(x)

Limit exist and continuous

Given that,

F(x)  =   (x√x)/(x - 6)²

To check whether it is continuous at x = 36

First taking LHL,

[tex]\lim_{h \to \00}[/tex]    [(36-h)√(36-h)]/((36-h) - 6)²

[tex]\lim_{h \to \00}[/tex]  [(36-0)√(36-0)]/((36-0) - 6)²

⇒   (36 x 6)/30²

⇒    6/25

Now, taking RHL

[tex]\lim_{h \to \00}[/tex]    [(36+h)√(36+h)]/((36+h) - 6)²

[tex]\lim_{h \to \00}[/tex]  [(36+0)√(36+0)]/((36+0) - 6)²

⇒   (36 x 6)/30²

⇒    6/25

The value at x = 36

F(-1)  =  [(36)√(36)]/((36) - 6)²

⇒   (36 x 6)/30²

⇒    6/25

RHL = LHL = F(36)

Hence Proved,  the function is continuous at x = 36

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The area of the lawn that will be watered by the sprinkler is 191.1 square feet, rounded to the nearest tenth.

Area is a measurement of the size of a two-dimensional surface, such as a square, rectangle, or circle. It is calculated by multiplying the length of the surface by its width. Area can also be measured in terms of square units, such as square feet or square meters. Area is an important concept in mathematics and is used to measure the size of shapes and objects. It is also used to find the total area of a group of shapes.

To calculate the area, use the formula A = r2θ, where A is the area, r is the radius, and θ is the central angle in radians. In this scenario, the radius is 10 ft and the central angle is 30°, or π/6 radians. Plugging these numbers into the formula, we get A = (10 ft)2(π/6 radians), or 191.1 square feet.

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The slope-intercept form of the function described by this table?

x 0 1 2 3

y 3 8 13 18. is y = 5x + 3

The slope-intercept form of a linear equation is given by y = mx + b, where m is the slope of the line and b is the y-intercept. To find the equation that fits the table, we can use the two points (0, 3) and (1, 8) to calculate the slope and y-intercept.

First, calculate the slope:

m = (8 - 3) / (1 - 0) = 5

Next, use the point-slope form of a line to find the y-intercept:

y - 3 = 5(x - 0)

Expanding the right-hand side:

y - 3 = 5x

Adding 3 to both sides:

y = 5x + 3

So, the slope-intercept form of the function described by the table is:

y = 5x + 3

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Answer:

y=1/6x+3

Step-by-step explanation:

THe equation of the graph is y=mx+c

so,

m= { (-3)-(-4) } / (-1)-(-7) = 1/6

intercept = (0,y)

1/6= (y+3)/1

y=3

so, the equation ⇒y=1/6x+3.

The value of the number that occupies the hundreds place in 25,450 is 400

Calculating a set's size, or the number of elements in a finite collection of objects is the process of counting.

In a counting sequence, there are different units used such as:

Hence, in the value of 25,450,

The value that is found in the hundreds place is 400

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The English translation is:

What is the value of the number that occupies the hundreds place in 25,450

Answer:

(2, -5)

Step-by-step explanation:

-7x=y=-19

y=7x-19

-2x+3(7x-19)=-19

-2x+21x-57=-19

19x-57=-19

19x=38

x=2

-2(2)+3y=-19

-4+3y=-19

3y=-15

y=-5

For x=2, the functions g(x)=[tex]f^{-1}(x)[/tex] are true.

Functions are the fundamental part of the calculus in mathematics. The functions are the special types of relations. A function in math is visualized as a rule, which gives a unique output for every input x.

The given function is f(x)=(x-1)/4 and g(x)=2x+5.

Here, y=(x-1)/4

Interchange x with y and y with x, we get

x=(y-1)/4

4x=y-1

y=4x+1

[tex]f^{-1}(x)[/tex]=4x+1

Now, g(x)=[tex]f^{-1}(x)[/tex]

2x+5=4x+1

2x=4

x=2

Therefore, for x=2, the functions g(x)=[tex]f^{-1}(x)[/tex] are true.

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The linear function that predicts the number of cars that will be sold at a price of x dollars is y = -0.05x + 11580.

Each term in a linear equation has an exponent of 1, and when this algebraic equation is graphed, it always produces a straight line. It is called a "linear equation" for this reason.

To determine the linear function that predicts the number of cars sold we need to find the slope of the equation.

The two coordinated given are:

(x1, y1) = (52000, 8980)

(x2, y2) = (57000, 8730)

The slope of the equation is given by:

m = (y2 - y1) / (x2 - x1)

m = (8730 - 8980) / (57000 - 52000)

m = - 0.05

Substituting the value of the coordinates and slope in the point-slope form we have:

y - y1 = m (x - x1)

y - 8980 = -0.05(x - 52000)

y - 8980 = -0.05x + 2600

y = -0.05x + 11580

Hence, the linear function that predicts the number of cars that will be sold at a price of x dollars is y = -0.05x + 11580.

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The amount to be deposited for 10 years is $2000.

Given that,

Simple interest  is 5%

The amount to be in the account is $3000

The time period is 10 years.

The formula associated with these values is

A = P(1+rt)

where A = amount after 10 years

P = amount to be deposited in the account

r = rate of interest

t = time period

By applying the formula,

3000 = P (1+ (5% × 10))

3000 = P (1+ (0.05 × 10)

3000 = P (1 + 0.5)

3000 = P (1.5)

P = 3000/1.5

P = 2000 $

Therefore, $2000 should be deposited in an account in order to have $3000 in the account after 10 years.

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1:4

it would be 1 to the ratio of 4 because its 4 numbers listed only once which means youll have a chance of one out of 4 to get one number four different times

The distance betwen the spots are

From the question, we have the following parameters that can be used in our computation:

The coordinates of the spots

The distance between them is then calculated as

Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

Using the above as a guide, we have the following:

AD = √[(-4 + 4)² + (3 - 1)²] = 2

DF = √[(-4 + 4)² + (1 + 3)²] = 4

EG = √[(5 - 5)² + (1 + 3)²] = 4

AC = √[(-4 - 1)² + (3 - 3)²] = 5

BD = √[(2 + 4)² + (1 - 1)²] = 6

The above values represent the distance betwen the spots

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The percent of the total calories in each brownie comes from fat is 34%

Percentage is described as a number or ratio expressed as a fraction of 100.

It is often represented with the percent sign, "%"

Abbreviations like "pct" and "pc" are used to denoted it.

Percentage is a dimensionless number, that is, it has no unit of measurement.

From the information given, we have that;

207 calories of fat/606 calories in a brownie × 100/1

Divide the values, we get;

0. 34 × 100/1

Multiply the values

34%

Hence, the value is 34%

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R and S are points on the number line. If RS= 11 and S lies at 5, where could R be located 16 or -6.

A number line is a visual representation of numbers arranged in a continuous sequence along a straight line. It is used to represent the real numbers and to help visualize relative sizes and order of numbers. The numbers on a number line are placed at equal intervals, and the distance between two numbers represents the magnitude of the difference between them.

A typical number line starts from a specific number on the left, such as zero, and extends to the right, with positive numbers increasing as you move to the right and negative numbers decreasing as you move to the left. Points on the number line correspond to specific numbers, and the position of a point on the line represents the value of that number.

The number line is a useful tool in mathematics for solving problems, such as determining the solution to an equation, estimating the value of a decimal or fraction, or visualizing the magnitude of a negative or positive number. It is also used in geometry to represent the positions of points and to understand concepts such as distance and midpoint.

If S lies at 5 on the number line and RS = 11, then R could be located at either 5 + 11 = 16 or 5 - 11 = -6. So, R could be at either 16 or -6 on the number line.

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Answer:

3.2 miles

Step-by-step explanation:

To approximate the height of the mountain, we need to use the tangent function in trigonometry. We have two angles of elevation and the distance between the observer and the mountain, which we can use to create a right triangle.

Let h be the height of the mountain and d be the distance from the observer to the foot of the mountain.

At the first observation, we have:

tan(3.5°) = h/d

And at the second observation, when the observer is 11 miles closer to the mountain:

tan(9°) = h/(d-11)

Dividing the two equations, we have:

tan(9°) / tan(3.5°) = h/(d-11) / h/d

Solving for h, we have:

h = (d * tan(9°)) / (tan(3.5°) + 1)

We don't have an exact value for d, so we can use an estimate based on the difference in distance between the two observations.

Since d-11 = 11 miles, we have:

d = 11 * 2 = 22 miles

Using this estimate and the given angles, we can approximate the height of the mountain:

h = (22 * tan(9°)) / (tan(3.5°) + 1) = (22 * 0.156434465) / (0.066012183 + 1)

= 22 * 0.156434465 / 1.066012183 = 22 * 0.146838446 = 3.23 miles

The height of the mountain is approximately 3.2 miles to one decimal place.

There are 2 observations remaining in the data

The complete question is added as an attachment

From the question, we have the following parameters that can be used in our computation:

The table of values

From the table of values. we have

Number of observations = 5

Observations with missing values = 3

So, we have

Remaining =5 - 3

Evaluate

Remaining = 2

Hence, the remaining observations is 2

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Answer:

162/5 hours.

Step-by-step explanation:

Let's call the normal depth of the river "D". Since 2/3 of the way to go back to normal is still flooded, the current depth of the water is D + 2/3D = 5/3D.

The flood waters recede at a rate of 1/9 the flood level per hour, so the change in water level each hour is -1/9 * 5/3D = -5/27D.

To find the number of hours it takes to go from the current depth (5/3D) to the normal depth (D), we need to divide the change in depth (-5/27D) by the change in depth per hour (-5/27D): (5/3D - D) / (-5/27D) = (5/3D - D) / (-5/27D) = -(5/3D - D) / (5/27D) = (2/3D) / (5/27D).

This can be simplified to 2/3 * 27/5 = 162/5 hours. The answer is 162/5 hours.

Answer: Since the flood waters still have 2/3 of the way to go back down to reach the normal level, it means that 1/3 of the flood level has already receded.

If the flood level recedes at a rate of 1/9 the flood level per hour, then it will take 1/9 * 1 hour = 1 hour to recede 1/9 of the flood level.

Therefore, to recede 2/3 of the flood level, it will take 2/3 * 1/9 hours = 2/27 hours.

Since there are 60 minutes in 1 hour, the time it takes for the flood waters to reach the normal level in minutes is: 2/27 hours * 60 minutes/hour = 20 minutes.

Therefore, it will take 20 more minutes for the water to reach its normal depth.

Step-by-step explanation:

Answer: 16.65

Step-by-step explanation:

Answer:

[tex](3+\sqrt{7},7+2\sqrt{7})[/tex]

[tex](3-\sqrt{7},7-2\sqrt{7})[/tex]

Step-by-step explanation:

Given equations:

[tex]\begin{cases}y=2x+1\\y=x^2-4x+3\end{cases}[/tex]

Substitute the second equation into the first equation:

[tex]\implies x^2-4x+3=2x+1[/tex]

Rearrange so that the terms in x are on the left side of the equation and the constants are on the right:

[tex]\implies x^2-4x-2x=1-3[/tex]

[tex]\implies x^2-6x=-2[/tex]

Add the square of half the coefficient of the term in x to both sides:

[tex]\implies x^2-6x+\left(\dfrac{-6}{2}\right)^2=-2+\left(\dfrac{-6}{2}\right)^2[/tex]

[tex]\implies x^2-6x+9=-2+9[/tex]

[tex]\implies x^2-6x+9=7[/tex]

Factor the perfect square trinomial on the left side of the equation:

[tex]\implies x^2-3x-3x+9=7[/tex]

[tex]\implies x(x-3)-3(x-3)=7[/tex]

[tex]\implies (x-3)(x-3)=7[/tex]

[tex]\implies (x-3)^2=7[/tex]

Square root both sides of the equation:

[tex]\implies x-3=\pm \sqrt{7}[/tex]

Add 3 to both sides of the equation:

[tex]\implies x=3\pm\sqrt{7}[/tex]

To find the y-values, substitute the found x-values into the first equation:

[tex]x=3+\sqrt{7}\implies y=2(3+\sqrt{7})+1=7+2\sqrt{7}[/tex]

[tex]x=3-\sqrt{7}\implies y=2(3-\sqrt{7})+1=7-2\sqrt{7}[/tex]

Therefore, the solutions to the system of equations are:

-14 is less than 16. All negative numbers are less than positives!

x = 2

y = 2 [tex]\sqrt{2}[/tex]

Special right triangles are triangles with specific angle measurements whose side lengths have patterns.

45-45-90 Triangle

The triangle above is a 45-45-90 triangle since it has two 45-degree angles. For this explanation, let a and b represent the legs of a right triangle and c be the hypotenuse. In this type of triangle, the legs are equal, so a=b. The hypotenuse is [tex]\sqrt{2}[/tex] times the leg length. So, c = a√2.

Finding x and y

In this problem, x is one of the legs. This means that x = 2 since 2 is the length of the leg. The y-value represents the hypotenuse, so we can multiply the leg by √2. Thus, y = 2√2.

The equation, in slope-intercept form is y = 0.75x + 1.75

The equation for total cost can be represented in slope-intercept form as:

y = mx + b

where y is the total cost, x is the number of tacos,

m is the slope (the cost per taco), and b is the y-intercept (the cost of the drink).

So in this case,

m = 0.75 (the cost per taco),

b = 1.75 (the cost of the drink), and

x is the number of tacos.

So, the equation is:

y = 0.75x + 1.75

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Complete question

Chin is sells food at a local taco truck. She sells a soda which costs $1.75. Tacos cost $0.75 each.

What is the equation for Chin’s charges needed to solve the system and find the cost of the initial and additional hours?

The number of red jelly beans and blue jelly beans are 15 and 40 respectively.

Billy bought a bag of jelly beans from the store. Billy counted the jelly beans when he got home and counted a total of 55. The bag contained red and blue jelly beans. The number of blue jelly beans was 5 less than three times the number of red jelly beans.

Therefore, let's find the jelly beans of each colour.

Using equation,

let

x = number of blue jelly beans

y = number of red jelly beans

Therefore,

x + y = 55

x = 3y - 5

Therefore,

x + y = 55

-x + 3y = 5

add the equations

4y = 60

y = 60 / 4

y = 15

x = 55 - 15

x = 40

Therefore,

number of blue jelly beans = 40

number of red jelly beans = 15

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The measures of the interior angles of the triangle are: A=45°, B=110° and C=25°.

Triangles are geometric figures with three (3) sides. They can present different classifications regarding their sides or angles.

The sum of any triangle's angles will always be 180°, independent of its classification.

The question gives that the angles of the given  triangle are:

A=x

B= x+65°

C=25°

If A+B+C= 180° with the given values for angles B and C you can write:

x+x+65+25=180

2x+90=180

2x=180-90

x=90/2

x=45°

Thus, from the given information  you have:

A=45°

B= 45+65°=110°

C=25°

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The solution to the equation 3 / 5x-2 = 2 / 3x-1 is x = 1.

Given the equation in the question;

3 / 5x-2 = 2 / 3x-1

To solve for x, cross multiply the equation

3 / 5x-2 = 2 / 3x-1

3( 3x - 1 ) = 2( 5x - 2 )

Apply distributive property

3( 3x - 1 ) = 2( 5x - 2 )

3×3x + 3×-1 = 2×5x + 2×-2

9x - 3 = 10x - 4

Subtract 10x from both sides

9x - 10x - 3 = 10x - 10x - 4

9x - 10x - 3 = - 4

Add 3 to both sides

9x - 10x - 3 + 3 = - 4 + 3

9x - 10x  = - 4 + 3

-x = -1

Divide both sides by -1

-x/-1 = -1/-1

x = 1

Therefore, the value of x is 1.

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