Given:
The function is:
[tex]f(x)=x^2-16[/tex]
To find:
The inverse of the function.
Solution:
We have,
[tex]f(x)=x^2-16[/tex]
Step 1: Substitute [tex]f(x)=y[/tex].
[tex]y=x^2-16[/tex]
Step 2: Interchange x and y.
[tex]x=y^2-16[/tex]
Step 3: Isolate y.
[tex]x+16=y^2[/tex]
[tex]\pm \sqrt{x+16}=y[/tex]
[tex]y=\pm \sqrt{x+16}[/tex]
Step 4: Substitute [tex]y=f^{-1}(x)[/tex].
[tex]f^{-1}(x)=\pm \sqrt{x+16}[/tex]
Therefore, the inverse function of the given function is [tex]f^{-1}(x)=\pm \sqrt{x+16}[/tex].
There’s 10 in total I need help with
Answer:
8√2
Step-by-step explanation:
this one is a right triangle so both side are equal since the angle is 45 degrees
Plz help with these.
Answer:
5. [tex]a ( x + y^{2} + z )[/tex]
6. [tex]2a ( x + y + z)[/tex]
7. [tex]4 ( x +4y)[/tex]
8. [tex]-5 ( x + y )[/tex]
9. [tex]7a ( a + b)[/tex]
10. [tex]-2 ( x + 2y + 3z)[/tex]
11. [tex]bx ( a + y)[/tex]
12. [tex]-x (x^{2} + x + 1)[/tex]
this took long but i hope the answers are correct :)
Welp what is 103883+293883=? please helppp mah mum in city :')
[tex]\huge{\mathbb{\tt { PROBLEM:}}}[/tex]
Help What is 103883+293883=?
[tex]\huge{\mathbb{\tt { ANSWER:}}}[/tex]
397766[tex]{\boxed{\boxed{\tt { SOLUTION:}}}}[/tex]
[tex] \: \: \: 103883 \\ \frac{+293883}{ \: \: \: \: 397766} [/tex]
----------------------------------------------------------------------------------------------------[tex]\huge{\mathbb{\tt { WHAT \: IS \: ADDITION \: ?}}}[/tex]
Addition is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or sum of those values combined#CarryOnLearning
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XxKim02xXuse our rules for differentiating e x to show that cosh'(x) = sinh(x) sinh' (x) = cosh(x)
[tex]\sinh'(x) = \cosh(x)$.[/tex]
Hence, we have shown that [tex]\cosh'(x) = \sinh(x)$ and $\sinh'(x) = \cosh(x)$[/tex] using the rules for differentiating [tex]e^x$.[/tex]
What are Hyperbolic Functions?
Hyperbolic functions are a set of mathematical functions that are analogs of the trigonometric functions. While trigonometric functions are defined based on the unit circle, hyperbolic functions are defined using the hyperbola.
To show that [tex]\cosh'(x) = \sinh(x)$ and $\sinh'(x) = \cosh(x)$[/tex] using the rules for differentiating [tex]e^x$:[/tex]
[tex]\textbf{1. Derivative of $\cosh(x)$:}[/tex]
Starting with the definition of [tex]\cosh(x)$:[/tex]
[tex]\[\cosh(x) = \frac{1}{2}(e^x + e^{-x})\][/tex]
Taking the derivative with respect to x using the chain rule and the derivative of [tex]e^x$:[/tex]
[tex]\cosh'(x) &= \frac{1}{2}\left(\frac{d}{dx}(e^x) + \frac{d}{dx}(e^{-x})\right) \\\\&= \frac{1}{2}(e^x - e^{-x}) \\\\&= \sinh(x)[/tex]
Therefore, [tex]\cosh'(x) = \sinh(x)$.[/tex]
[tex]\textbf{2. Derivative of $\sinh(x)$:}[/tex]
Starting with the definition of [tex]\sinh(x)$:[/tex]
[tex]\[\sinh(x) = \frac{1}{2}(e^x - e^{-x})\][/tex]
Taking the derivative with respect to x using the chain rule and the derivative of [tex]$e^x$[/tex]:
[tex]\sinh'(x) &= \frac{1}{2}\left(\frac{d}{dx}(e^x) - \frac{d}{dx}(e^{-x})\right) \\\\&= \frac{1}{2}(e^x + e^{-x}) \\\\&= \cosh(x)[/tex]
Therefore, [tex]\sinh'(x) = \cosh(x)$.[/tex]
Hence, we have shown that [tex]\cosh'(x) = \sinh(x)$ and $\sinh'(x) = \cosh(x)$[/tex] using the rules for differentiating [tex]e^x$.[/tex]
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Explain Cantor's Theorem, that is, the fact that A and P(A) have different sizes, for every given set A. Summarize the proof of this result, pointing out the main ideas. What consequence
Cantor's Theorem states that the cardinality of a set A is strictly less than the cardinality of its power set P(A), for every set A. In other words, there is no bijection between A and P(A).
The proof of Cantor's Theorem relies on a diagonalization argument. Suppose there is a bijection f between A and P(A). We can use f to construct a subset B of A that is not in the image of f.
To do this, we define B as follows: for each element x in A, if x is not in the set f(x), then we add x to B. In other words, B contains all elements of A that are not in their corresponding set in P(A) under f.
Now, we show that B is not in the image of f. Suppose that there exists some element y in A such that f(y) = B. Then, we have two cases: either y is in B or y is not in B.
If y is in B, then y is not in f(y), since y was added to B precisely because it is not in its corresponding set in P(A) under f. But this contradicts the assumption that f(y) = B.
If y is not in B, then y is in f(y), since y is not in B precisely because it is in its corresponding set in P(A) under f. But this also contradicts the assumption that f(y) = B.
Therefore, we have shown that B is not in the image of f, which contradicts the assumption that f is a bijection between A and P(A). Thus, there can be no such bijection, and Cantor's Theorem follows.
The consequence of Cantor's Theorem is that there are different sizes of infinity, which has profound implications for mathematics and philosophy. It shows that there are sets that are "larger" than others, and that there is no "largest" infinity. This has led to the development of set theory as a foundational branch of mathematics, and has influenced debates about the nature of infinity in philosophy.
Which of the following statements about the polynomial function F(x)=x^3+2x^2-1 is true
The true statement about the polynomial function is (d) 0 relative minimum
How to determine the true statement about the polynomial functionFrom the question, we have the following parameters that can be used in our computation:
f(x) = x³ + 2x² - 1
Differentiate and set the function o 0
So, we have
3x² + 4x = 0
Factor the expression
So, we have
x(3x + 4) = 0
Next, we have
x = 0 or x = -4/3
So, we have
f(0) = (0)³ + 2(0)² - 1 = -1
f(-4/3) = (-4/3)³ + 2(-4/3)² - 1 = 0.2
This means that it has a relative minimum at x = 0
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Write the sentence as an equation
the product of 48 and g, reduced by 26 is equal to 345 subtracted from the quantity g times 150
Please help me if you know. Please give me an answer
9514 1404 393
Answer:
6 3/10 pounds
Step-by-step explanation:
The weight change will be found by multiplying the rate of change by the time.
∆w = (-1.8 lb/h)(3.5 h) = -6.3 lb
The total change in weight after 3 1/2 hours is 6 3/10 = 6.3 pounds.
Furaha pushes his rectangle next to Rahma's rectangle to form a new, longer rectangle. Draw an area model to show the new rectangle. Label the side lengths.
Answer:
See attachment
Step-by-step explanation:
Given
See attachment for complete question
Required
Determine the new model
From the question;
Furaha's model is: 6 by 4
Rahma's model is: 7 by 4
When Furaha's rectangle is pushed next to Rahma's, the new model becomes: (6 + 7) by 4
i.e. 13 by 4
See attachment 2
Good morning guys, I need help with a math problem ..8x+3y-2x-4y-6x
Answer:
8x+3y-2x-4y-6x = -y
Step-by-step explanation:
8x+3y-2x-4y-6x = ?
combine like terms:
(8x - 2x - 6x) + (3y - 4y) = ?
The x total is 0 and the y total is -1y
Answer:
-1y
Step-by-step explanation:
8x-2x-6x=0
3y-4y=-1y
That means the solution I think is -1y.
I say this because the X gets completely cancelled, and then the only thing left is -1y. If the X still had a number in front of it, it would then be like (This Is An Example: 2x = -1y) That's what it would look like if the X wasn't completely cancelled out. Have a good day. And if you see any fault in my answer please let me know so I can get better with problems like these. Thanks.
Aidan bought a pizza cut into 5 slices. If he ate one slice for lunch, what percentage of the pizza remained uneaten?
Answer:
80%
Step-by-step explanation:
You started of with 5/5 once one slice was eat it became 4/5
you must then convert 4/5 into a percent
To convert a fraction to a percent, divide the numerator by the denominator. Then multiply the decimal by 100.
so 4 divided by 5 =0.8
0.8 x 100= 80
80%
36. Given x is the score in the distribution. The interval of scores containing within one standard deviation from the mean is a < x < b. solve for a.
37. Given x is the score in the distribution. The interval of scores containing within one standard deviation from the mean is a < x < b. solve for b.
Given x is the score in the distribution. The interval of scores containing within one standard deviation from the mean is a < x < b. The solution for a and b are; a = x - SD and b = x + SD.
Let's see how we can solve these problems using the standard deviation and distribution.
36. To solve for a:
One standard deviation from the mean in a normal distribution includes about 68 percent of the scores.
Therefore, we know that:
P(mean - SD < x < mean + SD) = 68%. Where, SD = standard deviation of the distribution.
Therefore, to find a, we need to subtract SD from the mean.
So, a = mean - SD.
Distribution:
a < x < b
P(mean - SD < x < mean + SD) = 68%
Mean is x in this case; so:
P(x - SD < x < x + SD) = 68%
Now, solve for a by subtracting SD from x:
x - SD = a = x - SD
37). To solve for b.
From the previous problem,
we have: P(x - SD < x < x + SD) = 68%
To find b, we need to add SD to x
So, b = x + SD
Substitute the values of SD and x to get the value of b.
Distribution: a < x < b
P(mean - SD < x < mean + SD) = 68%
P(x - SD < x < x + SD) = 68%
So,
a = x - SD and b = x + SD.
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36. The value of a is µ - σ for the interval of scores containing within one standard deviation from the mean is a < x < b.
37. The value of b is µ + σ for the interval of scores containing within one standard deviation from the mean is a < x < b.
Explanation:
Given that x is the score in the distribution. The interval of scores containing within one standard deviation from the mean is a < x < b. We need to find the values of a and b.
Formula used:
µ ± σ,
where µ = mean
σ = standard deviation.
Using this formula we can write:
µ - σ < x < µ + σ
36.
Given that x is the score in the distribution.
The interval of scores containing within one standard deviation from the mean is a < x < b.
Substitute the above formula in the given expression: µ - σ < x
a = µ - σ
µ - σ = a + σ
a = µ - σ
Thus, the value of a is µ - σ.
37.
Given that x is the score in the distribution.
The interval of scores containing within one standard deviation from the mean is a < x < b.
solve for b.
Substitute the above formula in the given expression:
xa < µ + σ
b = µ + σ
µ + σ = b - σ
b = µ + σ
Thus, the value of b is µ + σ.
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please help will give brainlist if its correct
Answer:
I think it's B
Step-by-step explanation:
x = -2
2 + x > -8
2 + -2 > -8
0 > -8
Which One Doesn't Belong?
Answer:
I THINK C I’m not totally sure because it has an end point visible
Step-by-step explanation:
Can anyone help find x?
Answer:
x = 112°
Step-by-step explanation:
To find the value of x, you need to understand the properties of vertical and supplementary angles and know that the sum of angles inside a triangle is 180°.
Solve the given initial-value problem. (Enter the first three nonzero terms of the solution.) (x + 3)y" + 2y = 0, y(0) = 1, y'(0) = 0 1- . 2 3 x + 12 + ...
The solution to the given initial-value problem is a power series given by y(x) = 1 - 2x^3 + 3x^4 + O(x^5). As x increases, higher powers of x become significant, and the series must be truncated at an appropriate order to maintain accuracy .
y(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + a_4x^4 + ..., where a_0, a_1, a_2, ... are constants to be determined. We then differentiate the series term-by-term to find the derivatives y' and y''. Differentiating y(x), we have
y' = a_1 + 2a_2x + 3a_3x^2 + 4a_4x^3 + ..., and differentiating once more, we find y'' = 2a_2 + 6a_3x + 12a_4x^2 + ...Substituting these expressions into the given differential equation, we have:
(x + 3)(2a_2 + 6a_3x + 12a_4x^2 + ...) + 2(a_0 + a_1x + a_2x^2 + a_3x^3 + a_4x^4 + ...) = 0
Given the initial conditions y(0) = 1 and y'(0) = 0, we can use these conditions to find the values of a_0 and a_1. Plugging in x = 0 into the power series, we have a_0 = 1. Differentiating y(x) and evaluating at x = 0, we get a_1 = 0.Therefore, the power series solution is y(x) = 1 + a_2x^2 + a_3x^3 + a_4x^4 + ..., where a_2, a_3, a_4, ... are yet to be determined.
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The balanced scale represents the equation: 3x + 1 = x + 3
If one x block is subtracted from the right side and three numbered blocks are subtracted from the left side. What process will
balance the scale?
subtract one x block from the left side and subtract three numbered blocks
from the left side
subtract one x block from the left side and subtract three numbered blocks
from the right side
subtract one x block from the right side and subtract'three numbered
blocks from the left side
subtract one x block from the right side and subtract three numbered
blocks from the right side
Answer:
1 block is required in the process to balance the scale
Step-by-step explanation:
In order to get the process that will balance the scale, we need to solve the given expression for x as shown;
3x + 1 = x+ 3
Subtract x from both sides
3x+1-x = x+3 - x
3x - x + 1 = 3
2x + 1 = 3
Subtract 1 from both sides
2x + 1 - 1 = 3 -1
2x = 2
Divide both sides by 2
2x/2 = 2/2
x = 1
Hence 1 block is required in the process to balance the scale
4 of 21 (4 complete)
HW Score: 16%, 4 of 25
X 11.2.33 Assigned Media
Question Help
*
The results of a medical test show that of 39 people selected at random who were given the test, 36 tested negative and 3 tested positive. Determine the odds against
a person selected at random from these 39 people testing negative on the test.
Answer:
The odds against a person selected at random from these 39 people testing negative on the test is 92.31%.
Step-by-step explanation:
In the group of 39 randomly selected people:
# of people tested negative: 36
36 / 39 = 92.31%
2. If m arc VW - 62" and marc YZ -25°, then what is the
measure of
Help Starr worksheet review i wanna know what to do
4h + 14 > 38
What’s the answer
Answer:
Inequality Form:
h > 6
Interval Notation:
( 6 , ∞ )
Step-by-step explanation:
Isolate the variable by dividing each side by factors that don't contain the variable.
Choose ALL the lines PARALLEL to the following line: y = 2/9x - 7
y = 9/2x-12
y = 2/9x + 8
-2y = 9x + 8
9y = 2x -18
Answer: y = 2/9x + 8 and -2y = 9x + 8
Step-by-step explanation: Hope this help :D
Parallel lines have the same slope so y = 2/9x + 8 and 9y = 2x -18 will be parallel to the line y = 2/9x - 7 so option (B) and (D) will be correct.
What are parallel lines?Two lines in the same plane that are equally spaced apart and never cross each other are said two lines in the same plane that are equally spaced apart and never cross each other to be parallel lines.
Parallel lines are those lines in which slopes are the same and the distance between them remains constant.
The equation of a linear line is given by
y = mx + x where m is slope
So,
y = 2/9x - 7 have slope as 2/9
Now since parallel lines have the slope same so all lines whose slope matches with 2/9 will be parallel to this.
So,
y = 2/9x + 8 has slope of 2/9
9y = 2x -18 ⇒ y = 2/9 x - 2 has slope of 2/9
Hence "y = 2/9x + 8 and 9y = 2x -18 will be parallel to the line y = 2/9x - 7"
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The average sum of differences of a series of numerical data from their mean is:
a. Zero
b. Varies based on the data series
c. Variance
d. other
e. Standard Deviation
The average sum of differences of a series of numerical data from their mean is zero (option a).
This property holds true for any data set when calculating the mean deviation (also known as the average deviation) from the mean. The mean deviation is calculated by taking the absolute difference between each data point and the mean, summing them up, and dividing by the number of data points.
However, it's important to note that this property does not hold true when using squared differences, which is used in the calculation of variance and standard deviation. In those cases, the average sum of squared differences from the mean would give the variance (option c) or the squared standard deviation (option e).
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Let x be proba bility with a random variable density function fca) =c(3x² + 4 ) ( ocx S3 Х - Let Y=2x-2, where У is the random variable in the above to find the density function fy(t) of y. Y. Make to specify the region where fy (t) #o. sure О
The density function fy(t) of the random variable Y, where Y = 2x - 2, can be determined by transforming the density function of the random variable X using the given relationship.
To find fy(t), we first need to find the inverse relationship between X and Y. From Y = 2x - 2, we can solve for x:
x = (Y + 2) / 2
Next, we substitute this expression for x in the density function of X, fX(x):
fX(x) = c(3x² + 4)
Substituting (Y + 2) / 2 for x, we have:
fX((Y + 2) / 2) = c[3((Y + 2) / 2)² + 4]
Simplifying the expression:
fX((Y + 2) / 2) = c(3/4)(Y² + 4Y + 4) + 4c
Expanding and simplifying further:
fX((Y + 2) / 2) = (3/4)cY² + 3cY + (3/4)c + 4c
Combining like terms:
fX((Y + 2) / 2) = (3/4)cY² + (12c + 3c)Y + (3/4)c + 4c
Now, we can see that fy(t), the density function of Y, is a quadratic function of Y. The specific coefficients and constants will depend on the values of c.
It is important to note that we need to specify the region where fy(t) is defined. Since fy(t) is derived from fX(x), we need to ensure that the transformation (Y = 2x - 2) is valid for the range of x values where fX(x) is defined.
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i need help with this questionn
Given the differential equation:
dy/dx -xy = -2 (x2 ex – y2)
with the initial condition y(0) = 1, find the values of y corresponding to the values of x0+0.2 and x0+0.4 correct to four decimal places using Heun's method.
The value of y corresponding to x₀ + 0.2 is approximately 0.6701 and the value of y corresponding to x₀ + 0.4 is approximately 0.5650 using Heun's method .
The differential equation using Heun's method, we will approximate the values of y at x₀ + 0.2 and x₀ + 0.4 based on the initial condition y(0) = 1.
Heun's method involves using the slope at two points to estimate the next point. The algorithm for Heun's method is as follows:
Given the initial condition y(x₀) = y₀, let h be the step size.
Set x = x₀ and y = y₀.
Compute k₁ = f(x, y) = -xy + 2(x² × eˣ - y²), where f(x, y) is the given differential equation.
Compute k₂ = f(x + h, y + hk₁).
Update y = y + (h/2) × (k₁ + k₂).
Update x = x + h.
Using the given initial condition y(0) = 1, we'll apply Heun's method to find the values of y at x₀ + 0.2 and x₀ + 0.4.
Initial condition
x₀ = 0
y₀ = 1
Step size
h = 0.2 (given)
Iterating through the steps until we reach x = 0.4:
x = 0, y = 1
k₁ = -0 × 1 + 2(0² × e⁰ - 1²) = -1
k₂ = f(0.2, 1 + 0.2×(-1)) = f(0.2, 0.8) = -0.405664
y = 1 + (0.2/2) × (-1 + (-0.405664)) = 0.7978688
x = 0.2, y = 0.7978688
k₁ = -0.2 × 0.7978688 + 2(0.2² × [tex]e^{0.2}[/tex] - 0.7978688²)
= -0.1777845
k₂ = f(0.4, 0.7978688 + 0.2×(-0.1777845))
= f(0.4, 0.7633118)
= -0.2922767
y = 0.7978688 + (0.2/2) × (-0.1777845 + (-0.2922767))
= 0.6701055
x = 0.4, y = 0.6701055
k₁ = -0.4 × 0.6701055 + 2(0.4² × [tex]e^{0.4}[/tex] - 0.6701055²)
= -0.1027563
k₂ = f(0.6, 0.6701055 + 0.2×(-0.1027563))
= f(0.6, 0.6495543)
= -0.2228019
y = 0.6701055 + (0.2/2) × (-0.1027563 + (-0.2228019))
= 0.5649933
Therefore, the value of y corresponding to x₀ + 0.2 is approximately 0.6701 (correct to four decimal places) and the value of y corresponding to x₀ + 0.4 is approximately 0.5650 (correct to four decimal places).
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Plz help! Dont answer if you cant help
Answer:
42.09 cubic units
Step-by-step explanation:
[tex]\frac{4.11*5.12}{2} *4[/tex]
=42.0864, which rounds to 42.09
Note: The 6.57 is not needed to solve this problem
Kendra Buys three bracelets for $48 which table shows the correct amount she would need to pay to buy nine or 13 bracelets at the same price per bracelet
Answer:
The answer is B
Step-by-step explanation:
Divide 48 by 3 which gives you the total amount for one bracelet then you have to multiply the amount by 9 and 13 and then find your answer in the letters.
The proportion relationship is as follows;
number of bracelet total cost($)
3 48
9 144
13 208
Proportional relationshipProportional relationship is one in which two quantities vary directly with each other.
Therefore, we can establish a proportional relationship between the number of bracelets and it cost.
Hence,
let
x = number of bracelet
y = cost of the bracelets
Therefore,
y = kx
where
k = constant of proportionality
48 = 3k
k = 48 / 3
k = 16
Let's find the cost for 9 or 13 bracelet
y = 12x
y = 16(9) = 144
y = 16(13) = 208
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( 74 GUIDED Name:
PRACTICE Using Dot Plots to Make Inferences
1.
Joseph asks 10 of his friends how many baseball trading cards each friend has.
The data is shown in the dot plot. How many friends have more than five cards?
1
2
3
8
9
10
4 5 6 7
baseball trading cards
11
A. 3
C. 10
B. 5
Answer:
2 friends have more than 5 cards
Step-by-step explanation:
Incomplete question;
I will answer this question with the attached dot plot
The horizontal axis represents the friends, the vertical represents the number of baseball trading cards and the dots represent the frequency
So, we have:
[tex]Friend\ 1 = 2[/tex]
[tex]Friend\ 2 = 3[/tex]
[tex]Friend\ 3 = 7[/tex]
[tex]Friend\ 4 = 4[/tex]
[tex]Friend\ 5 = 2[/tex]
[tex]Friend\ 6 = 6[/tex]
[tex]Friend\ 7 = 2[/tex]
[tex]Friend\ 8 = 5[/tex]
[tex]Friend\ 9 = 1[/tex]
[tex]Friend\ 0 = 0[/tex]
The friends that has more than 5 are:
[tex]Friend\ 3 = 7[/tex]
[tex]Friend\ 6 = 6[/tex]
Hence, 2 friends have more than 5 cards
how to find what is the value of the correlation coefficient?
The value of the correlation coefficient is represented by the symbol "r." It is a statistical measure that determines the degree of correlation or association between two variables.
There are various methods of calculating r, but the most common one is the Pearson correlation coefficient. To calculate the Pearson correlation coefficient, follow these steps:
Step 1: Collect the data for the two variables you want to determine the correlation for. The data should be continuous and normally distributed.
Step 2: Calculate the mean of both variables.
Step 3: Calculate the standard deviation of both variables.
Step 4: Calculate the covariance of the two variables using the formula below: `Cov(X, Y) = Σ [(Xi - Xmean) * (Yi - Ymean)] / (n-1)
`Step 5: Calculate the correlation coefficient using the formula below: `r = Cov(X, Y) / (SD(X) * SD(Y))` where r is the correlation coefficient, Cov is the covariance, SD is the standard deviation, X is the first variable, Y is the second variable, Xi and Yi are the individual values of X and Y, X mean and Y mean are the means of X and Y, and n is the number of observations. The resulting value of r ranges from -1 to +1. A value of -1 indicates a perfect negative correlation, a value of 0 indicates no correlation and a value of +1 indicates a perfect positive correlation.
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