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Absolute value is the distance between a number and 0 on the real number line. It is written like |x| with x being a number. So to jog memory the absolute value of -5 or |-5| is 5, and the absolute value of 5 or |5| is also 5. This is because distance can only be positive (this can be proven by using the Pythagorean theorem to graph right triangles on a 2 dimensional X/Y plane). Absolute value is also a function because no X values repeat when it is graphed. This is seen when it is graphed as any variation of the parent function F(x) = |x| or y = |x|. Moreover, Absolute value is a piece wise function because it has a limiting domain. The inverse of absolute value or F^-1(x) is not a function. The question is: Is absolute value an operation? For those who forget what an operation is, I will explain. A mathematical operation is something like x (multiplication), + (addition), - (subtraction) , / (division), and so on. For this debate I will be using the operation known as the square and square root. As some of you may already know, absolute value equations will often have two solutions when solved. For example, If I wanted to find the the x-intercepts of the function F(x) = |x - 1| - 3, I would have two solutions. When the function is graphed, you will see two x- intercepts at x = 4 and x = -2. This can be found algebraically by replacing y ( f(x) ) with 0 like so :
y = |x - 1| - 3
0 = |x - 1| - 3
3 = |x - 1|
+/- 3 = x - 1 ( I will come back to this as an example later)
x = 4 and x = -2
I am proposing that absolute value is a operation. Operations can often be written in multiple ways. For example, take x^2 and its inverse the square root of x. The square root of x is an operation, but it can be written as multiple operations like x^(1/2) with the multiple operations being in the exponent. What if I were to do the same thing with absolute value? What if I wrote |x| as the square root of x^2 or x^(2/2) or any other even root of x raised to the same root? By getting rid of the absolute value in the equation it would yield a positive or negative answer because I eventually square rooted both sides of the equation like so:
3 = |x - 1|
3 = (x - 1) ^(2/2)
9 = (x - 1) ^(2/1)
+/- 3 = x -1 (You get the plus or minus (+/-) because -3^2 and 3^2 both equal 9)
x = 4 and x = -2
What do you guys think? Is absolute value an operation?
PS: Sorry for my grammar and long post about math lol
y = |x - 1| - 3
0 = |x - 1| - 3
3 = |x - 1|
+/- 3 = x - 1 ( I will come back to this as an example later)
x = 4 and x = -2
I am proposing that absolute value is a operation. Operations can often be written in multiple ways. For example, take x^2 and its inverse the square root of x. The square root of x is an operation, but it can be written as multiple operations like x^(1/2) with the multiple operations being in the exponent. What if I were to do the same thing with absolute value? What if I wrote |x| as the square root of x^2 or x^(2/2) or any other even root of x raised to the same root? By getting rid of the absolute value in the equation it would yield a positive or negative answer because I eventually square rooted both sides of the equation like so:
3 = |x - 1|
3 = (x - 1) ^(2/2)
9 = (x - 1) ^(2/1)
+/- 3 = x -1 (You get the plus or minus (+/-) because -3^2 and 3^2 both equal 9)
x = 4 and x = -2
What do you guys think? Is absolute value an operation?
PS: Sorry for my grammar and long post about math lol
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