Absolute Value Function

Absolute Value Function Definitions and Examples

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    What is The Absolute Value Function?

    Introduction

    The absolute value function is a mathematical function that returns the absolute value of a number. The absolute value of a number is the distance of the number from zero on a number line. The absolute value function is written as |x|.

    What is the absolute value function?

    The absolute value function is a mathematical function that returns the absolute value of a number. The absolute value of a number is the magnitude of the number without regard to its sign. For example, the absolute value of -5 is 5, and the absolute value of 5 is also 5.

    The absolute value function can be written as |x|, and the absolute value of a number x can be calculated as |x| = x if x is positive, and |x| = -x if x is negative.

    How to graph the absolute value function

    To graph the absolute value function, start by graphing y=|x|. This will give you a “V” shape. To complete the graph, draw a line from the origin (0,0) to the point (0,1). This will give you a full picture of the absolute value function.

    How to find the inverse of the absolute value function

    To find the inverse of the absolute value function, we need to find the function’s inverse function. To do this, we first need to determine the function’s domain and range. The domain of the absolute value function is all real numbers, while the range is all non-negative numbers.

    Next, we need to find the inverse function’s domain and range. To do this, we need to switch the x and y values of the points on the graph of the absolute value function. This will give us a new graph with a reversed domain and range. The inverse function’s domain will now be all non-negative numbers, while the range will be all real numbers.

    Now that we have determined the inverse function’s domain and range, we can begin to graph it. To do this, we simply plot points that satisfy the inverse function’s equation. For example, if we want to find out what point (2, 3) satisfies the inverse function, we would plug 2 in for x and 3 in for y in the equation y = |x|. We would then solve for x to get x = 2. This means that (2, 3) satisfies the equation y = |x| and is therefore on the graph of the inverse absolute value function.

    How to solve absolute value equations

    There are a few different ways to solve absolute value equations, depending on what type of equation you have. If you have a linear equation, meaning the absolute value is by itself and not nested inside another function, then you can solve it using the properties of absolute value. First, you need to isolate the absolute value expression on one side of the equation. Then, you can split the equation into two cases, when the expression is positive and when it is negative. For each case, you will end up with a different linear equation that you can solve using regular methods.

    If you have a quadratic equation, meaning the absolute value is nested inside another function, then you will need to use the Quadratic Formula to solve it. The Quadratic Formula is a specific way to solve any quadratic equation, and it involves taking the square root of both sides of the equation. Once you have done that, you will end up with two linear equations that you can solve using regular methods.

    Applications of the absolute value function

    The absolute value function is a mathematical function that returns the absolute value of a number. The absolute value of a number is the distance of the number from zero on a number line. The absolute value function can be written as |x|=x if x?0 and |x|=-x if x<0.

    The absolute value function is used in many different fields, such as algebra, calculus, and statistics. In algebra, the absolute value function can be used to simplify equations. In calculus, the absolute value function is used to find the maximum and minimum values of a function. In statistics, the absolute value function is used to calculate the standard deviation of a data set.

    Conclusion

    In mathematics, the absolute value or modulus of a real number x, denoted |x|, is the non-negative value of x without regard to its sign. That is, |x| = x when x is positive, and |x| = ?x when x is negative (in which case |?x| = ?(?x) = x). For example, the absolute value of 3 is 3, and the absolute value of ?3 is also 3. The absolute value function expresses one of the fundamental concepts in mathematical analysis—distance from 0 along the real number line.

    Absolute Value Function

    Plots

    Plots

    Plots

    Alternate form assuming x>0

    x

    Alternate form assuming x is real

    sqrt(x^2)

    Root

    x = 0

    Properties as a real function

    R (all real numbers)

    {y element R : y>=0} (all non-negative real numbers)

    even

    Derivative

    d/dx(abs(x)) = x/abs(x) (assuming a function from reals to reals)

    Indefinite integral assuming all variables are real

    integral abs(x) dx = (x^2 sgn(x))/2 + constant

    Global minimum

    min{abs(x)} = 0 at x = 0

    Alternative representations

    abs(x) = x/sgn(x) for x!=0

    abs(x) = sqrt(x x^*)

    abs(x) = sqrt(-x^2 + 2 x Re(x))

    Series representations

    abs(x) = 2/π - (4 sum_(k=1)^∞ ((-1)^k T_(2 k)(x))/(-1 + 4 k^2))/π for (x element R and -1<x<1)

    abs(x) = sum_(k=0)^∞ ((-1)^k (1/2 + 2 k) P_(2 k)(x) (-1/2)_k)/((1 + k)!) for (x element R and -1<x<1)

    abs(x) = ( sum_(k=0)^∞ ((-1)^k H_(2 k)(x) (-1/2)_k)/((2 k)!))/sqrt(π) for (x element R and -1<x<1)

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