Mathematics

Absolute Value Function

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15 December 2020                

Read time: 5 minutes

Introduction

Distances in large space can be measured in all directions. As such, it is useful to think about distance in terms of absolute value.

galaxies that are billions of light-years away

  • Until the 1920s, the spiral nebulae were believed to be clouds of dust and gas in our own galaxy, some tens of thousands of light-years away. Then, astronomer Hubble proved that these objects are galaxies in their own right, at distances of millions of light-years.
     
  • Today, astronomers can detect galaxies that are billions of light-years away. Distances within the universe are often measured in all directions. So, it is useful to consider distance as an absolute value function. In this section, we will continue our investigation of absolute value functions.
     
  • The absolute value function is usually thought of as providing the distance the number is from zero on a number line. Algebraically, regardless of the input value is, the output is the value without regard to sign. Knowing this, we will use absolute value functions to solve some kinds of real-world problems. The basic form of the absolute value function \(f(x)=|x|.\)

Using Absolute Value Function to Determine Resistance

Electrical parts, such as resistors and capacitors, associate with specified values of their operating parameters: resistance, capacitance, etc. However, because of imprecision in manufacturing, the actual values of these parameters vary somewhat from piece to piece, even when they are supposed to be the same. The best that manufacturers can do is to try to guarantee that the variations will stay within a specified range.

Absolute Value Function to Determine Resistance

Suppose we have a resistor rated at 680 ohms, use the absolute value function to find the range of absolute values possible of the actual resistance.

We can find that 5% of 680 ohms is 34 ohms. The absolute value function of the difference between the actual and nominal resistance should not exceed the stated variability, so, with the resistancein \(R\) ohms, \(|R-680|\le34\).

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Absolute Value Function

The absolute value function often defined as a piecewise function.

\[f(x) = |x| = \left\{ \begin{array}{l}x\,\,{\text{if}}\,\,x \ge 0\\x\,\,\text{if}\,\,x < 0\end{array} \right.\]

We look at the expression |x| like this: “the absolute value of x.” This is also the absolute value parent function. All other absolute value equations are derived from this.

  • Treat absolute value functions like parentheses. If there is an operation inside the absolute value symbol, evaluate that operation first.
  • The absolute value function of a number or an absolute value expression is always positive or zero. It cannot be negative. With absolute value, we are only interested in how far a number is from zero, not the direction.

The absolute value function finds the distance of a number on the number line from 0 irrespective of the direction from zero. Since it is a distance it can only take positive values.

  • Let us take the number 5. Counting from the origin or 0, it takes us 5 numbers to reach 5, hence the distance which is the absolute value would be 5.

     

  • The output of the absolute value of \(-5\) is \(5.\)

     

absolute value of  5 is 5

This is because we take 5 number wise steps from 0 to -5 in a similar way to while calculating distance for 5. This leads to |-5| = |5| = 5.

  • What would be the value of |2 + \(-7\)|?

    This is equivalent to \-5\. And we have proven it to be equal to 5.

    distance of sum from 0: 5 units

  • The absolute value of 0 is 0. (This is why we don't say that the absolute value function on a number is positive: Zero is neither negative nor positive.)

The absolute value symbol for the absolute value function is two straight lines surrounding the number or absolute value expression for which you wish to indicate absolute value.

 

Let us take an example of |x| = 2. This can be represented by the figure given below:

Absolute value is 2.

What about absolute value inequality?

Let us try to represent |x|>2. This represents all the numbers having their absolute value greater than 2. 

two sets of values for inequalities

In this, you get two sets of values for inequalities with |x| > some number or with |x| = some number.

Similarly, |x|<2 will represent all the value of x whose absolute value function gives out a number less than 2. This can be visualized as given below:

values for absolute inequalities

Here are some examples of how the modulus function works on inequalities:

  • For |x| < 5, -5 < x < 5

  • For |x| = 8, -8 = x, x = 8

  • For  |x + 8| < 75, -83 < x < 67

For the absolute value function f(x)=x  there is no restriction on x  so the domain is unrestricted. But the range will be affected since we cannot have negative values as output.

absolute value function


 Properties of Absolute Value

The Absolute Value Function and the properties of absolute value are the most used functions in mathematics. Its definition and some of the properties of absolute value are given below.

Absolute Value Function and the properties of absolute value

The following are properties of absolute value function that needs to be looked at.

  • For any of the two real numbers x and y, we have |xy| = |x| |y| . Suppose x < 0 and y ≥ 0. Then xy is ≤ 0 and we have |xy| = − (xy) = (−x) y = |x| |y| .
  • For any real number x, and any nonnegative number a, we have |x| ≤ a if and only if −a ≤ a ≤ x. This is verified by considering the two possible cases x ≥ 0 and x < 0.

    We consider the case of x ≥ 0. So now think of x ≥ 0 and |x| ≤ a.Then we have −a ≤ 0 ≤ x = |x| ≤ a.

    That is, if |x| ≤ a, then −a ≤ x ≤ a. If we think in the opposite way suppose x ≥ 0 and −a ≤ x ≤ a. Then we have |x| = x ≤ a. The argument for the case x < 0 is similar and left to the reader.

  • For any real number x, and any nonnegative number a, we have |x| ≥ a if and only if x ≥ a or x ≤ −a. There are again two cases to consider: x positive or x negative. So suppose x ≥ 0. If |x| ≥ a, now we have x = |x| ≥ a.

    On the other hand if x ≥ a, then we have |x| = x ≥ a. Conversely suppose x < 0 and |x| ≥ a, then we have −x = |x| ≥ a x ≤ −a . In that case where x < 0, if we have x ≤ −a, then |x| = −x ≥ − (−a) = a

    Note: When x < 0 the condition x ≥ a cannot be true, since we are assuming x ≥ 0. simplest case being when both x and y are nonnegative; in this case we have |x + y| = x + y = |x| + |y|

  • For any two real numbers x and y we have |x + y| ≤ |x| + |y|. The proof of this is again handled by considering the four possible cases determined by the signs of x and y.

    The than or equal relation can be replaced with strict absolute inequality.

    For instance, |x| < a if and only if −a < x < a.


Summary

The absolute value function is used in the real world to define the DIFFERENCE or change from one point to another. They use it in physics and calculus to define certain situations: you want to know how far something traveled when it went 5 ft. one way and 5 ft. back. Well, it didn't actually go anywhere because it is back where it started. Kind of like adding -5 to +5. You get zero. But with an absolute value you can say, it traveled 10 ft. altogether even though it didn't go anywhere.

In the blog, we covered have covered:

  • Absolute value function 
     
  • Absolute value parent function
     
  • Properties of absolute value

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Frequently Asked Questions (FAQs)

What does the modulus function mean?

The modulo (or "modulus" or "mod") is the remainder after dividing one number by another. Example: 100 mod 9 equals 1. Because 100/9 = 11 with a remainder of 1.

Why is modulus function value always positive?

The absolute value of a number is always positive. Consider the number of jumps it would take to get to zero from a number. It would take five jumps to get to zero from -5, so the absolute value function on -5 is 5.

Absolute value parent function?

The absolute value function often defined as a piecewise function.

\[f(x) = |x| = \left\{ \begin{array}{l}x\,\,{\text{if}}\,\,x \ge 0\\x\,\,\text{if}\,\,x < 0\end{array} \right.\]

Absolute value symbol?

|x| with the two vertical lines.


External References 

To know more about Absolute Value Function, visit these blogs:


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