This article will be about absolute values. This topic is somewhat tricky for those new to this topic.
The Absolute Value
The absolute value of a real number is the non-negative part of . It is expressed as . Sometimes it is referred to the modulus and can be thought of as a distance from zero.
The simplest case of absolute values is with numbers. Here are some examples.
So far we have dealt with absolute value of numbers and not variables. We now generalize and present the formal definition.
Here is a visual of the absolute function centered at .
The absolute function is an example of a piecewise function. This type of function depends on the value of . It is not like linear functions where it is a smooth line for all values. If we had then the picture above would be upside down and would look like a hat.
We now consider intercepts with the variable inside the absolute value. If we had then the origin point at (0,0) from would be shifted to the right 7 units to the point (7, 0) for . If we had then the origin point at (0,0) from would be shifted to the left 2 units to the point (2, 0) for . Refer to the picture below as a visual aid.
Any vertical shifts would have numbers outside the absolute value function. For example, would shift the every point from up by 3 units.
Domain and Range
For the absolute function , the domain is all real numbers for . The range for is for all positive numbers and zero.
There are properties associated with the absolute value function. Here are a few.
These two properties come in handy when dealing with inequalities. (The constant is a real number.)
For example if we have , we can solve for as follows:
Here is another example where we have . The second property is used.
There are more things that can be said about the absolute value of . The topic of absolute values and complex numbers are not mentioned here.