Understanding the Absolute Value Function: Properties, Graphing, and Applications

Absolute Value Function

The absolute value function is a mathematical function denoted by |x|, where x is any real number

The absolute value function is a mathematical function denoted by |x|, where x is any real number. The absolute value of a number represents the distance of that number from zero on the number line.

The absolute value function has the following properties:

1. Range: The range of the absolute value function is always non-negative. The output value is either zero or a positive value.

2. Symmetry: The absolute value function is symmetric about the y-axis. This means that the absolute value of a positive number is the same as the absolute value of its negative counterpart. For example, |3| = |-3| = 3.

3. Definition: The absolute value function is defined as follows:
– If x is greater than or equal to zero, then |x| = x.
– If x is less than zero, then |x| = -x (change the sign).

4. Graph: The graph of the absolute value function is V-shaped and opens upwards. The vertex of the graph is at the point (0, 0).

To illustrate the graph of the absolute value function, consider the following examples:

Example 1: Graphing |x|
To graph the function |x|, we can create a table of values and plot the points on a graph.

Table of values:
x |x|
-3 3
-2 2
-1 1
0 0
1 1
2 2
3 3

Plotting these points on a graph will result in a V-shaped graph, symmetric about the y-axis, with the vertex at the point (0, 0).

Example 2: Evaluating |x| for a given value of x
Let’s evaluate |(-4)|. Since -4 is less than zero, we use the definition of the absolute value function and change the sign of -4, giving us |-4| = 4. Therefore, the absolute value of -4 is 4.

Example 3: Solving absolute value equations
To solve an absolute value equation, we need to consider the two possible cases based on the definition of the absolute value function.

For example, if we have the equation |2x + 3| = 7, we need to consider both the positive and negative cases.

Positive case:
2x + 3 = 7
Solving this equation gives us x = 2.

Negative case:
-(2x + 3) = 7
Simplifying, we get -2x – 3 = 7
Solving this equation gives us x = -5.

So, the solutions to the absolute value equation |2x + 3| = 7 are x = 2 and x = -5.

It is important to understand the properties and applications of the absolute value function in order to effectively solve problems and interpret mathematical expressions involving absolute values.

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