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# Absolute Value Function

The absolute value function is a rule that involves taking the absolute value of the input variable. Here are some examples:

 f(x) = |x| f(x) = |2x| - 5 f(x) = |2x - 3|

Since we can find the absolute value of any real number, the domain of these absolute value functions is all real numbers. That is, the interval (-, +). The range will depend on the individual function.

To graph an absolute value function, first calculate several ordered pairs. Then, plot the ordered pairs on a Cartesian coordinate system. Finally, connect the points with a line.

Example

Make a table of five ordered pairs that satisfy the function f(x) = |x|. Then, use the table to graph the function.

Solution

To make a table, select 5 values for x. Weâ€™ll let x = -6, -3, 0, 3, and 6.

Substitute those values of x into the function and simplify.

 x f(x) = |x| (x, y) -6 -3 0 3 6 f(-6) = |-6| = 6 f(-3) = |-3| = 3 f(3) = |3| = 3 f(0) = |0| = 0 f(6) = |6| = 6 (-6, 6) (-3, 3) (3, 3) (0, 0) (6, 6)

Now, plot the points and connect them. Notice the shape of the graph. This is characteristic of the absolute value function, although sometimes the vee is upside down, like this . The point of the vee is called the vertex.

The domain of f(x) = |x| is all real numbers since we can find the absolute value of any real number.

To find the range, consider the y-values on the graph. The smallest y-value is 0. So, the range is y 0. This is the interval [0, +).

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