Absolute Value Function
Example
Graph the function f(x) = x  3.
Solution
Since this is an absolute value function, we first find
the vertex. To do this, set the quantity inside the
absolute value symbols equal to 0.

x  3 = 0 
Solve for x. 
x = 3 
So, the xvalue of the vertex is 3. Therefore, to create a table for this
function we will use x = 3 as one input value and choose xvalues on
either side of 3.
Weâ€™ll let x = 2, 0, 3, 6, and 8.
Substitute those values of x into the function and simplify.
x 
f(x)
=  x  3 
(x, y) 
2
0
3
6
8 
f(2) = 2  3 = 5
f(0) = 0  3 = 3
f(6) = 6  3 = 3
f(3) = 3  3 = 0
f(8)= 8  3 = 5 
(2, 5) (0, 3)
(3, 0)
(6, 3)
(8, 5) 
Now, plot the points and connect them.
Notice that the graph of
f(x) = x  3 is an inverted
. This is because of the negative sign
in front of the absolute value.
The domain of f(x) = x  3 is all real numbers because we can find the
absolute value of any real number.
To find the range, examine the yvalues on the graph. The yvalues consist
of all real numbers less than or equal to 0.
Thus, the range is y ≤ 0. This is the interval (∞, 0].
Notes:
In f(x) = x  3, the expression x  3 is
linear. That is why the graph on each side
of x = 3 is a straight line.
Be careful with the negative signs when
evaluating functions like
f(x) = x  3. First, find the value of x  3. Then
multiply the result by 1.
For example:
f(4) = 4 3 = 7 = (7) = 7
f(3) = 3  3 = (0) = 0
