### Absolute Values

```THE
ABSOLUTE VALUE
FUNCTION
Properties
of
The Absolute Value Function
Vertex
(2, 0)
f (x)=|x -2| +0
vertex
(x,y) = (-(-2), 0)
Maximum or Minimum?
a=1>0
graph opens up
Minimum y = 0
Domain and Range
Domain
The domain of the Absolute value function is
the set of real numbers
Range
1.
If the function concave up, the range is
greater than or equal to the minimum
2.
If the function concave down, the range is
less than or equal to the maximum
Equations Involving Absolute Value
Functions
If |x|= a where a is a positive real number
Then either x = a or x = -a
Examples
If |x| = 3 then either x = 3 or x = -3
If|x+1| = 5 then either x+1 = 5 or x+1 = 5
If |2x-3| = 4 then either 2x - 3 = 4 or 2x - 3 = -4
Inequalities Involving Absolute Value
Functions
If |x| < a where a is a positive real
number
Then
–a < x < a
If |x|  a
then –a  x  a
Examples
If |x| < 3
If |x+1|

then r -3 < x < 3
5
If |2x-3| < 4
then -5  x + 1  5
-6  x  4
then either -4 < 2x - 3 < 4
Solution of Equations Involving
Absolute Value Functions
The solution of two functions f(x) and
g (x) is their point of intersection
If f (x) = |x-1| and g (x) = 3
Then the solution of f (x) and g (x) is
the point (s)
|x – 1| = 3
Example
Solve the problem and graph
f (x) = |2x - 1|
g (x) = 2
Let f (x) = g (x)
If |2x – 1| = 2
Then 2x – 1 = 2
or 2x -1 = -2
+1 1
+1 +1
2x = 3
2x = - 1
x = 3/2
x = -1/2
Solve the problem and graph
(cont.)
Vertex (1/2,0)
Y- intercept x = 0
|2(0)-1| = 1
X – intercept y = 0
|2x-1| = 0
2x – 1 =0
x=½
Note the x intercept is the x coordinate of
the vertex.
Solve the problem and graph
Concave up a = 2
Minimum y = 0
Domain: R
Range: y  0
Points of intersection (-0.5, 2)
(1.5, 2)
Graph of y = |2x – 1| and y = 2
Points of intersections are (-.5, 2) and (1.5, 2)
Solve the problem and graph
(cont.)
f (x) = |2x – 1| < g (x) = 2
|2x -1| < 2
From the graph we can see that
|2x -1| < 2 f (x) is below the line y =2
When -.5 < x < 1.5
```