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HIGH SCHOOL
DIFFERENTIAL CALCULUS
COURSE
1.3 SUPPORT 1 – ABSOLUTE VALUE FUNCTIONS
1.3.1. Define the absolute value function.
The absolute value or modulus of a real number is its numerical value without its sign. The notation of
absolute value of a value was introduced first by Karl Weierstrass in 1841.
The absolute value of a negative number makes it positive. Because of how the absolute value function
behaves, it is important to include also negative inputs in your T-chart when graphing the function. The
absolute value function is defined as f ( x)  x , f :  ,   0, which means that the function is
taking any real number for its domain and is getting only zero or positive numbers as range. (Remember:
f : Domain  Range )
For example, if we want to graph this function f ( x)  x , a general error could be made if we use only
positive x values in the table of values.
X
f ( x)  x
0
0
1
1
2
2
3
3
Using only this table of values, the function could be mistakenly graphed.
y
y







x











Plotting the
points with this
table of values,
we get a wrong
graph for the
function

x















Be careful, when you work with absolute value functions, always give values in the negative and positive side. So:
X
f(x)
We plot the new points and we get:
-3
3
-2
2
-1
1
0
0
1
1
2
2
3
3
y
y







This is the
graph of
f ( x)  x
x











x
















1.3.2. Explore the effect of the absolute value when it is applied over any type of function
use of a graphing utility.
by the
If we are trying to apply the absolute value of other functions, the effect is the same.
Domain
Domain
y
x   , 
y




x   , 




x
x
Range

















Range



y   3,  
y  0,  






y  x2  3
y  x2  3
y
y
Domain
x   0,  






Domain
x   0,  


x
x
Range

















Range


y   , 

y   0,  






y  ln x
y  ln x
y
y
Domain
x   ,  








Domain
x   ,  
x
x
Range
y   1, 1




















y  sin x





Range
y  0, 1

y  sin x
From these functions we can observ that the absolute-value will flip the negative part of the graph up into the
positive values (above the x-axis).
1.3.3. Graph the following absolute value functions, with absolute value.
1.3.4. Determine the domain and range of a function with absolute value.
Given the function f ( x)  x  x  2
2
To graph the function
f ( x)  x 2  x  2 we have to decide the vertex first. Completing the square, we find
2
1 9

1 9
that the function it can be written as f ( x )   x    so the vertex is V  ,   .
2 4

2 4
Using the transformations of functions, the basic function
f ( x)  x 2 is moved
1
9
to the right and
2
4
downwards.
Graphing the quadratic function , we get:
y
y





x











Applying the absolute
value of this function,
the negative side of the
function, below the x
the x axis

x












axis of symmetry

axis of symmetry

The domain of this function is:
The domain of this function is:
x   , 
x   , 
The range is:
The range is:
 9

y    ,  
 4

y  0,  


