HIGH SCHOOL
DIFFERENTIAL CALCULUS
COURSE
1.2 PIECEWISE FUNCTIONS
1.2.1 Define and graph piecewise functions
1.2.2 Determine the domain and range of a piecewise function
A piecewise defined function is a function which is defined symbolically by using two or more
different functions for different parts of their domain.
Example1: A simple piecewise-defined function is the absolute value function, f ( x)  x which can be
 x, if x  0
written as f ( x)  
 x, if x  0
In this example, when the x   ,0  , we use the function f ( x)   x and using the table of values,
we get
x
y=-x
-4
4
-3
3
-2
2
-1
1
On the other hand, when the x  0,   , we use the function f ( x)  x and using the table of values, we
get
x
y=x
1
1
2
2
3
3
4
4
Now if we graph this piecewise-defined function, we get:

y




x













You can observe, that in the point x=0 , the function f ( x)   x is not included Ο and the function
f ( x)  x it is included . Remember, if a vertical test line is touching many not included points and
only one included point, that point is an included point. If the vertical test line is touching two or more
than two included points, in this case we are not talking about a function.
Example2: Graph the following piecewise function:
 x 2  4 if x  1
f ( x)  
 x  3 if x  1
in other words, the function f ( x)  x 2  4 when the x is less or equal than ONE and f ( x)  x  3 when
the x is greater than ONE
To graph a piecewise function, first we have to graph each component function
Let´s graph first the function f ( x)  x 2  4 . Using the table of values we get:
x
y  x2  4
-2
-1
0
1
0
-3
-4
-3
y = x^2-4

y




x













Now let´s graph the other function f ( x)  x  3 . Using the table of values (or tabulating), we get:
x
1
2
3
4
y=x-3
-2
-1
0
1
Now we have to check the intervals where these functions exist. We get this graph:
y = x^2-4; -4.000000 <= x <= 1.000000

y = x-3; 1.000000 <= x <= 7.000000
y




x













The domain of this function is x  R and the range is y   4,  
Example3: Graph the following piecewise-defined functions:
Function 1
 1
if
x


,

1


 x

Function 2
f ( x)   x  2 if x   1,1
 ln x if x  1,  
Function 3


Tabulating each function, we get
Function 1
Function 2
x
x
1
y  x2
y
x
-1
1
-4
-0.25
0
2  1.41
-3
-0.(3)
1
3  1.73
-2
-0.5
-1
-1
Lets graph each function

Function 3
y  ln x
x
1
ln1=0
2
ln2≈0.69
3
ln3≈1.1
4
ln4≈1.38
y


y=sqrt(x+2)

sqrt(3)
y=lnx

x









y=1/x




The domain of this function is x  R and the range is y   1,0
 0, 