Definition of Piecewise Functions
A piecewise function is usually defined by more than one formula: a fomula for each interval.
Example 1:
f( x ) =  x if x <= 2
= x if x > 2
What the above says is that if x is smaller than or equal to 2, the formula for the function is f( x ) = x and if x is greater than 2, the formula is f( x ) = x. It is also important to note that the domain of function f defined above is the set of all the real numbers since f is defined everywhere for all real numbers.
Example 2:
f( x ) = 2 if x > 3
= 5 if x < 3
The above function is constant and equal to 2 if x is greater than 3. function f is also constant and equal to 5 if x is less than 3. It can be said that function f is piecewise constant. The domain of f given above is the set of all real numbers except 3: if x = 3 function f is undefined.
Example 3:
Functions involving absolute value are also a good example of piecewise functions.
f( x ) =  x 
Using the definition of the absolute value, function f given above can be written
f( x ) = x if x >= 0
= x if x < 0
The domain of the above function is the set of all real numbers.
Example 4:
Another example involving absolute vaule.
f( x ) =  x + 6 
The above function may be written as
f( x ) = x + 6 if x >= 6
=  (x + 6) if x < 6
The above function is defined for all real numbers.
Example 5:
Another example involving more than two intervals.
f( x ) = x^{ 2}  3 if x <= 10
=  2x + 1 if 10 < x <= 2
=  x^{ 3} if 2 < x < 4
= ln x if x > 4
The above function is defined for all real numbers except for values of x in the interval (2 , 2] and x = 4.
Example 6: f is a function defined by
f( x ) = 1 if x <= 2
= 2 if x > 2
Find the domain and range of function f and graph it.
Solution to Example 6:
Function f is defined for all real values of x. The domain of f is the set of all real numbers. We will graph it by considering the value of the function in each interval.
In the interval ( inf , 2] the graph of f is a horizontal line y = f(x) = 1 (see formula for this interval above). Also this interval is closed at x = 2 and therefore the graph must show this : see the "closed point" on the graph at x = 2.
In the interval (2 , + inf) the graph is a horizontal line y = f(x) = 2 (see formula for this interval above). The interval (2 , + inf) is open at x = 2 and the graph shows this with an "open point". Function f can take only two values: 1 and 2. The range is given by {1, 2}
Example 7: f is a function defined by
f( x ) = x^{ 2} + 1 if x < 2
=  x + 3 if x >= 2
Find the domain and range of function f and graph it.
Solution to Example 7:
The domain of f is the set of all real numbers since function f is defined for all real values of x.
In the interval ( inf , 2) the graph of f is a parabola shifted up 1 unit. Also this interval is open at x = 2 and therefore the graph shows an "open point" on the graph at x = 2.
In the interval [2 , + inf) the graph is a line with an x intercept at (3 , 0) and passes through the point (2 , 1). The interval [2 , + inf) is closed at x = 2 and the graph shows a "closed point". From the graph, we can observe that function f can take all real values. The range is given by ( inf, + inf).
Example 8: f is a function defined by
f( x ) = 1 / x if x < 0
= e^{ x} if x >= 0
Find the domain and range of function f and graph it.
Solution to Example 8:
The domain of f is the set of all real numbers since function f is defined for all real values of x.
In the interval ( inf , 0) the graph of f is a hyperbola with vertical asymptote at x = 0.
In the interval [0 , + inf) the graph is a decreasing exponential and passes through the point (0 , 1). The interval [0 , + inf) is closed at x = 0 and the graph shows a "closed point".
As x becomes very small, 1 / x approaches zero. As x becomes very large, e^{ x} also approaches zero. Hence the line y = 0 is a horizontal asymptote to the graph of f.
From the graph of f shown below, we can observe that function f can take all real values on ( inf , 0) U (0 , 1] which is the range of function f.
Example 9: f is a function defined by
f( x ) = 1 if x <= 1
= 1 if 1 < x <= 1
= x if x > 1
Find the domain and range of function f and graph it.
Solution to Example 9:
The domain of f is the set of all real numbers.
In the interval ( inf , 1], the graph of f is a horizontal line y = f(x) = 1. Closed point at x = 1 since interval closed at x = 1.
In the interval (1 , 1] the graph is a horizontal line. There should a closed point at x = 1 but read below.
In the interval (1 , + inf) the graph is the line y = x. There should an open point at x = 1 since the interval is open at x = 1. But a closed point (see above) and an open point at the same location becomes a "normal" point.
From the graph of f shown below, we can observe that function f can take all real values on {1} U [1 , + inf) which is the range of function f.
More references and links on graphing.
Graphing Functions
