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.
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