Ex.1) 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.
