Piecewise+functions

 __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.

http://www.analyzemath.com/Graphing/piecewise_functions.html [|Try these]