Section 1: Piecewise-Defined Functions

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 4.1 Piecewise-Defined Functions 335 Version: Fall 2007 4.1 Piecewise-Defined Functions In preparation for the definition of the absolute value function, it is extremely important to have a good grasp of the concept of a piecewise-defined function. However, before we jump into the fray, let’s take a look at a special type of function called a constant function . One way of understanding a constant function is to have a look at its graph. l⚏ Example 1. Sketch the graph of the constant function f ( x ) = 3 . Because the notation f ( x ) = 3 is equivalent to the notation y = 3 , we can sketch a graph of f by drawing the graph of the horizontal line having equation y = 3 , as shown in Figure 1 . x 10 y 10 f ( x ) = 3 Figure 1. The graph of a constant func- tion is a horizontal line. When you look at the graph in Figure 1 , note that every point on the horizontal line having equation f ( x ) = 3 has a y -value equal to 3. We say that the y -values on this horizontal line are constant , for the simple reason that they are constantly equal to 3. The function form works in precisely the same manner. Look again at the notation f ( x ) = 3 . Note that no matter what number you substitute for x in the left-hand side of f ( x ) = 3 , the right-hand side is constantly equal to 3. Thus, f ( 5) = 3 , f ( 1 / 2) = 3 , f ( 2) = 3 , or f ( π ) = 3 . The above discussion leads to the following definition. Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/ 1
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336 Chapter 4 Absolute Value Functions Version: Fall 2007 Definition 2. The function defined by f ( x ) = c , where c is a constant (fixed real number), is called a constant function . Two comments are in order: 1. f ( x ) = c for all real numbers x . 2. The graph of f ( x ) = c is a horizontal line. It consists of all the points ( x, y ) having y -value equal to c . Piecewise Constant Functions Piecewise functions are a favorite of engineers. Let’s look at an example. l⚏ Example 3. Suppose that a battery provides no voltage to a circuit when a switch is open. Then, starting at time t = 0 , the switch is closed and the battery provides a constant 5 volts from that time forward. Create a piecewise function modeling the problem constraints and sketch its graph. This is a fairly simple exercise, but we will have to introduce some new notation. First of all, if the time t is less than zero ( t < 0 ), then the voltage is 0 volts. If the time t is greater than or equal to zero ( t 0 ), then the voltage is a constant 5 volts. Here is the notation we will use to summarize this description of the voltage. V ( t ) = 0 , if t < 0 , 5 , if t 0 (4) Some comments are in order: The voltage difference provide by the battery in the circuit is a function of time. Thus, V ( t ) represents the voltage at time t . The notation used in ( 4 ) is universally adopted by mathematicians in situations where the function changes definition depending on the value of the independent variable. This definition of the function V is called a “piecewise definition.” Because each of the pieces in this definition is constant, the function V is called a piecewise constant function.
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