Piecewise+Functions+-+B1

A piecewise function is a function thatis defined on a sequence of intervals. In other words, it is a function that is divided into pieces for a set interval of the graph. Piecewise functions are usually expressed:
 * What is a piecewise function?**

Put simply, piecewise functions are the functions you have already studied. The unique thing about piecewise functions is that several different types of standard functions (linear, quadratic, absolute value, etc.) are contained in one piecewise function.
 * How is a piecewise function similar to (or different from) the functions you have already studied?**


 * Here's a link with an easy to understand tutorial on how to graph a piecewise function (don't mind the accent)**
 * [|http://www.youtube.com/watch?v=9aXoOhNK_-s]**


 * Given a graph, how do you write the expression for a piecewise function?**
 * 1) To express a piecewise function, you must identify each section of the graph. In this case the open and closed green dots mark the start and finish of each function.
 * 2) Going from left to right find the equation of each section as if it were by itself. This graph includes: y=x+2, y=x^2 and y=3.
 * 3) The next step is to express the interval of the graph as a whole that each individual piece takes up. Remember: a closed circle indicates inclusive in the function, and a closed circle indicates non-inclusive. If a piece has one side of it without a dot or circle, it indicates infinity. The intervals for this graph are (Infinity, -1], [-1,2], (2,infinity)
 * 4) Now, match the intervals with thier equations and express them in the following way:




 * How do you determine the domain of a piecewise functions?**


 * Ex:**



Definitions

Domain: The set of all possible input values, x. Range: The set of all possible output values, y.

1. To find the domain of a piecewise function you must first find what values of x do NOT exist. In the above example x does not exist from -2 to 0. Therefore the domain, in integral notation, is (-infinity, -2), (0, infinity).

2. To find the range of a piecewise function you again must find what values for y do not exist. Looking at the example you must figure out where each equation begins and ends in relation to the y-axis. The graph shows that between -2 and -1 there are no y values. Therefore the domain in integral notation is (-infinity, -2), (-1, infinity).

3. Often times piecewise functions are missing __pieces__ of its domain and range. As long as the graph passes the vertical line test it is still a function.


 * How to write piecewise functions from an absolute value graph?**

How to write piecewise functions from an absolute value graph:


 * The easiest way to create a piecewise function from an absolute value graph is divide the graph into two parts.




 * Here we can see that the graph is actually two lines that intersect at a given point
 * One line has a hole at the given value while the other line ends at the same value
 * Therefore, find the slope and equation of each line and write it as shown:
 * o As we can see in the above graph, Line one has an equation of
 * o Line two has an equation of
 * We then put the equation together into a piecewise function that looks like:


 * Why is an absolue value graph a piecewise function?**

In the graph of f(x)=abs(x), there are two lines connected at the origin, rather than just one graph. Since the value of y can never be negative, it stays above the x axis. So the graph is the combination of f(x)=-x when x<0, and f(x)=x when x>0.

[]
 * Sources:**