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In addition to the revision notes for Piecewise Functions on this page, you can also access the following Functions learning resources for Piecewise Functions
| Tutorial ID | Title | Tutorial | Video Tutorial | Revision Notes | Revision Questions |
|---|---|---|---|---|---|
| 16.5 | Piecewise Functions |
In these revision notes for Piecewise Functions, we cover the following key points:
By definition, a piecewise-defined function is a special type of function that is described not by a single equation, but by two or more ones.
The general structure of a piecewise function made by two equations is shown below.
In a certain sense, piecewise functions are composite functions connected in series (the combination takes place in a sequential way, i.e. one after another). The term "piecewise", means "one piece at a time".
The components of a piecewise function are analysed separately in their corresponding part of the domain. This action also includes plotting the graph for each part separately. When we are asked to find the value of a piecewise function at a given point, we consider only the equation that corresponds to the part of the domain this point belongs.
The domain of a piecewise function is obtained by taking the total of all intervals or segments that represent the individual domain of each function's piece.
As for the range, first, we identify the ranges of each part of the function. These ranges are eventually combined to form the overall range.
Some functions are undefined for certain values of the independent variable x. The graphs of such functions contain two different parts that have no common points, i.e. they are not connected with each other.
Basically, there are two types of functions in regard to their graph: continuous and non-continuous. By definition, a function is continuous if its graph has no interruptions. In other words, if you can plot a graph without taking your hand off the sheet, then you are dealing with the graph of a continuous function. If you have no other choice but to sketch the graph in two or more steps by taking off the hand from the sheet, then you are dealing with a non-continuous function.
You can check the continuity of a piecewise function by finding its value at the boundary (limit) point x = a. If the two pieces give the same output for this value of x, then the function is continuous.
Piecewise functions may contain more than two parts but the approach used is the same as when they have two pieces only. The only difference is that you must have to check the function's continuity (if required) at more than one point.
As for the graph, you must sketch the graph of each individual function in its domain and then look for any common point. It is worth pointing out the fact that if any piece of such function does not include an endpoint, we represent this endpoint on the graph through a blank dot. However, if the function is continuous, this blank dot is filled by the endpoint of the other piece of the original function.
Some piecewise functions have no variable in one or more parts of them. This means that part of the graph is a constant function. The corresponding graph of that part is a horizontal line. All inputs in that part of the function give the same output, so we don't have to find the limit values, as all of them are the same.
The absolute value of a number shows its distance from the origin regardless of the direction in the number axis. Therefore, the absolute value of a number is always positive, as it shows only how far from the origin a number is, regardless of the direction.
Since the expression inside the symbols of absolute value can take two values - one positive and one negative - an absolute value function will split into two different parts according to the scheme below
where a is the value which makes the expression equal to zero. Therefore, absolute value functions can be considered as piecewise functions, given that they can be written in two parts.
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