In this lesson, our instructor Vincent Selhorst-Jones teaches Piecewise Functions with analogies and many examples. He will go over the notations for these functions and give step-by-step instructions on how to graph them. Youll also learn about continuous and step functions before diving into four fully-worked out examples.
Up until now, all the functions we've seen used a single "rule" over their entire domain. No matter what goes into the function (as long as it's in the domain), the same process happens to the input. In this lesson, though, we'll examine functions where
the process changes depending on what goes into the function. This is the idea behind piecewise functions.
As a non-math example, we could imagine cooking potatoes in different ways depending on the size of the potato. If it's small, you boil and mash the potato. But if it's over a certain size, you cut it up and turn it into fries. In both cases, something
happens to the potato. But the transformation is different depending on the type.
We use the following notation for piecewise functions:
That is, given some input, we first check which category it belongs to, then use the corresponding transformation. [Check out the video for some concrete examples of this.]
We graph piecewise functions the same way as other functions: a series of points ( x, f(x) ). The difference is that the rule determining where x maps to can change depending on which x we're looking at. Often it looks like the graph "changes" at switchovers
between rules: when x switches from one category to another, the shape/location of the curve can change.
An important graphical note is that we show inclusion with a solid circle. We show exclusion with an empty circle. That way, when we have two categories like x < 1 and 1 ≤ x, we can see which curve in the graph "owns" x=1.
This is a great time to bring up the idea of a continuous function. We'll occasionally refer to this idea in this course, and it will come up often in Calculus. It's hard to formally define continuous with symbols and numbers right now,
but we can understand what it means graphically. All the below mean the same thing:
All the parts of its graph are connected.
Its graph could be drawn without ever having to lift your pencil off the paper.
There are no "breaks" in the graph.
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.