Drawing piecewise defined functions

masterwu (59)in #steemiteducation • 7 years ago

One of the quirky aspects of functions is they don't have to be defined by just a single expression, nor do they have to be continuous. They can be a combination of 2 or more expressions, and those expressions don't even have to be unique.

For example, let's take a look at the function...

At first glance, it looks like we can simplify this equation as the numerator is the difference of 2 squares. And thus...

So it looks like out chart is going to be a simple straight line, right?...

Or is it?

What happens at x = 2? The denominator in the original equation equals 0.

What is division by 0? Division by 0 in mathematics is undefined, and as such, the function at x = 2 is also undefined. So we have a "hole" in the function at this point.

Graphically, we draw this function as follows...

Figure 1.
Created with: www.desmos.com/calculator

And hence, the function can be written or interpreted as...

Or y = x + 2 everywhere, except for when x = 2, where we have a point discontinuity. We have one piece (y = x + 2) to the left of the discontinuity, and we have another piece (y = x + 2) to the right of the discontinuity.

Let's have a look at graphing another piecewise defined function. For example...

Now, when we overlay the two functions, we have the 2 functions intersecting each other...

Figure 2.
Created with: www.desmos.com/calculator

Now, at x = -1, we are told that y = 2 + (-1) = 1 according to the prescribed conditions. Thus we have a function that is defined for the domain...

Applying the conditions, we can plot the piecewise defined function as follows...

Figure 3.
Created with: www.desmos.com/calculator

The function shown in Figure 3 is what we call piecewise continuous, since we can draw the entire function without metaphorically lifting the pencil or pen off the sheet. However it is not a smooth function since it has a clear edge at x = -1.

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