We will determine whether the following function is continuous or discontinuous using a graphing calculator. Note that a function is continuous if its graph has no breaks, holes, or gaps.

f (x) = \frac{x ^{2} - x}{x}

Let's draw the graph of the function! To draw a graph on a calculator, we first press the

Y =

button and type the function in one of the rows. Having written the function, we can push

GRAPH

to draw it.

When we draw a graph on a graphing calculator,the default viewing window is the standard viewing window, which has the following settings.

However, to see if the graph has a hole, the dimensions of the window must be proportional with the dimensions of the calculator screen in terms of the number of pixels. For our screen, which is $63$ pixels high and $95$ pixels wide, we use a viewing window setting as follows.

Even if it is hard to see there is a hole at $(0, - 1),$ so the function is discontinuous. We can also verify this by using the table feature of the calculator. To do so, we push $2 ND$ and $WINDOW .$ Then, we change the settings as follows.

Having changed the settings, we push $2 ND$ and $GRAPH$ to get a table of values of the function.

Since the function is undefined at

x = 0,

we can verify that it is discontinuous.

Extra

Determining a Viewing Window

To think about how to determine a viewing window size, let's consider the following viewing window — which is $63$ pixels high and $95$ pixels wide.

In this screen, there are $47$ pixels on both sides of the $y -$ axis. Note that the $y -$ axis is also counted as $1$ pixel. Therefore, in total, there are $95$ pixels horizontally. By the same logic, there are $63$ pixels vertically.

With the viewing window settings we can conclude that each pixel maps to a single integer value on each axis. Thus, to see the pixels at the corresponding $x -$ values, the dimensions of the window must be proportional with the dimensions of the screen in terms of the number of pixels.

Exercise

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