Mathematical Practices
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Note that a function is continuous if its graph has no breaks, holes, or gaps.
No, see solution.
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)=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.
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 2ND and WINDOW. Then, we change the settings as follows.
Having changed the settings, we push 2ND 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.
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.