Lesson Rational Expressions

Discontinuities of a Rational Function
Similar to the maze game, graphs of rational functions can have a hole, or break, in the graph which is known as a point of discontinuity. Let's graph the rational function f(x) = using GeoGebra.
1.Open Geogebra.
2.In the "View" menu, make sure that "Axes," "Grid," "Algebra View," and "Input Bar" are all checked.
3.Enter the function f(x) = (x2 + 2x)/(x + 2) in the "Input" field. You must include the parentheses around the expression in the numerator and denominator. Otherwise, the incorrect function will be graphed.

4.After entering the function, a graph will appear.


But wait! It appears as if this is a linear function. Not only that, there does not appear to be any breaks in the function. Let's simplify the rational expression to see what happens.


f(x)
=



Factor the GCF from the numerator.


f(x)
=



Cancel the common factors between the numerator and denominator.



f(x)

=





f(x)
=
x

If the function notation is replaced by y, it is easier to see that the graph of y = x is almost identical to the graph f(x) = . But what is the difference?
Recall that there are restrictions placed on a rational function since a fraction may never have a denominator equal to 0. If it does, the fraction is undefined. What value for x makes the function f(x) = undefined?
The denominator of x + 2 cannot equal 0. This means that x cannot equal –2.


x + 2
≠
0


–2

–2


x
≠
–2

Take another look at the graph of the function. At the x-coordinate –2, there is a corresponding y-coordinate of –2. Select the "New Point" icon, , and try to place a point at (–2, –2). What happens?


Notice that a point does not appear. But there is a new "Dependent Object" in the Algebra Window on the left, labeled "undefined." This indicates that the point (–2, –2) does not exist on the graph and is, therefore, a point of discontinuity! So even though the graph may look like there are no holes, the algebra tells
us otherwise!
Let's indicate the discontinuity on the graph by typing in (–2, –2) in the Input field.


A closed circle appears. However, you know that this point does not exist on the graph. You also know from your knowledge of inequalities that an open circle represents a point not included on a graph, while a closed circle represents a point that is included. Turn this circle into an open circle by following these steps:
1.Right-click on the point and scroll down to "Object Properties." Under the "Show Label" drop-down menu you can change the label to reflect the value of the point instead of a name.

2.Under the "Style" tab, change the "Point Style" to an open circle.


The correct graph of f(x) = is the line f(x) = x with an open circle at (–2, –2).

The zeros of the function can be found by substituting 0 for f(x) in the simplified rational function...

f(x) = x
0 = x
...or by identifying the x-intercept on the graph of the function.


In either case, the zero of the function is zero!
Take a look at a couple more examples to be sure you understand these concepts.