Understanding Removable Discontinuities

When students learn about functions in Grade 10 Algebra II, one important idea to grasp is removable discontinuities.

But what does "removable discontinuity" mean?

A removable discontinuity happens at a point on a graph where there is a "hole." This hole is caused by a part of the function that can be canceled out. It often shows up in rational functions (those that have a fraction). Even though the function isn't defined at that hole, we can change it so that it works smoothly at that point.

How to Spot Removable Discontinuities

Here are some simple ways to help identify removable discontinuities:

1. Factor the Function

One great way to find a removable discontinuity is by factoring the function. Here’s an example:

$f(x) = \frac{x^2 - 1}{x - 1}$

First, we factor the top (the numerator):

$f(x) = \frac{(x - 1)(x + 1)}{x - 1}$

Now we can see that we can cancel the $(x - 1)$ from both the top and the bottom. This gives us:

$f(x) = x + 1$ for all $x \neq 1$

So, the function is not defined at $x = 1$, but we can choose $f(1) = 2$. This means there’s a removable discontinuity at this point.

2. Use Limits to Check Continuity

To better understand the discontinuity, students can use limits. For the function above, we calculate:

$\lim_{x \to 1} f(x) = \lim_{x \to 1} (x + 1) = 2$

Since the limit is defined, we can fill in the hole at $x = 1$ by setting $f(1) = 2$. This way, the function becomes continuous.

3. Graph the Function

Drawing helps a lot! By sketching the graph of our function, students can see the "hole" at $x = 1." A picture can show what's happening with continuity and discontinuity very clearly. The graph of$f(x)$ would look like a straight line with a gap at the point (1, 2).

4. Look at Piecewise Functions

Sometimes, functions are made up of different pieces, which can also create removable discontinuities. Here’s an example of a piecewise function:

\begin{cases} \frac{x^2 - 4}{x - 2} & \text{for } x \neq 2 \\ 3 & \text{for } x = 2 \end{cases} $$ By simplifying $g(x)$, we get: $$ g(x) = \frac{(x - 2)(x + 2)}{x - 2} = x + 2 $$ for $x \neq 2$ Once again, $g(x)$ has a removable discontinuity at $x = 2$. We check the limit: $$ \lim_{x \to 2} g(x) = 4 $$ But since $g(2) = 3$, there’s a hole at that point. If we set $g(2) = 4$, the discontinuity would go away. #### 5. Practice Makes Perfect Finally, practice is really important. Students should try solving many different functions to find removable discontinuities. By working on many examples, they can get better at factoring, using limits, and understanding graphs. ### Conclusion Understanding removable discontinuities is a key skill for grasping how functions work. By practicing techniques like factoring, using limits, graphing, and looking at piecewise functions, students can improve their understanding and confidence. With enough practice, they’ll learn how to spot and fix these discontinuities with ease!