Rational functions can exhibit puncture points (also called ‘holes’).
As you'll learn in Calculus, puncture points are an example of a ‘removable discontinuity’.
Here's the idea.
The function $\displaystyle R(x) := \frac{x^3-2x^2}{x-2}\,$ certainly
looks like a typical rational function.
Upon closer inspection, though, we see that there's an extra factor of
$\,1\,$ in the formula:
$$
\frac{x^3 - 2x^2}{x - 2} \quad = \quad \frac{x^2(x-2)}{x-2} \quad = \quad x^2\cdot\frac{x-2}{x-2}
$$
Therefore, $\,\displaystyle R(x) = \frac{x^3-2x^2}{x-2}\,$ has exactly the same outputs as
the much simpler function, $\,P(x) := x^2\,$,
except that the function
$\,R\,$ it isn't defined when $\,x = 2\,$.
The graphs of both $\,P\,$ and $\,R\,$ are shown below—the puncture point (hole) in $\,R\,$ is caused by that extra factor of $\,1\,$.
$P(x) = x^2$
|
$\displaystyle R(x) = \frac{x^3 - 2x^2}{x-2} = x^2\cdot\frac{x-2}{x-2}$
|
Puncture points are studied in more detail in a future section.
by practicing the exercise at the bottom of this page.
When you're done practicing, move on to:
Finding Vertical Asymptotes