3.6: Rational Functions
This is the last type of function we will study in this chapter!
A rational function has the form \[r(x) = \dfrac{P(x)}{Q(x)}\]where $P$ and $Q$ are polynomials.
This one is not: \[f(x) = \dfrac{\sqrt{x} - 2}{x^4 - x^2 - 2}\] because the numerator has an exponent of $\frac{1}{2}$. Polynomials only have whole numbers for powers.
Before looking into rational functions, let's answer "how to approach a number?"
(from the left)
$x\rightarrow a^+$(from the right)
(from the left)
$x\rightarrow a^+$(from the right)
In Calculus, the first week will discuss this "approaching a number" idea in more detail.
From this example, the two behaviors \[\dfrac{1}{\text{BIG}} = small \qquad \qquad \dfrac{1}{\text{small}} = \text{BIG}\] correspond to horizontal and vertical asymptotes, respectively.
In Calculus, these are called limits! But for us, we will call them asymptotes.
Vertical/Horizontal Asymptotes
The line $y = b$ is a horizontal asymptote if $y \rightarrow b$ as $x \rightarrow \pm \infty$.
Finding asymptotes for rational functions is straightforward.
- To find vertical asymptotes, set denominator $= 0$ and solve for $x$.
- To find horizontal asymptotes, there are three cases, depending on the leading terms:
- If $n < m$, then $y = 0$ is the horizontal asymptote.
- If $n = m$, then $y = \dfrac{a_n}{b_m}$.
- If $n > m$, there are no horizontal asymptotes.
Here's what the graph looks like:
Notice how the horizontal asymptote can be crossed, but the vertical ones cannot.
Graphing this example is difficult by hand. You won't be expected to do so.
However, you will need to know how to graph rational functions which have holes by hand.
Holes
We first saw what holes are back in Section 2.7.
Holes are the zero of the common factor in the numerator and denominator.