Definition 8.0.1.
A rational function is a function \(f(x)\) that can be written in the form
\begin{equation*}
f(x)=\frac{M(x)}{N(x)}
\end{equation*}
where \(M(x)\) and \(N(x)\) are both polynomial functions and \(N(x) \neq 0\text{.}\)
Chapter 8 LO8: Rational Functions
In LO 7, we learned about polynomial functions. More specifically, we learned how to find and describe the end behavior of polynomial functions, find \(x\)-intercepts and \(y\)-intercepts, and determine their behavior (bouncing off the \(x\)-axis or crossing through the \(x\)-axis). In LO 8, we will learn about what happens when we divide one polynomial function by another.
An important question we should ask ourselves is this: Why are we focusing on division but ignoring addition, subtraction, and multiplication? Let’s look at some examples, using \(f(x)=x^2+3x\) and \(g(x)=x^2-1\text{.}\)
If we add these two functions, we get...
\begin{align*}
f(x)+g(x) \amp = (x^2+3x)+(x^2-1) \\
\\
\amp =x^2+3x+x^2-1 \\
\\
\amp =2x^2+3x-1
\end{align*}
This shows that the result of adding two polynomial functions is just another polynomial function, and we already know how to handle polynomial functions.
The same thing happens when we subtract polynomial functions...
\begin{align*}
f(x)-g(x) \amp =(x^2+3x)-(x^2-1) \\
\\
\amp = x^2+3x-x^2+1 \\
\\
\amp =3x+1
\end{align*}
... or multiply polynomial functions:
\begin{align*}
f(x) \cdot g(x) \amp = (x^2+3x)(x^2-1) \\
\\
\amp = x^4+3x^3-x^2-3x
\end{align*}
When we add, subtract, or multiply two polynomial functions together, what we get as a result is just another polynomial function, so we don’t need any new tools to handle these.
But let’s look at what happens when we divide \(f(x)\) by \(g(x)\text{:}\)
\begin{equation*}
\frac{f(x)}{g(x)}=\frac{x^2+3x}{x^2-1}
\end{equation*}
We really haven’t encountered a function that looks like this before, so we’ll need to investigate this kind of function and figure out how to handle it.
The name rational functions is based on the root word ratio, which means fraction.
Rational functions are a special type of fraction in which both the numerator and denominator of the fraction are polynomials.
In LO 8, we will learn how to determine the long-run and short-run behavior of rational functions, we will learn how to sketch graphs from rational function equations, and we will learn how to write rational function equations from graphs.
