INTRODUCTION TO RATIONAL FUNCTIONS

Polynomials aren't immediately useful for modeling the following behaviors:


outputs that approach a specific real number
as $\,x\rightarrow\infty\,$ (or $\,x\rightarrow -\infty$)

Why not?
For non-constant polynomials,
when inputs get arbitrarily large,
the outputs always get arbitrarily large.


outputs that ‘blow up’ (go to $\,\pm\infty\,$)
as the input approaches a finite number

Why not?
The only time that polynomial outputs
can get arbitrarily large
is when $\,x\rightarrow\pm\infty\,$.

Polynomials can still be useful in modeling these situations—we just need to allow them in the denominator!
This produces a rational function, which is the subject of this section.

RATIONAL FUNCTION definition; domain
A function is a rational function if and only if it can be written as a ratio of polynomials, where the denominator is not always zero.

Equivalently, a function $\,f\,$ is a rational function if and only if it can be written in the form $$f(x) = \frac{P(x)}{Q(x)}$$ for polynomials $\,P\,$ and $\,Q\,$, where $\,Q\,$ is not the zero function.

The domain of a rational function is the set of all real numbers for which the denominator is nonzero.

EXAMPLE: a rational function; its domain

The function $\displaystyle\,f(x) = \frac{x^2 - 1}{2x + 3}\,$ is a rational function, since both the numerator and denominator are polynomials.

The domain of $\,f\,$ is the set of all real numbers $\,x\,$ for which the denominator is nonzero.
There's only one place where the denominator is equal to zero: $$ \begin{gather} 2x + 3 = 0\cr 2x = -3\cr x = -\frac{3}{2} \end{gather} $$ Therefore, the number $\,-\frac{3}{2}\,$ must be excluded from the domain of $\,f\,$.

Recall that the notation ‘$\text{dom}(f)\,$’ is used to denote the domain of $\,f\,$.
The domain can be conveniently described using either set-builder notation or interval notation:

Using set-builder notation: $$\text{dom}(f) = \{x\ |\ x\ne -\frac{3}{2}\}$$ The domain of $\,f\,$ consists of two intervals of real numbers—the interval to the left of $-\frac{3}{2}\,$ and the interval to the right of $-\frac{3}{2}\,$.
Recall that the union symbol, ‘$\cup\,$’, is used to ‘combine’ two sets into a bigger set.
Thus, using interval notation, we can alternately write: $$ \text{dom}(f) = (-\infty,-\frac{3}{2}) \cup (-\frac{3}{2},\infty) $$

NOTES ABOUT RATIONAL FUNCTIONS:

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
introduction to asymptotes
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
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(MAX is 20; there are 20 different problem types.)