INTRODUCTION TO RATIONAL FUNCTIONS

LESSON READ-THROUGH
by Dr. Carol JVF Burns (website creator)
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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 $$\cssId{s14}{f(x) = \frac{P(x)}{Q(x)}}$$ 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: $$ \begin{gather} \cssId{s21}{2x + 3 = 0}\cr \cssId{s22}{2x = -3}\cr \cssId{s23}{x = -\frac{3}{2}} \end{gather} $$ 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: $$\cssId{s28}{\text{dom}(f) = \{x\ |\ x\ne -\frac{3}{2}\}}$$ 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 alternatively write: $$ \cssId{s34}{\text{dom}(f) = (-\infty,-\frac{3}{2}) \cup (-\frac{3}{2},\infty)} $$

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.
PROBLEM TYPES:
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AVAILABLE MASTERED IN PROGRESS

(MAX is 20; there are 20 different problem types.)