Introduction
We have previously discussed the function . Note that both the numerator and the denominator are polynomials ( is a constant polynomial). We can study what happens when we replace those with arbitrary polynomials.
- (a)
- .
- (b)
- .
- (c)
- .
It is not a rational function, because the denominator is not a polynomial. Recall that a square root cannot distribute over addition or subtraction, so . This is a common error to make when first learning about rational functions, so be sure to watch out for it! Note that even though this does not define a rational function, it is defined for all possible values of , since for all . - (d)
- A polynomial function .
- (e)
- .
Domains of Rational Functions
While we said it already, it is worth emphasizing that a rational functions is not defined when the polynomial in the denominator is equal to zero. That is, if is a rational function and both and are polynomials, than the domain of is all values of except those where . Notice, that finding the domain of a rational function is going to require finding the -intercepts of a polynomial!
Therefore, and are not in the domain of our function . We can write this domain in interval notation as . From the fact that the the domain comes in four pieces, we can know that the graph of the rational function will also have four pieces. Here is what this function looks like when graphed:
That looks quite different from the graphs of functions we have studied thus far! We will continue to explore the behavior of rational functions in this unit so that we can understand and predict this graph shape.
Combining Rational Functions
When we add, subtract, multiply, or divide rational functions, we get another rational function. Let’s see why.
Adding and Subtracting Rational Functions
Given the rational functions and , we can rewrite them with a common denominator: and . Then, yielding another rational function, since both the numerator and denominator are polynomials.
This idea can be expanded to the sum of any two rational functions. Given two rational functions and , with , , , and being polynomials, then which has polynomials as its numerator and denominator and is therefore a rational function.
Furthermore, by replacing above with , we can see that subtracting two rational functions also yields a rational function. Note that the sum and difference of two rational functions are only defined when both rational functions are defined.
Multiplying and Dividing Rational Functions
Let’s look at multiplication. Given the rational functions and , we can write their product as , which has polynomials for its numerator and denominator, and is thus a rational function again.
Given two rational functions and , with , , , and being polynomials, then which has polynomials as its numerator and denominator and is therefore a rational function.
Furthermore, by swapping the roles of and , we can see that dividing two rational functions also results in a rational function. Note that the product of two rational functions is only defined when both rational functions are defined. In addition, the quotient of two rational functions is only defined when the dividend and the reciprocal of the divisor are defined.