Definition

A rational function is a function of the form,

where P(x) and Q(x) are polynomials. Some examples of rational functions include:

As you may have guessed, many of the functions we have studied so far fall into the broad class of rational functions. For example, some rational functions are power functions and vice versa. Polynomial functions are special cases of rational functions where Q(x) = 1 in the above definition.

Domain and Range

The domain of a rational function depends on the denominator Q(x). In particular, the domain of

f(x) = P(x) / Q(x)

is all real numbers x such that Q(x) ≠ 0. Notice that we need to remove values of x that are zeros of Q(x) since division by zero is undefined. For example, the domain of the function
f(x) = 1/ x is {x | x ≠ 0}.

Note of caution

It is not always possible to determine the domain of a rational function. For example, if Q(x) is of high degree we generally cannot find all of its zeros.

In other cases, we can find the domain of f(x) = P(x) / Q(x) by finding the zeros of Q(x) as discussed in the sections on quadratic functions and polynomials. For example, the domain of the function,

consists of all real numbers x such that x² − 6x + 3 ≠ 0. To determine which values of x to remove we set,

x² − 6x + 3 = 0,

and solve for x using the quadratic formula as,

Thus, we conclude that the domain of,

is the set,

{x | x ≠ 3 + √6 or x ≠ 3 − √6},

or using interval notation,

(−∞, 3 − √6) ∪ (3 − √6, 3 + √6) ∪ (3 + √6, ∞) .

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In the next section we will learn about asymptotes of rational functions.

Asymptotes