Rational Functions

What is a Rational Function?

Rational functions are functions that can be expressed as the ratio of two polynomials. A function that is rational follows the form:

[latex]\Large{f(x) = \frac{p(x)}{q(x)}}[/latex]

Where p(x) and q(x) are polynomials. An example of this kind of function is shown in figure 3, which can be represented as:

[latex]\Large{f(x) = \frac{x-1}{x-3}}[/latex]

How can we find the domain and range of rational functions? Well, if you recall, the domain of a function is all values of x which give a defined value of y. The x-values for which these kind of functions are undefined are called vertical asymptotes.

Knowing this, we can say that the denominator of a rational function cannot be equal to zero, since anything divided by zero is undefined. So, the domain of a rational function is all real numbers excluding the vertical asymptotes.

The range of a function that is rational is a bit more difficult to determine, since we need to find the horizontal asymptotes (HA) of the function.

Note:

There are 3 cases that we may come across:

1. If N > D then, HA at y = 0

2. If N = D then, HA at y = ratio of leading coefficients

3. If N < D then, HA does not exist

WHERE:
N - Degree of numerator
D - Degree of denominator

Example 1.5

Find the vertical and horizontal asymptotes for the function f(x) = (x-1) ⁄ (x-3). State the domain and range.

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