Fractional functions

A fractional function is a synonym for a rational function.

That is, a function written as:

 

 \( \large f(x) = \frac{P(x)}{Q(x)} \)

 

When analyzing a fractional function, one typically proceeds in this order:

First find the domain, then the zeros, then the intersection with the y-axis, and finally the asymptotes.

This gives a coherent description of the graph.

 

Domain:

The domain is the set of all allowed x-values for the function.

For a fractional function, the domain is all real x where \(\large Q(x)\neq 0 \).

If both numerator and denominator become 0 at the same x (common factor), the point is still excluded from the domain and appears as a hole in the graph.

If \(\large Q(x)=0 \) but \(\large P(x)\neq 0 \), one typically gets a vertical asymptote.

 

Zeros:

The graph has zeros at the points where the numerator is 0, provided that the denominator is not also 0 at the same point.

 

Intersection with the y-axis:

Found by inserting \( \large x = 0 \), if this lies within the domain.

 

Asymptotes:

Vertical asymptotes typically occur at the \( \large x \)-values where the denominator is 0. Horizontal or oblique asymptotes are found by comparing the degrees of numerator and denominator or by polynomial division.

 

An example is the function:

 

$$ \large f(x) = \frac{x^2 - 1}{x^2 - 4} $$

 

Here the numerator is \( \large x^2 - 1 \), which gives zeros at \( \large x = 1 \) and \( \large x = -1 \).

The denominator \( \large x^2 - 4 \) becomes 0 at \( \large x = -2 \) and \( \large x = 2 \), so the function is not defined there.

 

The domain is therefore all real \( \large x \) except \( \large -2 \) and \( \large 2 \). The graph has vertical asymptotes at \( \large x = -2 \) and \( \large x = 2 \). 

Since the degree of numerator and denominator is the same, a horizontal asymptote is obtained at \( \large y = 1 \). The intersection with the y-axis is found at \( \large x = 0 \): \( \large f(0) = \frac{-1}{-4} = \frac{1}{4} \).

 

With this procedure, one gets an overall view of the behavior of fractional functions.