• We will be studying the behaviour of simple rational functions.

In the case where \( h(x) = k, k \in \mathbb{R}, k\neq{0} \) (i.e., a constant polynomial of degree \(0\)), the rational function reduces to the polynomial function \( f(x) = \dfrac{1}{k}g(x) \).

Examples of rational functions include:

Each consists of a polynomial in the numerator and denominator. Restrictions are stated to ensure the denominator does not equal \(0\).

Functions such as \( p(x) = \dfrac{x}{\sqrt{x} - 1} \) and \( r(x) = \dfrac{\lvert x + 1 \rvert}{x^2 + 1} \) are not rational functions, since the denominator in \( p(x) \) and the numerator in \( r(x) \) are not valid polynomials. The first rational function example, \( y = \dfrac{1}{x^2 - 2x - 3}\), is the reciprocal of the quadratic function \( y = x^2 - 2x - 3 \).

In this module, we will investigate the behaviour of the graph of rational functions of the form \( y = \dfrac{1}{f(x)} \), where \(f(x) \) is a linear or quadratic polynomial function.

How does the graph of \( y = x^2 - 2x - 3 \) relate to the graph of its reciprocal, \( y = \dfrac{1}{x^2 - 2x - 3} \)?

We begin by considering the graph of the linear function \(y=x\) and its related reciprocal function \(y=\dfrac{1}{x}\).

The function \(y=x\) has an \(x\)-intercept at \(x=0\).

The function \(y=\dfrac{1}{x}\) is undefined at \(x=0\).

The graph of \(y=\dfrac{1}{x}\) has a vertical asymptote at \(x=0\).

Recall that an asymptote is a line that the graph of a function approaches but does not touch, for some values of \(x\) in the domain of the function.

The function \( y = x \) is an increasing function for all \( x \in \mathbb{R} \).

The function \( y = \dfrac{1}{x} \) is decreasing on its domain (that is, for all \( x \in \mathbb{R}, x\neq{0} \)).

The graphs of both functions are found in the \(1^{st}\) and \(3^{rd}\) quadrants.

In other words, both functions are negative in the interval \( x \in (-\infty, 0) \) and positive in the interval \( x \in (0, \infty) \).

The graphs of these functions intersect at \((-1,-1)\) and \((1,1)\).

The function \( y = x \) has opposite end behaviours with \( y \rightarrow -\infty \) as \( x \rightarrow -\infty \) and \( y \rightarrow \infty \) as \( x \rightarrow \infty \).

The function \( y = \dfrac{1}{x} \) has a horizontal asymptote of \( y = 0 \), since \( y \rightarrow 0 \) as \( x \rightarrow \pm \infty \).

Both functions are odd functions which are symmetrical about the origin.

Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0. https://ggbm.at/rsrtg3fu

Knowing the graph of \( y = f(x) \) and understanding the general relationship between a function and its reciprocal function \( y = \dfrac{1}{f(x)} \) can help when sketching the graph of the reciprocal function.

The purpose of the investigation in this module is to identify a relationship between the key characteristics of a graph of a simple polynomial function, \( y = f(x) \), either linear or quadratic, and the graph of the related reciprocal function, \( y = \dfrac{1}{f(x)} \).