2 Rational functions

A rational function is formed by dividing one polynomial by another. Examples include

$R_1\left(x\right)=\frac{x+6}{x^2+1},R_2\left(t\right)=\frac{t^3-1}{2t+3},R_3\left(z\right)=\frac{2z^2+z-1}{z^2+z-2}$

For convenience we have labelled these rational functions $R_1$ , $R_2$ and $R_3$ .

The graphs of rational functions can take a variety of different forms and can be difficult to plot by hand. Use of a graphics calculator or computer software can help. If you have access to a plotting package or calculator it would be useful to obtain graphs of these functions for yourself. The next Example and two Tasks allow you to explore some of the features of the graphs.

Example 17

Given the rational function $R_1\left(x\right)=\frac{x+2}{x^2+1}$ and its graph shown in Figure 31 answer the following questions.

Figure 31 :

1. For what values of $x$ , if any, is the denominator zero?
2. For what values of $x$ , if any, is the denominator negative?
3. For what values of $x$ is the function negative?
4. What is the value of the function when $x$ is zero?
5. What happens to the function as $x$ gets larger and larger?

Solution

1. $x^2+1$ is never zero
2. $x^2+1$ is never negative, it is always positive
3. only when the numerator $x+2$ is negative which is when $x$ is less than $-2$
4. 2, because that is when the numerator $x+2=0$
5. $R_1$ approaches zero because the $x^2$ term in the denominator becomes very large. (This is seen

   by substituting larger and larger values e.g. $10,100,1000\dots$ )

Note that for large $x$ values the graph gets closer and closer to the $x$ axis. We say that the $x$ axis is a horizontal asymptote of this graph.

Answering questions such as

1. to (c) above will help you to sketch graphs of rational functions.

Task!

Study the graph and the algebraic form of the function $R_2\left(t\right)=\frac{t^3-1}{2t+3}$ carefully and answer the following questions. The following figure shows its graph (the solid curve). The dotted line is an asymptote.

Graph of $R_2\left(t\right)=\frac{t^3-1}{2t+3}$

1. What is the function value when $t=1$ ?
2. What is the value of the denominator when $t=-3∕2$ ?
3. What do you think happens to the graph of the function when $t=-3∕2$ ?

Answer

Note from the answers to parts (b) and (c) that we must exclude the value $t=-3∕2$ from the domain of this function because division by zero is not defined. At this point as you can see the graph shoots off towards very large positive values (we say it tends to positive infinity) if the point is approached from the left, and towards very large negative values (we say it tend to negative infinity) if the point is approached from the right. The dotted line in the graph of $R_2\left(x\right)$ has equation $t=-\frac{3}{2}$ . It is approached by the curve as $t$ approaches $-\frac{3}{2}$ and is known as a vertical asymptote .

Task!

Study the graph and the algebraic form of the function $R_3\left(z\right)=\frac{2z^2+z-1}{\left(z-1\right)\left(z+2\right)}$ carefully and try to answer the following questions. The graph of $R_3\left(z\right)$ is shown in the following figure.

Graph of $R_3\left(z\right)=\frac{2z^2+z-1}{\left(z-1\right)\left(z+2\right)}$

1. What is happening to the graph when $z=-2$ and when $z=1$ ?
2. Which values should be excluded from the domain of this function?
3. Substitute some values for $z$ (e.g. 10, 100 $\dots$ ). What happens to $R_3$ as $z$ gets large?
4. Is there a horizontal asymptote?
5. What is the name given to the vertical lines $z=1$ and $z=-2$ ?

Answer

The previous Examples are intended to give you some guidance so that you will be able to sketch rational functions yourself. Each function must be looked at individually but some general guidelines are given in Key Point 13.

It is particularly important for engineers to find values of the independent variable for which the denominator is zero. These values are are known as the poles of the rational function.

Task!

State the poles of the following rational functions:

1. $f\left(t\right)=\frac{t-3}{t+7}$
2. $F\left(s\right)=\frac{s+7}{\left(s+3\right)\left(s-3\right)}$
3. $r\left(x\right)=\frac{2x+5}{\left(x+1\right)\left(x+2\right)}$
4. $f\left(x\right)=\frac{x-1}{x^2-1}$ .

In each case locate the poles by finding values of the independent variable which make the denominator zero:

Answer

If you have access to a plotting package, plot these functions now.

Exercises

1. Explain what is meant by a rational function.
2. State the degree of the numerator and the degree of the denominator of the rational function $R\left(x\right)=\frac{3x^2+x+1}{x-1}$ .
3. Explain the term ‘pole’ of a rational function.
4. Referring to the graphs of $R_1\left(x\right),R_2\left(t\right)$ and $R_3\left(z\right)$ (on pages 66 - 68), state which functions, if any, are one-to-one and which are many-to-one.
5. Without using a graphical calculator plot graphs of $y=\frac{1}{x}$ and $y=\frac{1}{x^2}$ . Comment upon whether these graphs are odd, even or neither, whether they are continuous or discontinuous, and state the position of any poles.

Answer