Rational Functions and Their Roots

Objectives

At the end of this lesson, you should be able to

• Identify rational functions.
• Obtain the roots of rational functions.
• Obtain the asymptotes of rational functions.
• Identify the behavior of the rational functions near the asymptotes.
• Explain the changes of the signs of the rational functions.

Definition

It is important to note that for the function there exists Q(x) should be distinct from 0.

Examples:

The following are rational functions:

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

On the other hand, the funciton:

$f\left(x\right)=\frac{x}{\sqrt[2]{x}}$

is not a rational function therefore the denominator is not a polynomial.

Roots of a Rational Function

The previous signifies that, to find the roots of the polynomial function f, we have to solve the equation f(x)=0. So that the function exists, the denominator should be distinct from zero. In general to find the roots of the polynomial function $f\left(x\right)=\frac{\text{P(x)}}{\text{Q(x)}}$ , si P(x) y Q(x) you don't have a common factor, it is sufficient to solve P(x)=0.

Example 1:

Find the roots of the function $f\left(x\right)=\frac{x^3+x^2-2x}{2x^2-x-6}$

Remember that the denominator cannot be the same as zero. In general, to find the roots of the rational function it is only necessary to find the roots of the numerator. Factorizing the numerator we find:

$f\left(x\right)=\frac{x\left(x-1\right)\left(x+2\right)}{2x^2-x-6}$

In general:

The roots of the function $f\left(x\right)=\frac{x^3+3x^2-2x}{2x^2-x-6}$ are x=0, x=1 and x=-2

You can see these roots observing the graph of this function in the following:

Example 2:

Find the roots of the function $f\left(x\right)=\frac{x^3+5x^2+4x}{x^2+x-2}$

Factoing the numerator in the expression we find that:

$f\left(x\right)=\frac{x\left(x+1\right)\left(x+4\right)}{x^2+x-2}$

Like the denominator cannot be equal to zero. The function can be zero if one of the factors of the numerator is zero:

The roots of the function $f\left(x\right)=\frac{x^3+5x^2+4x}{x^2+x-2}$ are x=0, x=-1 y x=-4

You can visualize these roots observing the graph of this function, seen below:

Example 3:

Find the roots of the function $f\left(x\right)=\frac{x^2+1}{x-1}$

The zeros of the rational function are the zeros of the numerator. We note that in this case, the numerator does not have real zeros. Then,

$\begin{array}{c}\left(x^2+1\right)=0\\ x^2=-1\\ x=\pm \sqrt{-1}\\ x=\pm i\end{array}$

The roots of the function $f\left(x\right)=\frac{x^2+1}{x-1}$ are x=i and x=-i

The graph of this function is the following, like we saw that function does not touch the axis x:

Click on the following link to practive the form of finding the roots of a polynomial:

Asymptotes

We have seen that the roots of the numerator correspond to the roots of the rational function.

Example 1:

Consider the function $f\left(x\right)=\frac{1}{x-1}$. How does the function near the roots of the denomator act?

Equating the denominator to zero to find the roots, we find:

$\begin{array}{c}\left(x-1\right)=0\\ x=1\end{array}$

We see how the function behaves for values of x near the root of the denominator x=1.

We see that when x is close to 1 at the left the value of the function decreases rapidly. When x is near to 1 at the right, the value of the function increases rapidly.

In the lesson Graphs of Rational Functions we see with detail the effect in the graph of the function of the roots of the denominator. Observe the graph of the function in the close value to x=1.

Example2:

Consider the function $f\left(x\right)=\frac{1}{\left(x-1\right)\left(x-2\right)}$ . How does the function behave near the roots of the denominator?

Equating the denominator to zero to find their roots, we find:

We see how the function behaves for values of x near the root of the denominator x=1.

We see that when x is near to 1 at the left the value of the function increases rapidly. When x is near to 1 at the right, the value of the function decreases rapidly

We see how the function behaves for values of x near the root of the denominator x=2.

We see that when x is near to 2 at the left the value of the function decreases rapidly. When x is near to 2 at the right, the value of the function increases rapidly.

Observe the graph of the function in the values near to x=1 y x=2.

Changes of sign of rational functions

One rational function can change the sign of their roots and near the values for those that are not defined, that is to say, near the roots of the denominator.

Example 1:

Consider the rational function:$f\left(x\right)=\frac{x+1}{x-1}$ .

To obtain all the possible changes of signs, we find the roots of the numerator and the denominator:

Roots of the numerator:x=-1

Roots of the denominator:x=1

All these values determine the intervals in those that the function can change the sign. Near the asymptotes we take two values that are near the right and near the left.

The graph of this rational function is the following. Observe how the graph of the function is below the axis x in the intervals where we obtained the negative sign and below the axis x in the intervals where we found the positive sign.

Example 2:

Consider the ratinoal function:$f\left(x\right)=\frac{x^3+3x^2+2x}{x-1}$ .

To obtain all the possible changes of signs, we factor the numerator and the denominator:

$f\left(x\right)=\frac{x\left(x+1\right)\left(x+2\right)}{\left(x-1\right)}$

Roots of the numerator: x=-2, x=-1 y x=0

Roots of the denominator:x=1

All these values determine the intervals in the function that can change the sign. Near the asymptotes we take two values that are near the right and the left.

The graph of this rational function is the following. Observe how the graph of the function is below the axis x in the intervals where we find the negative sign and above the axis x in the intervals where we find the positive sign.

Summary

Now that we have completed this lesson, you should be able to:

• Identify rational functions.
• Obtain the roots of rational functions.
• Obtain the asymptotes of rational functions.
• Identify the behavior of the rational functions near the asymptotes.
• Explain the changes of signs of the rational functions.