1.8 Rational Functions and Zeros

Okay, we’re slowly settling into rational functions. Plus, we know their end behavior when one (or neither) polynomial dominates the other! These fellas are more complicated than polynomial functions, though. How do we find their zeros (or roots), then? 🌳

😎 Real Zeros

The real zeros of a rational function correspond to the real zeros of the numerator for such values in its domain. This means that if we want to find the real zeros of a rational function, we only need to look at the zeros of the polynomial in the numerator. 0️⃣

To understand this concept, let's consider an example. Suppose we have the rational function:

$r(x) = (x² - 4) / (x - 2)$

The numerator of this function is a polynomial of degree 2, which we can factor as:

$r(x) = [(x - 2)(x + 2)] / (x - 2)$

Now, we can see that the denominator has a zero at x = 2, which means that this value is not in the domain of the function. Therefore, we need to exclude it from our analysis. Now, we can simplify the function as:

$r(x) = x + 2$

From this form, we can easily see that the real zero of this rational function is -2, which corresponds to the real zero of the polynomial in the numerator, x + 2. Yay! 🎉

In general, we can use the same process to find the real zeros of any rational function:

1. Factor the polynomial in the numerator and denominator.
2. Exclude any values from the domain where the denominator is equal to zero.
3. Simplify the function and identify the real zeros of the polynomial in the numerator.

Zeros as Endpoints

The real zeros of both polynomial functions of a rational function r play an important role in determining the behavior of the function, specifically in relation to the inequalities $r(x) ≥ 0$ or $r(x) ≤ 0$. These zeros can either be endpoints of intervals or asymptotes, and they can help us determine where the function is positive or negative. 🐎

Let's break down this statement further! 🔨

First, we know that the real zeros of a polynomial function are the values of x for which the function equals zero. In the context of a rational function r, we have a numerator and a denominator polynomial. Each of these polynomials has real zeros, and the zeros of the numerator are particularly important in determining the real zeros of the rational function.

Next, we have to consider the rational function inequalities $r(x) ≥ 0$ or $r(x) ≤ 0$. These inequalities describe the intervals of x-values where the function is positive or negative, respectively. To determine these intervals, we need to look at the sign of the function on each side of the zeros of the polynomial. 👀

If a zero of the polynomial is an endpoint of an interval, then we need to determine the sign of the function at that endpoint. If a zero is an asymptote, then we need to determine the sign of the function on each side of the asymptote.

Zeros as Endpoints: Walkthrough

Let's consider an example to illustrate this concept. Suppose we have the rational function:

$r(x) = (x² - 4) / (x - 2)$

The numerator of this function has real zeros at $x = -2$ and $x = 2$. The denominator has a zero at $x = 2$, which means that the function is undefined at that value. Therefore, we exclude x = 2 from our analysis.

Now, we need to determine the sign of the function on either side of each of the zeros of the numerator. Let's start with $x = -2$. We evaluate the function at a value slightly less than -2, say $x = -3$:

$r(-3) = [(-3)² - 4] / (-3 - 2) = 5 / 5 = 1$

This tells us that the function is positive to the left of $x = -2$. Similarly, we can evaluate the function at a value slightly greater than -2, say $x = -1$:

$r(-1) = [(-1)² - 4] / (-1 - 2) = -3 / -3 = 1$

This tells us that the function is positive to the right of $x = -2$ as well. Therefore, $x = -2$ is not a zero of $r(x) ≤ 0$ or $r(x) ≥ 0$. 🙅🏽

Now, let's consider $x = 2$, which is an asymptote of the function. As x approaches 2 from the left (i.e., $x < 2$), the function becomes very large and negative. As x approaches 2 from the right (i.e., $x > 2$), the function becomes very large and positive. Therefore, we can conclude that the function is negative to the left of $x = 2$ and positive to the right of $x = 2$. This means that $x = 2$ is a zero of $r(x) ≤ 0$ and a non-zero of $r(x) ≥ 0$. 🍪

Image Courtesy of CK-12