Rational Functions and Zeros - AP PreCalculus Study Guide

The Core Idea: Rational Functions and Zeros

A rational function is a function formed by the ratio of two polynomial functions, commonly expressed as $f(x)=\frac{p(x)}{q(x)}$, where $p(x)$ is the numerator polynomial and $q(x)$ is the denominator polynomial. The fundamental concept of this topic is to identify the specific input values, $x$, for which the output of the function, $f(x)$, is equal to zero. These values are known as the "zeros" of the function.

Graphically, the zeros of any function correspond to the x-intercepts—the points where the function's graph crosses or touches the x-axis. For a rational function, the value of the fraction becomes zero only when the numerator is zero and the denominator is non-zero. This leads to a critical condition: a value $x=c$ can only be a zero of $f(x)$ if it makes the numerator zero ($p(c)=0$) without simultaneously making the denominator zero ($q(c)\ne 0$). If a value makes both the numerator and denominator zero, the function is undefined at that point, and it cannot be a zero.

Key Rules

The determination of zeros for a rational function is governed by a precise rule derived from the properties of fractions.

Definition of a Rational Function:

A rational function $f(x)$ is defined as:

$$f(x)=\frac{p(x)}{q(x)}$$

where $p(x)$ and $q(x)$ are polynomial functions, and $q(x)\ne 0$.

Rule for Finding Zeros:

The zeros of a rational function $f(x)=\frac{p(x)}{q(x)}$ are the real numbers $x=c$ that satisfy two conditions:

1. The value $c$ must be a zero of the numerator polynomial, $p(x)$. That is, $p(c)=0$.

2. The value $c$ must not be a zero of the denominator polynomial, $q(x)$. That is, $q(c)\ne 0$.

In essence, to find the zeros, you first find the roots of the numerator and then you must exclude any of these roots that are also roots of the denominator.

Understanding the Denominator's Role

The most critical nuance in finding the zeros of a rational function is understanding why a value that makes both the numerator and denominator zero is excluded. This concept is rooted in the fundamental properties of division.

Consider the expression $f(x)=\frac{p(x)}{q(x)}$. For $f(x)$ to be zero, the fraction itself must equal zero.

• A fraction of the form $\frac{0}{k}$ (where $k$ is any non-zero number) is equal to $0$. This is the situation we are looking for. If $x=c$ makes $p(c)=0$ and $q(c)=k\ne 0$, then $f(c)=\frac{0}{k}=0$, and $x=c$ is a zero.

• A fraction of the form $\frac{k}{0}$ (where $k$ is any non-zero number) is undefined. This corresponds to a vertical asymptote, a concept related to but distinct from zeros.

• A fraction of the form $\frac{0}{0}$ is an indeterminate form. When $x=c$ makes both $p(c)=0$ and $q(c)=0$, the function $f(x)$ has a point of discontinuity (often a "hole" in the graph) at $x=c$. Because the function is not defined at $x=c$, it cannot have a value of zero at that point. Therefore, $x=c$ cannot be a zero of the function.

This distinction is absolute. A zero of a function is a point in its domain that results in an output of zero. A value $x=c$ that makes the denominator zero is, by definition, not in the domain of the rational function. Therefore, such a value can never be a zero of the function, even if it also makes the numerator zero.

Core Concepts & Rules

• Zero Definition: The zeros of a rational function $f(x)$ are the input values $x$ for which $f(x)=0$.

• Graphical Interpretation: The real zeros of a rational function are the x-coordinates of the points where the function's graph intersects the x-axis (the x-intercepts).

• Primary Step: To find potential zeros, set the numerator polynomial, $p(x)$, equal to zero and solve for $x$.

• Verification Step: For each potential zero $c$ found from the numerator, you must verify that it does not also make the denominator, $q(x)$, equal to zero. That is, you must confirm that $q(c)\ne 0$.

• Exclusion Rule: If a value $x=c$ makes both the numerator and the denominator zero ($p(c)=0$ and $q(c)=0$), it is not a zero of the rational function. The function is undefined at that point.

Step-by-Step Example 1: A Basic Application

Find the zeros of the rational function $f(x)=\frac{x^2-9}{x+1}$.

Step 1: Identify the numerator $p(x)$ and the denominator $q(x)$.

• Numerator: $p(x)=x^2-9$

• Denominator: $q(x)=x+1$

Step 2: Find the zeros of the numerator by setting $p(x)=0$.

$$x^2-9=0$$

$$(x-3)(x+3)=0$$

The zeros of the numerator are $x=3$ and $x=-3$. These are our potential zeros for $f(x)$.

Step 3: Check if these potential zeros make the denominator $q(x)$ equal to zero.

• Check $x=3$:

  $q(3)=(3)+1=4$. Since $q(3)\ne 0$, $x=3$ is a zero of $f(x)$.

• Check $x=-3$:

  $q(-3)=(-3)+1=-2$. Since $q(-3)\ne 0$, $x=-3$ is a zero of $f(x)$.

Step 4: State the final answer.

The zeros of the function $f(x)=\frac{x^2-9}{x+1}$ are $x=3$ and $x=-3$.

Step-by-Step Example 2: An Exam-Style Application

Determine all zeros of the rational function $g(x)=\frac{x^2+x-6}{x^2-4}$.

Step 1: Factor both the numerator and the denominator completely.

• Factor the numerator, $p(x)=x^2+x-6$:

  We need two numbers that multiply to -6 and add to +1. These are +3 and -2.

  $p(x)=(x+3)(x-2)$

• Factor the denominator, $q(x)=x^2-4$:

  This is a difference of squares.

  $q(x)=(x+2)(x-2)$

Step 2: Identify the potential zeros from the factored numerator.

Set the factored numerator equal to zero:

$(x+3)(x-2)=0$

The potential zeros are $x=-3$ and $x=2$.

Step 3: Check each potential zero against the denominator.

• Check $x=-3$:

  Does $x=-3$ make the denominator zero?

  $q(-3)=(-3+2)(-3-2)=(-1)(-5)=5$.

  Since $q(-3)\ne 0$, $x=-3$ is a zero of $g(x)$.

• Check $x=2$:

  Does $x=2$ make the denominator zero?

  $q(2)=(2+2)(2-2)=(4)(0)=0$.

  Since $q(2)=0$, the value $x=2$ makes both the numerator and denominator zero. Therefore, it is not a zero of $g(x)$.

Step 4: Conclude and state the final answer.

The only zero of the function $g(x)=\frac{x^2+x-6}{x^2-4}$ is $x=-3$.

Using Your Calculator

While finding zeros of a rational function is primarily an analytical process, a graphing calculator is an excellent tool for verifying your results.

**To verify the zeros of g(x) = \frac{x^2 + x - 6}{x^2 - 4}` from Example 2:** 1. **Enter the Function:** Press the `Y=` button. In `Y1`, enter the function. Be careful with parentheses to ensure the correct order of operations. `Y1 = (X^2 + X - 6) / (X^2 - 4)` 2. **Graph the Function:** Press the `GRAPH` button. Use a standard viewing window (`ZOOM` -> `6:ZStandard`) to start. You should see the graph cross the x-axis at one point. 3. **Find the x-intercept (Zero):** - Press `2nd` then `TRACE` to open the `CALC` (Calculate) menu. - Select `2: zero`. - The calculator will ask for a "Left Bound?". Use the arrow keys to move the cursor on the graph to a point that is clearly to the left of the x-intercept, then press `ENTER`. - The calculator will ask for a "Right Bound?". Move the cursor to a point that is clearly to the right of the x-intercept, then press `ENTER`. - The calculator will ask for a "Guess?". Move the cursor close to the x-intercept and press `ENTER`. 4. **Interpret the Result:** The calculator will display the coordinates of the zero. It should show `X = -3` and `Y = 0`. This confirms that $x=-3 is a zero.

5. Investigate the Excluded Value: Notice that our analysis showed $x=2$ was not a zero. If you use the TRACE feature and type $2$ then ENTER, you will see that Y= is blank. This indicates the function is undefined at $x=2$. If you zoom in on the graph near $x=2$, you may see a small gap or "hole," visually confirming that the graph does not exist at that point and therefore cannot have a zero there.

AP Exam Quick Hit

Common Question Types

• Direct Analytical Question: You will be given a rational function, either in factored or polynomial form, and asked to find its zeros.

  • Example: "What are the zeros of the function $f(x)=\frac{2x^2-8}{x^2-x-6}$?"

• Graphical Interpretation: You will be shown the graph of a rational function and asked to identify its zeros from the graph.

  • Example: "The graph of a rational function $h(x)$ is shown. Based on the graph, what is a zero of $h(x)$?" (You would then identify the x-intercept from the provided image).

• Table-Based Question: You will be given a table of values for two functions, $p(x)$ and $q(x)$, and asked to determine the zeros of the rational function $f(x)=\frac{p(x)}{q(x)}$.

  • Example: "The table below gives values for the polynomials $p(x)$ and $q(x)$ at selected x-values. If $f(x)=\frac{p(x)}{q(x)}$, what is a zero of $f(x)$?"

- *Solution:* You would identify that $p(5)=0$ while $q(5)\ne 0$, making $x=5$ a zero. You would also note that since $p(-2)=0$ and $q(-2)=0$, $x=-2$ is not a zero.

Common Mistakes

• Ignoring the Denominator Condition: The most frequent error is to solve $p(x)=0$ and list all solutions as the zeros of the rational function, without checking if any of those values also make $q(x)=0$.

• Confusing Zeros with Undefined Points: Students may incorrectly state that the zeros of the denominator are also zeros of the rational function. Remember, where the denominator is zero, the function is undefined.

• Algebraic Errors in Factoring: Simple mistakes made while factoring the numerator or denominator polynomials will lead to incorrect potential zeros from the very first step. Always double-check your factoring.

• Cancellation Confusion: Students may correctly identify a common factor in the numerator and denominator (e.g., $(x-c)$), cancel it, and then forget that $x=c$ is a point of discontinuity and not a zero. The original, unfactored form of the denominator determines the function's domain.