Welcome to 1.10: Rational Functions and Holes!

In previous chapters, we looked at how rational functions behave when they hit zero or shoot off to infinity. But what happens when a function seems like it should be undefined, yet the graph looks almost perfectly smooth? That "glitch" in the system is what we call a hole. In this chapter, we will learn how to spot these holes, find exactly where they live, and understand why they happen.

Prerequisite Check: To master this chapter, you should be comfortable with factoring polynomials. If you can factor a quadratic like \(x^2 - 4\) into \((x - 2)(x + 2)\), you are already halfway there!

What Exactly is a "Hole"?

In mathematics, a hole is formally known as a removable discontinuity. Imagine you are driving down a road and there is one single, tiny pothole. If you could just drop one brick into it, the road would be perfectly smooth again. That is why it's called "removable"—the gap is only at a single point.

A hole occurs in a rational function \(f(x) = \frac{p(x)}{q(x)}\) when a value of \(x\) makes both the numerator and the denominator equal to zero. This happens because there is a common factor in the top and bottom that "cancels out."

Important Distinction:
- Vertical Asymptotes (VA): These happen when a factor is only in the denominator (see Chapter 1.9). The graph breaks and heads to infinity.
- Holes: These happen when a factor is in both the numerator and the denominator. The graph stays mostly connected but has a single missing point.

Step-by-Step: How to Find a Hole Analytically

To find the location of a hole, follow these three simple steps. Let’s use the example function: \(f(x) = \frac{x^2 - 9}{x^2 - 2x - 3}\)

Step 1: Factor Everything

Break down the numerator and the denominator into their linear factors.

\(f(x) = \frac{(x - 3)(x + 3)}{(x - 3)(x + 1)}\)

Step 2: Identify the Common Factor

Look for the factor that appears in both the top and the bottom. Here, the common factor is \((x - 3)\). Setting this factor to zero gives us the x-coordinate of our hole.

\(x - 3 = 0 \implies x = 3\)

Step 3: Find the y-coordinate (The "Simplified" Value)

To find where the hole is vertically, "remove" the common factor and see what's left of the function. This is often called the reduced function.

Reduced function: \(f_{red}(x) = \frac{x + 3}{x + 1}\)

Now, plug the x-coordinate we found in Step 2 into this reduced function:

\(f_{red}(3) = \frac{3 + 3}{3 + 1} = \frac{6}{4} = 1.5\)

Conclusion: The function has a hole at the point \((3, 1.5)\).

Visualizing the Hole

When you look at a graph of a rational function on a standard graphing calculator, you might not even see the hole! Because a hole is just a single point, the calculator often draws a solid line right over it.

However, in a professional mathematical sketch:

  • The hole is represented by an open circle \(\circ\).
  • The rest of the graph behaves exactly like the reduced function.

Analogy: Imagine a movie film. A vertical asymptote is like a missing scene where the camera cuts away. A hole is like a single frame of the film being snipped out. You almost don't notice it, but there's technically a gap there!

Holes vs. Vertical Asymptotes: The "Tie-Breaker" Rule

Students often ask: "What if a factor appears twice in the bottom but only once in the top?"

Let's look at \(g(x) = \frac{x - 2}{(x - 2)^2}\).

  • One \((x - 2)\) cancels out, which suggests a hole.
  • However, there is still an \((x - 2)\) left over in the denominator.
  • The Rule: If the factor remains in the denominator after simplifying, the Vertical Asymptote wins. There is no hole at \(x = 2\); there is a vertical asymptote instead.

Quick Review Box:
- Factor only in numerator \(\implies\) Zero (x-intercept).
- Factor only in denominator \(\implies\) Vertical Asymptote.
- Factor in both \(\implies\) Hole (as long as it doesn't remain in the denominator).

Common Mistakes to Avoid

1. Forgetting the y-coordinate: On the AP exam, if you are asked where a hole is located, provide the full coordinate point \((x, y)\), not just the \(x\)-value.

2. Solving the original function: Don't plug your hole's \(x\)-value into the original equation. You will get \(0/0\), which is undefined. Always use the reduced (simplified) version.

3. Misidentifying the hole as a zero: Just because \(x - 3\) is in the numerator doesn't mean there is an x-intercept at \(3\). If it's also in the denominator, the hole "overrides" the zero.

Key Takeaways

  • A hole (removable discontinuity) occurs when a rational function has a common factor in the numerator and denominator.
  • The x-coordinate of the hole is the value that makes that common factor zero.
  • The y-coordinate is found by plugging that \(x\)-value into the simplified version of the function.
  • In the "battle" for a point, a Vertical Asymptote usually overrides a hole if the factor remains in the denominator after simplification.

Did you know? If you use the "Table" feature on your graphing calculator and scroll to the \(x\)-value of a hole, the \(y\)-column will say "ERROR" or "UNDEF", even if the graph looks like a solid line!