Holes in Rational Functions

In AP Precalculus, holes (also called removable discontinuities) occur in rational functions when a function is undefined at a certain x-value, yet the overall behavior of the graph remains smooth around that point.

Understanding holes is essential for:

• Distinguishing different types of discontinuities

• Accurately graphing rational functions

• Interpreting function behavior

• Preparing for limits and continuity in Calculus

What Is a Hole?

A hole occurs at x=a if:

• Both numerator and denominator equal zero at x=a

• The factor causing this zero cancels

• The function is undefined at x=a, but the limit exists

This type of discontinuity is removable.

How Holes Are Created

Holes are created by common factors in the numerator and denominator.

Example

$$f(x)=\frac{x^2-4}{x-2}$$​

Factor:

$$\frac{(x+2)(x-2)}{x-2}$$​

Cancel:

$$f(x)=x+2 , x\neq2$$

• Hole at x=2

Finding the Location of a Hole

Step 1: Identify the x-value

Set the canceled factor equal to zero:

$$x-2=0 \longrightarrow x=2$$

Step 2: Find the y-value

Evaluate the simplified function at that x-value:

$$y=2+2=4$$

Hole location:

(2,4)

Holes vs. Vertical Asymptotes

Example with Multiple Discontinuities

$$f(x)=\frac{(x-1)(x+2)}{(x-1)(x-3)}$$​

Cancel:

$$f(x)=\frac{x+2}{x-3}$$​

• Hole at $x=1$

• Vertical asymptote at $x=3$

Hole y-value:

$$y=\frac{1+2}{1-3}=-\frac{3}{2}$$​

Holes and Limits

A hole exists when:

$$\lim_{x\to a}f(x)$$exists , but $$f(a)$$ is undefined

This concept directly prepares students for continuity in Calculus.

Graphing Rational Functions with Holes

To graph:

1. Factor numerator and denominator

2. Simplify

3. Identify holes

4. Identify asymptotes

5. Plot the simplified function

6. Mark holes as open circles

Common Mistakes

• Treating holes as x-intercepts

• Forgetting to exclude hole x-values from the domain

• Canceling terms instead of factors

• Ignoring the y-value of the hole

Applications and Interpretation

Holes often represent:

• Undefined measurements

• Missing data points

• Simplified models with restrictions

Understanding holes helps interpret model limitations.

Summary

• Holes occur due to canceled common factors

• They are removable discontinuities

• Hole location requires both x- and y-values

• Holes differ fundamentally from vertical asymptotes

• The concept prepares students for limits and continuity