Introduction

Rational functions might sound complicated, but they’re just fractions where one polynomial sits on top of another. For the AP® Precalculus exam, it’s crucial to grasp how these functions work, especially when it comes to identifying holes in their graphs. Learning how to find holes of a rational function helps in sketching accurate graphs and analyzing the function’s behavior. This is vital for success on the exam because this skill comes directly from section 1.10 Rational Functions and Holes.

What are Rational Functions?

A rational function is simply a fraction that has two polynomials: one in the numerator (top) and one in the denominator (bottom). Mathematically, it is expressed as:

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

…where p(x) and q(x) are polynomials, and q(x) \neq 0 .

Examples of Rational Functions

1. f(x) = \frac{x^2 - 4}{x - 2}
2. g(x) = \frac{x^2 + x + 1}{x^2 - 1}

Each function involves a polynomial divided by another polynomial.

Understanding Holes in Rational Functions

A hole in the graph of a rational function is like a missing piece or an unplugged point. It occurs when a factor is shared by the numerator and the denominator.If a value of x makes both the numerator and the denominator zero, this typically indicates a hole rather than an asymptote.

Vertical Asymptotes vs. Holes

Vertical asymptotes are values where the denominator is zero but not the numerator, leading to the function heading towards infinity. Conversely, a hole occurs when the zero cancels out in both the numerator and the denominator, leaving an “invisible” point on the graph.

How to Find Holes of a Rational Function

Here’s a step-by-step process to find holes:

1. Find Common Factors: Factor both the numerator and the denominator.
2. Check for Common Factors: Identify factors that appear in both the numerator and the denominator.
3. Check for Holes: If they share common factors, those zero values are potential holes.
4. Analyze Multiplicities: The hole’s multiplicity is determined by how many times the factor appears.

Example 1: Step-by-Step Solution

Let’s find the holes in f(x) = \frac{x^2 - 3x-10}{x - 5} .

1. Factor Numerator and Denominator:

• Numerator: x^2 - 3x-10 = (x -5)(x + 2)
• Denominator: x - 5

1. Identify Common Factors:

• Common Factor: (x - 5)

1. Determine Holes:

• The hole is at x = 5, as it cancels out.

How to Find Holes of Rational Functions: Exact Points

To locate the exact point of the hole, consider the value of the function as it approaches the hole. This involves applying limits to understand function behavior near that point.

Example 2: Step-by-Step Solution

Find the coordinates of the hole for f(x) = \frac{x^2 - 3x-10}{x - 5} .

1. Cancel Common Factors:

• Simplified function: f(x) = x + 2 (no longer undefined).

1. Apply Limit:

• Limit as x approaches 5: \lim\limits_{x \to 5} (x + 2) = 7
• Coordinate of the hole: (5, 7)

Quick Reference Chart: Vocabulary and Definitions

Conclusion

Understanding how to find holes of a rational function is a valuable skill. It helps in graphing these functions and interpreting their behavior. Practice these techniques often, and don’t hesitate to seek help if needed.

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