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AP Precalculus: Rational Functions

Master asymptotes, domain restrictions, graphing techniques, and solving rational equations

📐 Asymptotes 🕳️ Holes 🧮 Solving 📊 Graphing

📚 Understanding Rational Functions

Rational functions are ratios of polynomials, introducing new behaviors like asymptotes (lines the graph approaches but never crosses) and holes (single points removed from the graph). This guide covers everything you need for AP Precalculus success with rational functions.

1 Rational Function Form & Domain

A rational function is a function of the form \(f(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) \neq 0\).

General Form \[f(x) = \frac{P(x)}{Q(x)} = \frac{a_nx^n + \cdots + a_0}{b_mx^m + \cdots + b_0}\]

Finding the Domain

  • Set the denominator equal to zero: \(Q(x) = 0\)
  • Solve for \(x\) — these values are excluded from the domain
  • Domain: all real numbers except values that make \(Q(x) = 0\)
  • Write in interval notation, excluding restricted values
📌 Example

Find the domain of: \(f(x) = \frac{x + 3}{x^2 - 4}\)

Step 1: Set denominator = 0: \(x^2 - 4 = 0\)

Step 2: Factor: \((x-2)(x+2) = 0\) → \(x = 2\) or \(x = -2\)

Domain: \((-\infty, -2) \cup (-2, 2) \cup (2, \infty)\)

or: \(\{x \in \mathbb{R} \mid x \neq -2, 2\}\)

2 Asymptotes

An asymptote is a line that a graph approaches but never touches (or crosses only at specific points). Rational functions can have vertical, horizontal, and oblique (slant) asymptotes.

📏 Vertical Asymptote
How to find: Set \(Q(x) = 0\) after canceling common factors with \(P(x)\).

Equation form: \(x = a\)

Graph behavior: \(f(x) \to \pm\infty\) as \(x \to a\)
➖ Horizontal Asymptote
How to find: Compare degrees of \(P(x)\) and \(Q(x)\).

Equation form: \(y = k\)

Graph behavior: \(f(x) \to k\) as \(x \to \pm\infty\)
↗️ Oblique (Slant) Asymptote
When it exists: Degree of \(P\) = Degree of \(Q\) + 1

How to find: Divide \(P(x) \div Q(x)\); the quotient is the asymptote

Equation form: \(y = mx + b\)

Horizontal Asymptote Rules

deg(P) < deg(Q)
\(y = 0\)
(x-axis)
deg(P) = deg(Q)
\(y = \frac{\text{lead coef of } P}{\text{lead coef of } Q}\)
deg(P) > deg(Q)
No horizontal
(check for oblique)
📌 Examples

\(f(x) = \frac{3x + 1}{x^2 - 4}\): deg(1) < deg(2) → HA: \(y=0\)

\(f(x) = \frac{2x^2 + 1}{5x^2 - 3}\): deg(2) = deg(2) → HA: \(y = \frac{2}{5}\)

\(f(x) = \frac{x^2 + 2x}{x - 1}\): deg(2) > deg(1) → No HA, check for oblique

3 Finding Oblique (Slant) Asymptotes

An oblique asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator. Use polynomial division to find it.

Steps to Find Oblique Asymptote

  1. Verify deg(\(P\)) = deg(\(Q\)) + 1
  2. Perform polynomial long division: \(\frac{P(x)}{Q(x)}\)
  3. The quotient \(mx + b\) is the oblique asymptote (ignore the remainder)
  4. Write as \(y = mx + b\)
📌 Example

Find the oblique asymptote of: \(f(x) = \frac{x^2 + 3x + 2}{x - 1}\)

Check: deg(2) = deg(1) + 1 ✓

Divide: \((x^2 + 3x + 2) \div (x - 1)\)

\(= x + 4 + \frac{6}{x-1}\)

Oblique asymptote: \(y = x + 4\)

4 Holes (Removable Discontinuities)

A hole (or removable discontinuity) occurs at values of \(x\) where both the numerator and denominator equal zero — i.e., where a common factor cancels out.

🕳️ Hole
Factor cancels from both numerator and denominator

Graph has a single point missing
📏 Vertical Asymptote
Factor only in denominator (doesn't cancel)

Graph approaches ±∞

Steps to Find Holes

  1. Factor both the numerator \(P(x)\) and denominator \(Q(x)\) completely
  2. Identify any common factors
  3. Set the common factor = 0 to find the x-coordinate of the hole
  4. Cancel the common factor and substitute to find the y-coordinate
📌 Example

Find any holes in: \(f(x) = \frac{x^2 - 4}{x^2 - x - 2}\)

Factor: \(f(x) = \frac{(x-2)(x+2)}{(x-2)(x+1)}\)

Common factor: \((x-2)\) → Hole at \(x = 2\)

Cancel: \(f(x) = \frac{x+2}{x+1}\) (with hole at \(x = 2\))

Y-coordinate: \(y = \frac{2+2}{2+1} = \frac{4}{3}\)

Hole: \(\left(2, \frac{4}{3}\right)\)

Vertical asymptote: \(x = -1\) (doesn't cancel)

⚠️ Always Factor First!

Always factor completely before determining asymptotes. A factor in the denominator might cancel, creating a hole instead of an asymptote.

5 Graphing Rational Functions

To graph a rational function, identify all key features: intercepts, asymptotes, holes, and the general shape of the curve.

Complete Graphing Checklist

  1. Factor numerator and denominator completely
  2. Find holes: Common factors that cancel
  3. Find vertical asymptotes: Set simplified denominator = 0
  4. Find horizontal/oblique asymptote: Compare degrees
  5. Find x-intercepts: Set simplified numerator = 0
  6. Find y-intercept: Evaluate \(f(0)\)
  7. Plot additional points and sketch the curve
📌 Example: Graph \(f(x) = \frac{2x}{x^2 - 1}\)

Factor: \(f(x) = \frac{2x}{(x-1)(x+1)}\)

Holes: None (no common factors)

Vertical asymptotes: \(x = 1\) and \(x = -1\)

Horizontal asymptote: deg(1) < deg(2) → \(y=0\)

X-intercept: \(2x = 0\) → \(x = 0\) → point \((0, 0)\)

Y-intercept: \(f(0) = 0\) → point \((0, 0)\)

6 Solving Rational Equations

To solve a rational equation (an equation containing rational expressions), clear the fractions and solve the resulting polynomial equation. Always check for extraneous solutions!

Steps to Solve Rational Equations

  1. Find the LCD (Least Common Denominator) of all fractions
  2. Multiply every term on both sides by the LCD
  3. Simplify — the denominators should cancel
  4. Solve the resulting polynomial equation
  5. Check solutions — reject any that make a denominator zero (extraneous)
📌 Example 1

Solve: \(\frac{1}{x} + 2 = \frac{3}{x}\)

LCD: \(x\)

Multiply by LCD: \(x \cdot \frac{1}{x} + x \cdot 2 = x \cdot \frac{3}{x}\)

Simplify: \(1 + 2x = 3\)

Solve: \(2x = 2\) → \(x = 1\)

Check: \(\frac{1}{1} + 2 = 3\) and \(\frac{3}{1} = 3\) ✓

📌 Example 2: Extraneous Solution

Solve: \(\frac{x}{x-2} = \frac{2}{x-2} + 1\)

LCD: \(x - 2\)

Multiply: \(x = 2 + (x-2)\)

Simplify: \(x = 2 + x - 2 = x\)

This is always true, but \(x = 2\) makes denominators zero!

Solution: All real numbers except \(x = 2\)

⚠️ Extraneous Solutions

An extraneous solution is a value that appears as a solution but makes the original equation undefined. Always substitute back into the original equation to check!

7 Checking Inverse Rational Functions

Two functions \(f(x)\) and \(g(x)\) are inverses if composing them in either order gives \(x\). For rational functions, verify by computing both compositions.

Inverse Function Test \(f\) and \(g\) are inverses if and only if:
\(f(g(x)) = x\) for all \(x\) in domain of \(g\)
AND
\(g(f(x)) = x\) for all \(x\) in domain of \(f\)
📌 Example

Verify that \(f(x) = \frac{2x+1}{x-3}\) and \(g(x) = \frac{3x+1}{x-2}\) are inverses:

Check \(f(g(x))\):

\(f(g(x)) = \frac{2 \cdot \frac{3x+1}{x-2} + 1}{\frac{3x+1}{x-2} - 3} = \frac{\frac{6x+2+x-2}{x-2}}{\frac{3x+1-3x+6}{x-2}} = \frac{7x}{7} = x\) ✓

Check \(g(f(x))\): Similarly shows \(g(f(x)) = x\) ✓

Conclusion: \(f\) and \(g\) are inverses

💡 Finding Inverse of Rational Function

To find the inverse: (1) Replace \(f(x)\) with \(y\), (2) Swap \(x\) and \(y\), (3) Solve for \(y\). Cross-multiply and rearrange to isolate \(y\).

8 End Behavior of Rational Functions

The end behavior of a rational function describes what happens to \(f(x)\) as \(x\) approaches \(\pm\infty\). It's determined by the horizontal or oblique asymptote.

End Behavior by Degree Comparison

deg(P) < deg(Q)

As \(x \to \pm\infty\), \(f(x) \to 0\)
Graph approaches x-axis

deg(P) = deg(Q)

As \(x \to \pm\infty\), \(f(x) \to \frac{a_n}{b_m}\)
Graph approaches horizontal line

deg(P) = deg(Q) + 1

As \(x \to \pm\infty\), graph approaches oblique line \(y = mx + b\)

deg(P) > deg(Q) + 1

As \(x \to \pm\infty\), \(f(x) \to \pm\infty\)
Behaves like polynomial

📋 Quick Reference: Key Formulas

Vertical Asymptote

Set simplified \(Q(x) = 0\)
Form: \(x = a\)

Horizontal Asymptote

Compare degrees of \(P\) and \(Q\)
Form: \(y = k\)

Oblique Asymptote

When deg(P) = deg(Q) + 1
Quotient of \(P \div Q\)

Holes

Common factors that cancel
\((x, f(x))\) using simplified form

X-intercepts

Set \(P(x) = 0\)
(numerator = 0)

Domain

All real numbers except
where \(Q(x) = 0\)

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