📌 What is a Rational Function?

A rational function is a function that can be written as the quotient of two polynomials:

\( f(x) = \frac{P(x)}{Q(x)} \) where \( Q(x) \neq 0 \)

Where \( P(x) \) and \( Q(x) \) are polynomial functions and \( Q(x) \) is not the zero polynomial.

Finding Excluded Values:

Excluded values are values of \(x\) that make the denominator equal to zero.

📝 Examples - Excluded Values:

Example 1: Find excluded values of \( f(x) = \frac{3x + 2}{x - 5} \)

Set denominator = 0: \( x - 5 = 0 \)
Solve: \( x = 5 \)
Excluded value: \( x = 5 \)
Domain: All real numbers except \( x = 5 \)

Example 2: Find excluded values of \( f(x) = \frac{x + 1}{x^2 - 9} \)

Set denominator = 0: \( x^2 - 9 = 0 \)
Factor: \( (x - 3)(x + 3) = 0 \)
Solve: \( x = 3 \) or \( x = -3 \)
Excluded values: \( x = 3, -3 \)
Domain: All real numbers except \( x = 3 \) and \( x = -3 \)

Types of Asymptotes:

Finding Vertical Asymptotes:

1. Factor the numerator and denominator completely
2. Cancel any common factors (these create holes, not VAs)
3. Set the remaining denominator factors equal to zero
4. Solve for \( x \) — these are the vertical asymptotes

VA occurs at \( x = a \) where denominator = 0 (after canceling)

Finding Horizontal Asymptotes:

Compare degrees of numerator (n) and denominator (d):

📝 Examples - Finding Asymptotes:

Example 1: Find asymptotes of \( f(x) = \frac{2x + 3}{x - 4} \)

VA: Set denominator = 0 → \( x = 4 \)
HA: Degrees equal (both 1) → \( y = \frac{2}{1} = 2 \)

Example 2: Find asymptotes of \( f(x) = \frac{3x^2 - 5}{x^2 + 2x - 8} \)

VA: Factor denominator: \( (x + 4)(x - 2) = 0 \)
\( x = -4 \) and \( x = 2 \)
HA: Degrees equal (both 2) → \( y = \frac{3}{1} = 3 \)

Example 3: Find asymptotes of \( f(x) = \frac{x + 1}{x^2 + 1} \)

VA: \( x^2 + 1 = 0 \) has no real solutions → No VA
HA: Degree of numerator (1) < degree of denominator (2) → \( y = 0 \)

How to Evaluate:

1. Substitute the given value for the variable
2. Simplify the numerator
3. Simplify the denominator
4. Divide or simplify the fraction
5. Check: Make sure the denominator ≠ 0

📝 Examples - Evaluating:

Example 1: Evaluate \( \frac{x^2 - 4}{x + 3} \) when \( x = 2 \)

Substitute: \( \frac{2^2 - 4}{2 + 3} = \frac{4 - 4}{5} = \frac{0}{5} = 0 \)

Example 2: Evaluate \( \frac{3x + 5}{x^2 - 1} \) when \( x = -2 \)

Substitute: \( \frac{3(-2) + 5}{(-2)^2 - 1} = \frac{-6 + 5}{4 - 1} = \frac{-1}{3} \)

Steps to Simplify:

1. Factor the numerator completely
2. Factor the denominator completely
3. Cancel common factors that appear in both numerator and denominator
4. State restrictions: values that make the original denominator = 0

⚠️ Important: Only cancel FACTORS, not terms!

📝 Examples - Simplifying:

Example 1: Simplify \( \frac{15x^3}{25x^2} \)

\( = \frac{5 \cdot 3 \cdot x^2 \cdot x}{5 \cdot 5 \cdot x^2} = \frac{3x}{5} \), where \( x \neq 0 \)

Example 2: Simplify \( \frac{x^2 - 9}{x^2 + 5x + 6} \)

Factor: \( \frac{(x - 3)(x + 3)}{(x + 2)(x + 3)} \)
Cancel: \( \frac{x - 3}{x + 2} \), where \( x \neq -2, -3 \)

Example 3: Simplify \( \frac{2x^2 - 8}{x^2 - 4x + 4} \)

Factor: \( \frac{2(x^2 - 4)}{(x - 2)^2} = \frac{2(x - 2)(x + 2)}{(x - 2)^2} \)
Cancel: \( \frac{2(x + 2)}{x - 2} \), where \( x \neq 2 \)

Multiplication Rule:

\( \frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d} \)

📝 Example - Multiplication:

Multiply: \( \frac{x^2 - 4}{x + 3} \cdot \frac{x + 3}{x - 2} \)

Factor: \( \frac{(x - 2)(x + 2)}{x + 3} \cdot \frac{x + 3}{x - 2} \)
Cancel common factors: \( (x + 3) \) and \( (x - 2) \)
Result: \( x + 2 \), where \( x \neq 2, -3 \)

Division Rule:

\( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} \)

Multiply by the reciprocal of the divisor (flip and multiply)

📝 Example - Division:

Divide: \( \frac{x^2 - 1}{x + 2} \div \frac{x + 1}{x^2 - 4} \)

Flip and multiply: \( \frac{x^2 - 1}{x + 2} \cdot \frac{x^2 - 4}{x + 1} \)
Factor: \( \frac{(x - 1)(x + 1)}{x + 2} \cdot \frac{(x - 2)(x + 2)}{x + 1} \)
Cancel: \( (x + 1) \) and \( (x + 2) \)
Result: \( (x - 1)(x - 2) = x^2 - 3x + 2 \)

Rules:

Steps for Adding/Subtracting:

1. Factor all denominators
2. Find the LCD: Include each factor the greatest number of times it appears
3. Rewrite each fraction with the LCD
4. Add/subtract numerators, keep the LCD
5. Simplify if possible

📝 Example 1 - Same Denominator:

Add: \( \frac{3x}{x + 2} + \frac{5}{x + 2} \)

Same denominator, so add numerators:
\( = \frac{3x + 5}{x + 2} \)

📝 Example 2 - Different Denominators:

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

LCD: \( x(x + 1) \)
Rewrite: \( \frac{2(x + 1)}{x(x + 1)} + \frac{3x}{x(x + 1)} \)
Add: \( \frac{2(x + 1) + 3x}{x(x + 1)} = \frac{2x + 2 + 3x}{x(x + 1)} \)
Simplify: \( \frac{5x + 2}{x(x + 1)} \)

📝 Example 3 - More Complex:

Subtract: \( \frac{5}{x - 3} - \frac{2}{x + 1} \)

LCD: \( (x - 3)(x + 1) \)
Rewrite: \( \frac{5(x + 1)}{(x - 3)(x + 1)} - \frac{2(x - 3)}{(x - 3)(x + 1)} \)
Subtract: \( \frac{5(x + 1) - 2(x - 3)}{(x - 3)(x + 1)} \)
Expand: \( \frac{5x + 5 - 2x + 6}{(x - 3)(x + 1)} \)
Simplify: \( \frac{3x + 11}{(x - 3)(x + 1)} \)

What is a Complex Fraction?

A complex fraction is a fraction that contains one or more fractions in its numerator, denominator, or both.

\( \frac{\frac{a}{b}}{\frac{c}{d}} \)

Two Methods to Simplify:

📝 Example - Complex Fraction:

Simplify: \( \frac{\frac{2}{x}}{\frac{3}{x^2}} \)

Method 1: Multiply by reciprocal

\( = \frac{2}{x} \div \frac{3}{x^2} = \frac{2}{x} \cdot \frac{x^2}{3} = \frac{2x^2}{3x} = \frac{2x}{3} \)

Method 2: Multiply by LCD

LCD of \( x \) and \( x^2 \) is \( x^2 \)
\( = \frac{\frac{2}{x} \cdot x^2}{\frac{3}{x^2} \cdot x^2} = \frac{2x}{3} \)

What is a Rational Equation?

A rational equation is an equation that contains one or more rational expressions.

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

Steps to Solve:

1. Find the LCD of all denominators
2. Multiply every term by the LCD to clear fractions
3. Solve the resulting equation
4. Check all solutions in the original equation
5. Reject extraneous solutions (make denominator = 0)

⚠️ Always check for extraneous solutions!

📝 Example 1 - Simple Equation:

Solve: \( \frac{3}{x} = \frac{6}{x + 2} \)

Method: Cross Multiply

\( 3(x + 2) = 6x \)
\( 3x + 6 = 6x \)
\( 6 = 3x \)
\( x = 2 \)

Check: \( \frac{3}{2} = \frac{6}{4} \) → \( \frac{3}{2} = \frac{3}{2} \) ✓

Solution: \( x = 2 \)

📝 Example 2 - Multiple Terms:

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

Step 1: LCD = \( x(x - 1) \)

Step 2: Multiply every term by LCD

\( x(x - 1) \cdot \frac{1}{x} + x(x - 1) \cdot \frac{2}{x - 1} = x(x - 1) \cdot \frac{3}{x} \)
\( (x - 1) + 2x = 3(x - 1) \)
\( x - 1 + 2x = 3x - 3 \)
\( 3x - 1 = 3x - 3 \)
\( -1 = -3 \) (False!)

No Solution

📝 Example 3 - Extraneous Solution:

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

LCD = \( x - 2 \), multiply all terms:
\( x = 2 + (x - 2) \)
\( x = 2 + x - 2 \)
\( x = x \) (Always true, but...)

⚠️ Check: \( x = 2 \) makes the denominator zero in original equation!

No Solution (extraneous)

⚡ Quick Summary

• Rational function: \( f(x) = \frac{P(x)}{Q(x)} \), where \( Q(x) \neq 0 \)
• Excluded values: Set denominator = 0 and solve
• Vertical asymptote: \( x = a \) where denominator = 0 (after simplifying)
• Horizontal asymptote: Compare degrees of numerator and denominator
• To simplify: Factor completely, then cancel common factors
• Multiply: Factor, cancel, then multiply across
• Divide: Multiply by the reciprocal
• Add/Subtract: Find LCD, rewrite fractions, combine numerators
• Complex fractions: Divide or multiply by LCD
• Solving equations: Clear fractions with LCD, solve, CHECK for extraneous solutions

📚 Key Formulas & Rules