Rational Functions

📖

Core Concepts — Quick Notes

🔑 What IS a Rational Function?

A rational function is a fraction where both top and bottom are polynomials.

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

← This is the big rule! Bottom ≠ 0 ever!

⚡ Ultra-Short Memory Keywords

VERTICAL Set bottom = 0
Solve for x
That's your vertical asymptote

HORIZONTAL Compare TOP vs BOTTOM degree
(n vs m rule)

HOLE Cancel common factor from top & bottom → that x-value = HOLE

DOMAIN All real x EXCEPT where bottom = 0

LCD Least Common Denominator — find it first before adding/subtracting fractions!

📏 Horizontal Asymptote — The n vs m Rule

Given: \( f(x) = \dfrac{a_n x^n + \cdots}{b_m x^m + \cdots} \)

→ n < m: HA is y = 0 (x-axis)

→ n = m: HA is y = leading/leading = \(\dfrac{a_n}{b_m}\)

→ n > m: No horizontal asymptote (oblique instead)

Basic rational function \(f(x)=\frac{1}{x}\) — the parent graph

⚠️ Common Mistake Alert!
The graph APPROACHES the asymptote but never actually touches it (usually). Don't draw the curve crossing the asymptote!

➕

Adding & Subtracting Rational Expressions

📋 The 4-Step Process (Memorize This!)

① FACTOR — Factor each denominator completely

② LCD — Find the Least Common Denominator

③ REWRITE — Build equivalent fractions with LCD

④ COMBINE — Add/subtract numerators, keep denominator, simplify

✏️ Worked Example

📝 EXAMPLE

Simplify: \(\dfrac{3}{x+2} + \dfrac{1}{x-2}\)

Step 1 — Factor denominators: Already factored: \((x+2)\) and \((x-2)\)
Step 2 — LCD: \((x+2)(x-2)\) ← multiply them since no common factors
Step 3 — Rewrite:
\(\dfrac{3(x-2)}{(x+2)(x-2)} + \dfrac{1(x+2)}{(x-2)(x+2)}\)
Step 4 — Combine numerators:
\(\dfrac{3x-6+x+2}{(x+2)(x-2)} = \dfrac{4x-4}{(x+2)(x-2)} = \dfrac{4(x-1)}{(x+2)(x-2)}\)

🚨 Subtraction Trap!

When subtracting, distribute the negative sign to ALL terms in the second numerator!
\(\dfrac{A}{B} - \dfrac{C+D}{B} = \dfrac{A - C - D}{B}\) ← Don't forget the minus on D!

✏️

Practice Problems

Choose the best answer. Get it right → 🎉 celebration! Get it wrong → see the explanation.

— Section A: Graphing & Asymptotes 📊 —

Q1 · Graph

1. What is the vertical asymptote of \(f(x) = \dfrac{3}{x-5}\)?

Q2 · Graph

2. What is the horizontal asymptote of \(f(x) = \dfrac{2x+1}{x-3}\)?

Q3 · Graph

3. Find the domain of \(f(x) = \dfrac{x+1}{x^2 - 4}\).

Q4 · 🔥 Tricky

4. \(f(x) = \dfrac{x^2 - x - 6}{x - 3}\) — does this graph have a hole or a vertical asymptote at \(x=3\)?

Q5 · Graph

5. The horizontal asymptote of \(f(x) = \dfrac{x^2 + 1}{x^3 - 2}\) is:

— Section B: Adding Rational Expressions ➕ —

Q6 · Add

6. \(\dfrac{1}{x} + \dfrac{2}{x} = ?\)

Q7 · Add

7. \(\dfrac{2}{x+1} + \dfrac{3}{x+1} = ?\)

Q8 · Add

8. \(\dfrac{1}{x} + \dfrac{1}{3} = ?\) (find LCD first!)

Q9 · Add

9. \(\dfrac{x}{x+2} + \dfrac{2}{x+2} = ?\)

Q10 · 🔥 Tricky

10. \(\dfrac{3}{x+1} + \dfrac{2}{x-1} = ?\)

— Section C: Subtracting Rational Expressions ➖ —

Q11 · Sub

11. \(\dfrac{7}{x} - \dfrac{3}{x} = ?\)

Q12 · 🔥 Tricky

12. \(\dfrac{x}{x+3} - \dfrac{3}{x+3} = ?\) (Watch: can you simplify further?)

Q13 · Sub

13. \(\dfrac{5}{x+2} - \dfrac{1}{x} = ?\)

Q14 · 🔥 Sign Trap!

14. \(\dfrac{4}{x-1} - \dfrac{x+3}{x-1} = ?\) ← minus sign applies to whole numerator!

Q15 · Sub

15. \(\dfrac{2}{x^2-1} - \dfrac{1}{x+1} = ?\) Hint: factor \(x^2-1\) first!

— Section D: Mixed Practice 🎯 —

Q16 · Mixed

16. Simplify completely: \(\dfrac{x^2-9}{x^2-x-6}\)

Q17 · Mixed

17. \(\dfrac{1}{x+3} + \dfrac{1}{x^2+3x} = ?\) Factor that denominator first!

Q18 · 🔥 Tricky

18. Which graph feature appears at \(x = -2\) for \(f(x) = \dfrac{(x+2)(x-1)}{(x+2)(x+3)}\)?

Q19 · Mixed

19. \(\dfrac{2x}{x+1} - \dfrac{x-1}{x+1} = ?\)

Q20 · 🔥 Final Boss

20. \(\dfrac{3}{x^2-4} + \dfrac{1}{x+2} - \dfrac{2}{x-2} = ?\)

🧠

Quick Reference — Explain It Out Loud!

Use these 1-word prompts to help explain each concept to yourself or someone else:

🗣️ Say It Out Loud — Explanation Starters

→ VERTICAL ASYMPTOTE: "Set the denominator equal to zero because..."

→ HOLE: "Cancel the common factor, then plug in x because..."

→ HORIZONTAL ASYMPTOTE: "Compare the degrees: if top is smaller..."

→ LCD: "Find what both denominators divide into because..."

→ SIGN DISTRIBUTION: "Minus sign applies to ALL terms in the bracket because..."

📐 The n vs m Horizontal Asymptote Cheat

Condition	HA Location	Memory word
Top degree < Bottom degree	\(y = 0\)	ZERO
Top degree = Bottom degree	\(y = \frac{\text{lead}}{\text{lead}}\)	RATIO
Top degree > Bottom degree	None (oblique)	NONE

⚠️ Most Commonly Missed!
1. Forgetting to distribute the negative sign when subtracting rational expressions.
2. Confusing holes (cancel-able factors) with vertical asymptotes (non-cancel-able zeros of denominator).
3. Adding denominators instead of finding LCD — never add the denominators!

Graphing · Addition · Subtraction