📐 Concept diagram

03-04 - Rational Functions

Phase: 3 | Subject: 03-04 Prerequisites: 03-03-polynomial-functions.md Next subject: 03-05-exponential-and-logarithmic-functions.md

Learning Objectives

By the end of this subject, you will be able to:

1. Identify and find vertical, horizontal, and oblique asymptotes
2. Determine domain restrictions from denominators
3. Sketch rational function graphs
4. Decompose proper rational functions into partial fractions
5. Solve rational inequalities

Core Content

What is a Rational Function?

$R(x) = P(x) / Q(x)$ where P and Q are polynomials, Q(x) ≠ 0.

Vertical Asymptotes

Where denominator Q(x) = 0 (and numerator ≠ 0 at that point).

Example: f(x) = 1/(x - 2) has VA at x = 2.

Holes: If a factor cancels between numerator and denominator, there's a hole (removable discontinuity), not a VA.

Example: f(x) = (x - 1)/(x - 1)(x + 2) Factor cancels: f(x) = 1/(x + 2) with a hole at x = 1. VA at x = -2.

Horizontal Asymptotes

Compare degrees of P(x) and Q(x):

Example: f(x) = (2x² + 1)/(x² - 3) Degrees equal (2 = 2). HA: y = 2/1 = 2.

Oblique (Slant) Asymptotes

When deg(P) = deg(Q) + 1. Use polynomial long division.

Example: f(x) = (x² + 2x + 1)/(x - 1) Divide: x² + 2x + 1 by x - 1 Quotient: x + 3 Oblique asymptote: y = x + 3

Partial Fractions Decomposition

Express a proper rational function as a sum of simpler fractions.

Example: Decompose 1/(x² - 1)

1. Factor denominator: (x - 1)(x + 1)
2. Set up: 1/(x² - 1) = A/(x - 1) + B/(x + 1)
3. Multiply: 1 = A(x + 1) + B(x - 1)
4. Let x = 1: 1 = 2A, so A = 1/2
5. Let x = -1: 1 = -2B, so B = -1/2
6. Result: 1/(x² - 1) = 1/(2(x-1)) - 1/(2(x+1))

Repeated factors: 1/(x - 1)² = A/(x - 1) + B/(x - 1)²

Quadratic factors: 1/(x² + 1) = (Ax + B)/(x² + 1)

Key Terms

• 03 04 Rational Functions
• Correct: A)
• Correct: B)
• Correct: C)
• Correct: D)
• Example 1: Sketch f(x) = (x + 2)/(x - 1)
• Example 2: Partial fractions
• Example 3: Rational inequality
• Horizontal Asymptotes
• Oblique (Slant) Asymptotes
• Partial Fractions Decomposition
• Vertical Asymptotes

Worked Examples

Example 1: Sketch f(x) = (x + 2)/(x - 1)

1. VA: x = 1
2. HA: y = 1/1 = 1
3. y-intercept: (0, -2)
4. x-intercept: x = -2 (where numerator = 0)
5. Test: as x → 1⁺, f(x) → +∞. As x → 1⁻, f(x) → -∞.
6. As x → ±∞, f(x) → 1 from above or below.

Example 2: Partial fractions

Decompose (2x + 3)/(x² - x - 2)

1. Factor: x² - x - 2 = (x - 2)(x + 1)
2. Setup: (2x + 3)/((x - 2)(x + 1)) = A/(x - 2) + B/(x + 1)
3. 2x + 3 = A(x + 1) + B(x - 2)
4. x = 2: 7 = 3A, A = 7/3
5. x = -1: 1 = -3B, B = -1/3
6. Result: (7/3)/(x - 2) - (1/3)/(x + 1)

Example 3: Rational inequality

Solve (x - 2)/(x + 1) ≥ 0

1. Find critical points: numerator = 0 at x = 2, denominator = 0 at x = -1
2. Number line intervals: (-∞, -1), (-1, 2], [2, ∞)
3. Sign test: x < -1: (-)/(−) = +; -1 < x < 2: (-)/(+) = −; x > 2: (+)/(+) = +
4. Include x = 2 (expression = 0). Exclude x = -1 (undefined).
5. Solution: (-∞, -1) ∪ [2, ∞)

Quiz

Q1: What does the concept of Horizontal Asymptotes primarily refer to in this subject?

A) The definition and application of Horizontal Asymptotes B) A historical anecdote about Horizontal Asymptotes C) A visual representation of Horizontal Asymptotes D) A computational error related to Horizontal Asymptotes

Correct: A)

• If you chose A: Horizontal Asymptotes is defined as: the definition and application of horizontal asymptotes. The other options describe different aspects that are not the primary focus. Correct!
• If you chose B: This is incorrect. Horizontal Asymptotes is defined as: the definition and application of horizontal asymptotes. The other options describe different aspects that are not the primary focus.
• If you chose C: This is incorrect. Horizontal Asymptotes is defined as: the definition and application of horizontal asymptotes. The other options describe different aspects that are not the primary focus.
• If you chose D: This is incorrect. Horizontal Asymptotes is defined as: the definition and application of horizontal asymptotes. The other options describe different aspects that are not the primary focus.

Q2: What is the primary purpose of Oblique (Slant) Asymptotes?

A) It is primarily a historical notation system B) It is used to oblique (slant) asymptotes in mathematical analysis C) It is used only in advanced research contexts D) It replaces all other methods in this domain

Correct: B)

• If you chose A: This is incorrect. Oblique (Slant) Asymptotes serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose B: Oblique (Slant) Asymptotes serves the purpose described in the correct answer. The other options misrepresent its role. Correct!
• If you chose C: This is incorrect. Oblique (Slant) Asymptotes serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose D: This is incorrect. Oblique (Slant) Asymptotes serves the purpose described in the correct answer. The other options misrepresent its role.

Q3: Which statement about Partial Fractions Decomposition is TRUE?

A) Partial Fractions Decomposition is an advanced topic beyond this subject's scope B) Partial Fractions Decomposition is a fundamental concept covered in this subject C) Partial Fractions Decomposition is mentioned only as a historical footnote D) Partial Fractions Decomposition is not related to this subject

Correct: B)

• If you chose A: This is incorrect. Partial Fractions Decomposition is a fundamental concept covered in this subject. This subject covers Partial Fractions Decomposition as part of its core content.
• If you chose B: Partial Fractions Decomposition is a fundamental concept covered in this subject. This subject covers Partial Fractions Decomposition as part of its core content. Correct!
• If you chose C: This is incorrect. Partial Fractions Decomposition is a fundamental concept covered in this subject. This subject covers Partial Fractions Decomposition as part of its core content.
• If you chose D: This is incorrect. Partial Fractions Decomposition is a fundamental concept covered in this subject. This subject covers Partial Fractions Decomposition as part of its core content.

Q4: Based on the worked examples in this subject, what is the correct result?

A) 1/(2(x-1)) - 1/(2(x+1)) B) An unrelated numerical value C) The inverse of the correct answer D) A different result from a common mistake

Correct: A)

• If you chose A: The worked examples show that the result is 1/(2(x-1)) - 1/(2(x+1)). The other options represent common errors. Correct!
• If you chose B: This is incorrect. The worked examples show that the result is 1/(2(x-1)) - 1/(2(x+1)). The other options represent common errors.
• If you chose C: This is incorrect. The worked examples show that the result is 1/(2(x-1)) - 1/(2(x+1)). The other options represent common errors.
• If you chose D: This is incorrect. The worked examples show that the result is 1/(2(x-1)) - 1/(2(x+1)). The other options represent common errors.

Q5: How are Partial Fractions Decomposition and Vertical Asymptotes related?

A) Partial Fractions Decomposition and Vertical Asymptotes are completely unrelated topics B) Partial Fractions Decomposition and Vertical Asymptotes are closely related concepts C) Partial Fractions Decomposition is the inverse of Vertical Asymptotes D) Partial Fractions Decomposition is a special case of Vertical Asymptotes

Correct: B)

• If you chose A: This is incorrect. Both Partial Fractions Decomposition and Vertical Asymptotes are covered in this subject as interconnected topics.
• If you chose B: Both Partial Fractions Decomposition and Vertical Asymptotes are covered in this subject as interconnected topics. Correct!
• If you chose C: This is incorrect. Both Partial Fractions Decomposition and Vertical Asymptotes are covered in this subject as interconnected topics.
• If you chose D: This is incorrect. Both Partial Fractions Decomposition and Vertical Asymptotes are covered in this subject as interconnected topics.

Q6: What is a common pitfall when working with What Is A Rational Function??

A) What Is A Rational Function? is always computed the same way in all contexts B) What Is A Rational Function? has no common misconceptions C) The main error with What Is A Rational Function? is using it when it is not needed D) A common mistake is confusing What Is A Rational Function? with a similar concept

Correct: D)

• If you chose A: This is incorrect. Students often confuse What Is A Rational Function? with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose B: This is incorrect. Students often confuse What Is A Rational Function? with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose C: This is incorrect. Students often confuse What Is A Rational Function? with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose D: Students often confuse What Is A Rational Function? with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!

Q7: When should you apply Example 1: Sketch F(X) = (X + 2)/(X - 1)?

A) Example 1: Sketch F(X) = (X + 2)/(X - 1) is not practically useful B) Avoid Example 1: Sketch F(X) = (X + 2)/(X - 1) unless explicitly instructed C) Apply Example 1: Sketch F(X) = (X + 2)/(X - 1) to solve problems in this subject's domain D) Use Example 1: Sketch F(X) = (X + 2)/(X - 1) only in pure mathematics contexts

Correct: C)

• If you chose A: This is incorrect. Example 1: Sketch F(X) = (X + 2)/(X - 1) is a practical tool used throughout this subject to solve relevant problems.
• If you chose B: This is incorrect. Example 1: Sketch F(X) = (X + 2)/(X - 1) is a practical tool used throughout this subject to solve relevant problems.
• If you chose C: Example 1: Sketch F(X) = (X + 2)/(X - 1) is a practical tool used throughout this subject to solve relevant problems. Correct!
• If you chose D: This is incorrect. Example 1: Sketch F(X) = (X + 2)/(X - 1) is a practical tool used throughout this subject to solve relevant problems.

Practice Problems

1. VA and HA of f(x) = 1/(x² - 4) Answer: VA: x = ±2. HA: y = 0 (deg numerator < deg denominator).

2. HA of f(x) = (3x² + 1)/(2x² - 5) Answer: y = 3/2 (equal degrees, ratio of leading coefficients).

3. Decompose 1/(x(x + 1)) Answer: A/x + B/(x+1). 1 = A(x+1) + Bx. x=0: A=1. x=-1: B=-1. Result: 1/x - 1/(x+1).

4. Solve (x + 1)/(x - 2) > 0 Answer: Critical points: x = -1, x = 2. Test intervals: (-∞, -1): negative/(-3) = positive. (-1, 2): positive/(-1) = negative. (2, ∞): positive/positive = positive. Solution: (-∞, -1) ∪ (2, ∞).

Summary

Key takeaways:

• Vertical asymptotes where denominator = 0
• Horizontal asymptotes from degree comparison
• Oblique asymptotes from polynomial division (degree diff = 1)
• Holes from cancelled factors
• Partial fractions: split proper rational functions into simpler terms

Pitfalls

• Confusing holes with vertical asymptotes: If a factor cancels between numerator and denominator, it creates a HOLE (removable discontinuity), not a vertical asymptote. Always simplify the rational function before identifying asymptotes. Forgetting to check for cancellation is a classic mistake.
• Misapplying horizontal asymptote rules: When deg(P) = deg(Q), the HA is y = (leading coefficient of P)/(leading coefficient of Q), not y = 0. When deg(P) > deg(Q) by exactly 1, you get an oblique asymptote — not "no asymptote."
• Including denominator zeros in inequality solutions: When solving rational inequalities, points where the denominator is zero are NEVER included (the expression is undefined there), while points where the numerator is zero are included if the inequality is ≤ or ≥. Confusing these leads to invalid solution intervals.
• Setting up partial fractions incorrectly: For repeated linear factors like (x-1)², you need terms A/(x-1) + B/(x-1)², not just A/(x-1)². For irreducible quadratic factors like (x²+1), you need (Ax+B)/(x²+1), not just A/(x²+1).
• Forgetting to check the degree condition for partial fractions: Partial fraction decomposition requires a PROPER fraction where deg(numerator) < deg(denominator). If the degree condition isn't met, you must perform polynomial long division first.

Next Steps

Next up: 03-05-exponential-and-logarithmic-functions.md