Rational Functions

Grade 12 · algebra · 101 practice problems · read aloud

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Rational Functions: Your Complete Guide

1. What Are Rational Functions? 🤔

A rational function is a function of the form f(x) = P(x)/Q(x), where both P(x) and Q(x) are polynomials, and Q(x) ≠ 0. They're essential for modeling real-world scenarios involving rates, concentrations, and optimization problems.

2. Key Features to Analyze

  1. Domain: All real numbers EXCEPT where denominator = 0
  2. Intercepts: x-intercept when numerator = 0, y-intercept at f(0)
  3. Asymptotes: Vertical where denominator = 0, horizontal/oblique by comparing degrees
  4. Holes: Occur when factors cancel from numerator and denominator

3. Worked Examples

Example 1: f(x) = (x² - 4)/(x - 2)

Step 1: Factor numerator: (x-2)(x+2)/(x-2)

Step 2: Cancel common factor: x+2 (x≠2)

Step 3: Hole at x=2, domain: (-∞,2)∪(2,∞)

Step 4: No vertical asymptote (factor canceled)

Example 2: f(x) = (2x+1)/(x²-9)

Step 1: Domain: x ≠ ±3

Step 2: Vertical asymptotes: x=3 and x=-3

Step 3: Horizontal asymptote: y=0 (degree numerator < degree denominator)

Step 4: x-intercept: set numerator=0 → x=-1/2

4. Common Mistakes to Avoid 🚫

Mistake 1: Forgetting to state domain restrictions

Fix: Always find where denominator = 0 FIRST

Mistake 2: Confusing holes vs. vertical asymptotes

Fix: Holes occur when factors cancel; asymptotes when they don't

Mistake 3: Incorrect horizontal asymptotes

Fix: Compare degrees: same → ratio of leading coefficients, numerator smaller → y=0, numerator larger → no HA

5. Tips & Tricks

BOBO BOTN EATS DC: Bigger On Bottom, Zero; Bigger On Top, None; Equal, Divide Coefficients (for horizontal asymptotes)

Always factor completely before analyzing - this reveals holes and simplifies finding intercepts

Check your work: Use graphing technology to verify asymptotes and intercepts

6. Practice Strategies

  • Start with simple rational functions and gradually increase complexity
  • Practice identifying domains without graphing first
  • Create your own problems by choosing different numerator/denominator combinations
  • Mix practice with polynomial long division for oblique asymptotes

Master rational functions by understanding they're just fractions with variables - the same rules apply!

Practice problems

6 of the 101, worked through step by step — try them before opening the answer.

1 lim(x→∞) (3x² - 2x + 1)/(x² + 4) = ?

Hint: For rational functions where the degrees of numerator and denominator are equal, the limit at infinity equals the ratio of leading coefficients.

Show the answer

Answer: 3

  1. Identify the highest power of x in the denominator. Here, the highest power is x² in both numerator and denominator.
  2. Divide every term in the numerator and denominator by x². Numerator: 3x²/x² - 2x/x² + 1/x² = 3 - 2/x + 1/x² Denominator: x²/x² + 4/x² = 1 + 4/x² So the expression becomes: (3 - 2/x + 1/x²) / (1 + 4/x²)
  3. Take the limit as x → ∞. As x → ∞: - 2/x → 0 - 1/x² → 0 - 4/x² → 0 So we get: (3 - 0 + 0) / (1 + 0) = 3/1 = 3
  4. Conclusion. The limit is 3. Answer: 3

Let's find the limit step by step. We want: lim(x→∞) (3x² - 2x + 1)/(x² + 4)

2 lim(x→∞) (3x² - 2x + 5)/(2x² + x - 1) = ?

Hint: For rational functions where the degrees of numerator and denominator are equal, the limit as x approaches infinity equals the ratio of the leading coefficients.

Show the answer

Answer: 3/2

  1. Identify the highest power of x in the denominator. The denominator is 2x² + x - 1. The highest power of x here is x².
  2. Divide every term in the numerator and denominator by x². This is the standard technique for limits at infinity of rational functions (polynomial divided by polynomial). We divide each term by x²: Numerator: (3x²/x²) - (2x/x²) + (5/x²) = 3 - (2/x) + (5/x²) Denominator: (2x²/x²) + (x/x²) - (1/x²) = 2 + (1/x) - (1/x²) So the function becomes: [3 - (2/x) + (5/x²)] / [2 + (1/x) - (1/x²)]
  3. Take the limit as x approaches infinity. As x becomes very large (approaches infinity), any term with x in the denominator approaches 0. - 2/x approaches 0 - 5/x² approaches 0 - 1/x approaches 0 - 1/x² approaches 0
  4. Substitute these limiting values. The expression becomes: [3 - 0 + 0] / [2 + 0 - 0] = 3/2
  5. State the final answer. Therefore, the limit is 3/2. Explanation: For large values of x, the lower degree terms (-2x, +5 in the numerator and +x, -1 in the denominator) become insignificant compared to the x² terms. The limit is determined by the ratio of the coefficients of the highest degree terms, which are 3 and 2.

To find the limit as x approaches infinity of (3x² - 2x + 5)/(2x² + x - 1), we follow these steps:

3 lim(x→∞) (3x² + 2x - 5)/(2x² - x + 7) = ?

Hint: For rational functions where the degrees of numerator and denominator are equal, the limit at infinity equals the ratio of the leading coefficients.

Show the answer

Answer: 3/2

  1. Identify the highest power of x in the denominator. The denominator is 2x² - x + 7. The highest power of x here is x².
  2. Divide every term in the numerator and denominator by x². This is the standard method for limits at infinity of rational functions (polynomial divided by polynomial). We divide each term by x². Numerator: (3x²)/x² + (2x)/x² - (5)/x² = 3 + 2/x - 5/x² Denominator: (2x²)/x² - (x)/x² + (7)/x² = 2 - 1/x + 7/x² So the function becomes: (3 + 2/x - 5/x²) / (2 - 1/x + 7/x²)
  3. Take the limit as x approaches infinity. As x becomes very large (approaches infinity), any term with x in the denominator will approach 0. Specifically: 2/x approaches 0 5/x² approaches 0 1/x approaches 0 7/x² approaches 0
  4. Substitute these limiting values. Our expression becomes: (3 + 0 - 0) / (2 - 0 + 0) = 3/2
  5. State the final answer. Therefore, the limit is 3/2. Explanation: For large values of x, the lower degree terms (2x, -5 in the numerator and -x, +7 in the denominator) become insignificant compared to the x² terms. The limit is determined by the ratio of the coefficients of the highest power terms, which are 3 and 2.

To find the limit as x approaches infinity of (3x² + 2x - 5)/(2x² - x + 7), we follow these steps:

4 lim(x→∞) (3x² - 2x + 1)/(x² + 5x - 3) = ?

Hint: For rational functions where numerator and denominator have the same highest degree, the limit at infinity equals the ratio of leading coefficients.

Show the answer

Answer: 3

  1. Identify the highest power of x in the denominator** The denominator is x² + 5x - 3. The highest power is x². --- **
  2. Divide numerator and denominator by x²** This is the standard method for limits at infinity of rational functions. Numerator: (3x² - 2x + 1) / x² = 3 - 2/x + 1/x² Denominator: (x² + 5x - 3) / x² = 1 + 5/x - 3/x² So the expression becomes: [3 - 2/x + 1/x²] / [1 + 5/x - 3/x²] --- **
  3. Take the limit as x → ∞** As x → ∞: - 2/x → 0 - 1/x² → 0 - 5/x → 0 - 3/x² → 0 So the expression becomes: (3 - 0 + 0) / (1 + 0 - 0) = 3/1 = 3 --- **
  4. Conclusion** The limit is 3. --- **Final Answer:** 3

Let's find the limit step by step. We want: lim(x→∞) (3x² - 2x + 1)/(x² + 5x - 3) --- **

5 lim(x→∞) (3x² + 2x - 1)/(x² - 4x + 7) = ?

Hint: For rational functions where the degrees of numerator and denominator are equal, the limit at infinity equals the ratio of the leading coefficients.

Show the answer

Answer: 3

  1. Identify the highest power of x in the denominator. The denominator is x² - 4x + 7. The highest power of x here is x².
  2. Divide every term in both the numerator and the denominator by this highest power (x²). Numerator becomes: (3x²)/(x²) + (2x)/(x²) - (1)/(x²) = 3 + 2/x - 1/x² Denominator becomes: (x²)/(x²) - (4x)/(x²) + (7)/(x²) = 1 - 4/x + 7/x² So the function becomes: (3 + 2/x - 1/x²) / (1 - 4/x + 7/x²)
  3. Take the limit as x approaches infinity. As x becomes very large (approaches infinity), any term with x in the denominator will approach 0. Specifically: 2/x approaches 0 1/x² approaches 0 4/x approaches 0 7/x² approaches 0
  4. Substitute these limiting values into our expression. The limit becomes: (3 + 0 - 0) / (1 - 0 + 0) = 3/1 = 3 Therefore, the limit is 3. Explanation: When finding limits at infinity of rational functions (polynomial divided by polynomial), we compare the degrees of the numerator and denominator. Since both have the same degree (2), the limit is simply the ratio of the leading coefficients: 3/1 = 3.

To find the limit as x approaches infinity of (3x² + 2x - 1)/(x² - 4x + 7), we follow these steps:

6 lim_(x→∞) (3x² - 2x + 5)/(4x² + x - 1) = ?

Hint: For rational functions where numerator and denominator have the same highest degree, the limit at infinity equals the ratio of leading coefficients.

Show the answer

Answer: 3/4

  1. Identify the highest power of x in the denominator. The denominator is 4x² + x - 1. The highest power of x here is x².
  2. Divide every term in the numerator and denominator by x². This is the standard technique for limits at infinity of rational functions (polynomial divided by polynomial). We divide each term by x²: Numerator: (3x²/x²) - (2x/x²) + (5/x²) = 3 - (2/x) + (5/x²) Denominator: (4x²/x²) + (x/x²) - (1/x²) = 4 + (1/x) - (1/x²) So the expression becomes: [3 - 2/x + 5/x²] / [4 + 1/x - 1/x²]
  3. Analyze what happens to each term as x approaches infinity. As x becomes very large (approaches infinity): - Terms with x in the denominator (like 2/x, 5/x², 1/x, 1/x²) approach 0. - The constant terms (3 and 4) remain unchanged.
  4. Take the limit by applying the limit to each term. lim(x→∞) [3 - 2/x + 5/x²] / [4 + 1/x - 1/x²] = [3 - 0 + 0] / [4 + 0 - 0] = 3/4
  5. State the final answer. Therefore, the limit is 3/4. Explanation of the reasoning: When finding limits at infinity for rational functions, the terms with the highest power of x dominate the behavior. By dividing numerator and denominator by this highest power, we transform the expression into a form where we can easily see which terms vanish and which remain, giving us the limit value.

To find the limit as x approaches infinity of (3x² - 2x + 5)/(4x² + x - 1), we follow these steps:

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