Lesson 3: Rational Functions

Example 1: Identifying Rational Functions

Which of the following are rational functions?

a) f(x) = (x² + 3x - 2) / (x - 5)

Answer: YES - ratio of two polynomials

b) g(x) = (√x + 1) / (x² - 4)

Answer: NO - numerator contains √x, which is not a polynomial

c) h(x) = 7 / (x³ - 2x + 1)

Answer: YES - 7 is a polynomial (degree 0), denominator is a polynomial

d) k(x) = (2x³ - x) / 5

Answer: YES - this simplifies to a polynomial, which is a special case of a rational function

Example 2: Finding Domain

Find the domain of f(x) = (x + 3) / (x² - 9)

Step 1: Set denominator equal to zero

x² - 9 = 0

Step 2: Solve for x

x² = 9

x = ±3

Step 3: State the domain

Domain: All real numbers except x = -3 and x = 3

Interval notation: (-∞, -3) ∪ (-3, 3) ∪ (3, ∞)

Set notation: {x | x ≠ -3, x ≠ 3}

Example 3: Domain with Factoring

Find the domain of g(x) = 5 / (x² + 7x + 12)

Step 1: Set denominator equal to zero and factor

x² + 7x + 12 = 0

(x + 3)(x + 4) = 0

Step 2: Solve

x = -3 or x = -4

Domain: All real numbers except x = -3 and x = -4

Interval notation: (-∞, -4) ∪ (-4, -3) ∪ (-3, ∞)

Example 4: Finding Vertical Asymptotes

Find all vertical asymptotes of f(x) = (x + 2) / (x² - 4)

Step 1: Factor numerator and denominator

Numerator: x + 2 (already factored)

Denominator: x² - 4 = (x + 2)(x - 2)

Step 2: Cancel common factors

f(x) = (x + 2) / [(x + 2)(x - 2)] = 1 / (x - 2), x ≠ -2

Step 3: Find vertical asymptotes from remaining denominator

x - 2 = 0

x = 2

Vertical asymptote: x = 2

Note: x = -2 creates a hole, not a vertical asymptote (we'll discuss holes later)

Example 5: Multiple Vertical Asymptotes

Find all vertical asymptotes of h(x) = (3x - 1) / (x² - 5x + 6)

Step 1: Factor the denominator

x² - 5x + 6 = (x - 2)(x - 3)

Step 2: Check for common factors

Numerator: 3x - 1 (prime)

No common factors to cancel

Step 3: Set denominator factors equal to zero

x - 2 = 0 → x = 2

x - 3 = 0 → x = 3

Vertical asymptotes: x = 2 and x = 3

Example 6: No Vertical Asymptotes

Find vertical asymptotes of f(x) = (x² + 1) / (x² + 4)

Step 1: Set denominator equal to zero

x² + 4 = 0

x² = -4

Step 2: Solve

No real solutions (x² cannot be negative)

Answer: NO vertical asymptotes

This function is defined for all real numbers!

Example 7: Horizontal Asymptote (Case 1)

Find the horizontal asymptote of f(x) = (3x + 2) / (x² - 1)

Step 1: Identify degrees

Degree of numerator: 1

Degree of denominator: 2

Since 1 < 2, we have Case 1

Step 2: Apply the rule

Horizontal asymptote: y = 0

Interpretation: As x → ±∞, f(x) → 0

Example 8: Horizontal Asymptote (Case 2)

Find the horizontal asymptote of g(x) = (4x² - 3x + 1) / (2x² + 5)

Step 1: Identify degrees

Degree of numerator: 2

Degree of denominator: 2

Since 2 = 2, we have Case 2

Step 2: Find ratio of leading coefficients

Leading coefficient of numerator: 4

Leading coefficient of denominator: 2

Ratio: 4/2 = 2

Horizontal asymptote: y = 2

Interpretation: As x → ±∞, g(x) → 2

Example 9: Horizontal Asymptote (Case 3)

Find the horizontal asymptote of h(x) = (x³ - 2x) / (x² + 1)

Step 1: Identify degrees

Degree of numerator: 3

Degree of denominator: 2

Since 3 > 2, we have Case 3

Step 2: Apply the rule

No horizontal asymptote

Note: The function grows without bound as x → ±∞

Example 10: Complete Asymptote Analysis

Find all asymptotes of f(x) = (2x² + x - 6) / (x² - x - 2)

Step 1: Factor to find vertical asymptotes

Numerator: 2x² + x - 6 = (2x - 3)(x + 2)

Denominator: x² - x - 2 = (x - 2)(x + 1)

No common factors

Vertical asymptotes: x = 2 and x = -1

Step 2: Find horizontal asymptote

Both numerator and denominator have degree 2

Leading coefficients: 2 (numerator) and 1 (denominator)

Horizontal asymptote: y = 2/1 = 2

Example 11: Finding a Hole

Find any holes in f(x) = (x² - 4) / (x² + 5x + 6)

Step 1: Factor completely

Numerator: x² - 4 = (x - 2)(x + 2)

Denominator: x² + 5x + 6 = (x + 2)(x + 3)

Step 2: Identify common factors

Common factor: (x + 2)

Step 3: Cancel and simplify

f(x) = (x - 2) / (x + 3), x ≠ -2

Step 4: Find x-coordinate of hole

x + 2 = 0 → x = -2

Step 5: Find y-coordinate

Use simplified function: f(-2) = (-2 - 2) / (-2 + 3) = -4/1 = -4

Hole at: (-2, -4)

Vertical asymptote at: x = -3

Example 12: Distinguishing Holes from Asymptotes

Analyze f(x) = (x² - 9) / (x² + x - 12)

Step 1: Factor

Numerator: x² - 9 = (x - 3)(x + 3)

Denominator: x² + x - 12 = (x + 4)(x - 3)

Step 2: Cancel common factor (x - 3)

f(x) = (x + 3) / (x + 4), x ≠ 3

Step 3: Find hole from cancelled factor

x - 3 = 0 → x = 3

y-coordinate: (3 + 3)/(3 + 4) = 6/7

Hole: (3, 6/7)

Step 4: Find vertical asymptote from remaining denominator

x + 4 = 0 → x = -4

Vertical asymptote: x = -4

Step 5: Find horizontal asymptote

After simplification: degree 1/degree 1

Leading coefficients: 1/1

Horizontal asymptote: y = 1

Example 13: Complete Graphing Example

Sketch the graph of f(x) = (x + 1) / (x - 2)

Step 1: Domain

x - 2 ≠ 0, so x ≠ 2

Domain: (-∞, 2) ∪ (2, ∞)

Step 2: Vertical asymptote

x - 2 = 0 → x = 2

Step 3: Horizontal asymptote

Degrees equal (both 1), leading coefficients both 1

y = 1/1 = 1

Step 4: X-intercept

x + 1 = 0 → x = -1

X-intercept: (-1, 0)

Step 5: Y-intercept

f(0) = (0 + 1)/(0 - 2) = -1/2

Y-intercept: (0, -1/2)

Step 6: Test points

For x < -1: f(-2) = (-1)/(-4) = 1/4 (above x-axis)

For -1 < x < 2: f(1) = 2/(-1) = -2 (below x-axis)

For x > 2: f(3) = 4/1 = 4 (above x-axis)

Graph behavior:

• Approaches y = 1 as x → ±∞
• Has vertical asymptote at x = 2
• Crosses x-axis at (-1, 0)
• Crosses y-axis at (0, -1/2)

Example 14: Graphing with a Hole

Analyze and sketch f(x) = (x² - 1) / (x² - 3x + 2)

Step 1: Factor

Numerator: x² - 1 = (x - 1)(x + 1)

Denominator: x² - 3x + 2 = (x - 1)(x - 2)

Step 2: Cancel (x - 1)

f(x) = (x + 1) / (x - 2), x ≠ 1

Step 3: Find hole

x = 1, y = (1 + 1)/(1 - 2) = 2/(-1) = -2

Hole at (1, -2)

Step 4: Vertical asymptote

x = 2

Step 5: Horizontal asymptote

y = 1 (degrees equal, leading coefficients 1/1)

Step 6: Intercepts

X-intercept: x + 1 = 0 → x = -1, point (-1, 0)

Y-intercept: f(0) = 1/(-2) = -1/2, point (0, -1/2)

Key features:

• Hole (open circle) at (1, -2)
• Vertical asymptote: x = 2
• Horizontal asymptote: y = 1
• Intercepts: (-1, 0) and (0, -1/2)

Example 15: Average Cost Application

A company's cost to produce x items is C(x) = 5000 + 20x dollars. Find the average cost per item function and analyze it.

Step 1: Set up average cost function

Average cost = Total cost / Number of items

A(x) = (5000 + 20x) / x = 5000/x + 20

Step 2: Find vertical asymptote

x = 0 (cannot produce 0 items)

Step 3: Find horizontal asymptote

Rewrite: A(x) = 5000/x + 20

As x → ∞, 5000/x → 0

Horizontal asymptote: y = 20

Interpretation:

• As production increases, average cost approaches $20 per item
• The $5000 fixed cost gets spread over more items
• $20 represents the variable cost per item

Example values:

A(100) = 5000/100 + 20 = $70 per item

A(1000) = 5000/1000 + 20 = $25 per item

A(10000) = 5000/10000 + 20 = $20.50 per item

Example 16: Concentration Problem

A tank contains 100 gallons of pure water. A solution containing 2 lbs of salt per gallon flows in at 5 gallons/minute. The mixture flows out at the same rate. Find the concentration after t minutes.

Step 1: Set up the function

Amount of salt entering: 2 lbs/gal × 5 gal/min = 10 lbs/min

After t minutes: 10t lbs of salt in 100 gallons

Concentration: C(t) = 10t / 100 = t/10 lbs per gallon

Note: This is actually a linear function in this simplified model

More realistic model with mixing:

C(t) = 2(1 - e^-t/20) approaches 2 lbs/gal as t → ∞