Rational Functions and Asymptotes

What is a Rational Function?

Rational function: Ratio of two polynomials

Form: f(x) = P(x)/Q(x)

Where P(x) and Q(x) are polynomials, Q(x) ≠ 0

Examples:

• f(x) = 1/x
• f(x) = (x + 1)/(x - 2)
• f(x) = (x² - 4)/(x² - 9)

Domain: All real numbers except where Q(x) = 0

Vertical Asymptotes

Vertical asymptote: Vertical line x = a where function approaches ±∞

Occur when denominator = 0 (and numerator ≠ 0)

To find:

1. Factor numerator and denominator
2. Cancel common factors
3. Set remaining denominator = 0
4. Solve for x

Example 1: Simple Vertical Asymptote

Find vertical asymptotes: f(x) = 1/(x - 3)

Denominator = 0:

x - 3 = 0
x = 3

Vertical asymptote: x = 3

Example 2: Multiple Asymptotes

Find vertical asymptotes: f(x) = x/(x² - 4)

Factor denominator:

x² - 4 = (x - 2)(x + 2)

Set = 0:

x = 2  and  x = -2

Vertical asymptotes: x = 2 and x = -2

Example 3: After Canceling

Find vertical asymptotes: f(x) = (x² - 9)/(x² - x - 6)

Factor:

f(x) = (x - 3)(x + 3) / [(x - 3)(x + 2)]

Cancel (x - 3):

f(x) = (x + 3)/(x + 2)  [x ≠ 3]

Vertical asymptote: x = -2 only (x = 3 is a hole, not an asymptote)

Holes (Removable Discontinuities)

Hole: Point where function undefined, but could be "filled in"

Occur when factor cancels from numerator AND denominator

To find:

1. Factor completely
2. Identify common factors that cancel
3. x-value where that factor = 0 is the hole
4. Find y-value by evaluating simplified function

Example: Find Hole

Find hole in f(x) = (x² - 1)/(x - 1)

Factor:

f(x) = (x - 1)(x + 1)/(x - 1)

Cancel (x - 1):

f(x) = x + 1  [x ≠ 1]

Hole at x = 1

y-coordinate: Evaluate x + 1 at x = 1

y = 1 + 1 = 2

Hole at point (1, 2)

Horizontal Asymptotes

Horizontal asymptote: Horizontal line y = b as x → ±∞

Rules (compare degrees of numerator and denominator):

If degree(P) < degree(Q): y = 0

If degree(P) = degree(Q): y = (leading coefficient of P)/(leading coefficient of Q)

If degree(P) > degree(Q): No horizontal asymptote (may have slant asymptote)

Example 1: Degree of Numerator < Denominator

f(x) = 3/(x² + 1)

Degree of numerator: 0 Degree of denominator: 2

Horizontal asymptote: y = 0

Example 2: Equal Degrees

f(x) = (2x² + 3x - 1)/(x² - 4)

Both degree 2

Leading coefficients: 2 and 1

Horizontal asymptote: y = 2/1 = 2

Example 3: Numerator Higher Degree

f(x) = (x³ + 1)/(x² - 1)

Degree 3 > Degree 2

No horizontal asymptote

Slant (Oblique) Asymptotes

Slant asymptote: Diagonal line as x → ±∞

Occur when degree(P) = degree(Q) + 1

To find: Divide P(x) by Q(x); quotient is slant asymptote

Example: Find Slant Asymptote

f(x) = (x² + 2x - 1)/(x - 1)

Degree 2 = Degree 1 + 1, so slant asymptote exists

Divide:

         x + 3
       ________
x - 1 | x² + 2x - 1
        x² - x
        ________
            3x - 1
            3x - 3
            ______
                 2

Slant asymptote: y = x + 3

(Ignore remainder for asymptote)

Graphing Strategy

To graph rational function:

1. Factor numerator and denominator
2. Find domain: Exclude zeros of denominator
3. Find holes: Common factors that cancel
4. Find vertical asymptotes: Zeros of reduced denominator
5. Find horizontal/slant asymptote: Compare degrees
6. Find x-intercepts: Zeros of numerator (not holes)
7. Find y-intercept: f(0) if defined
8. Plot points in each region
9. Sketch approaching asymptotes

Example: Complete Graph Analysis

Graph: f(x) = (x² - 4)/(x² - 1)

Step 1: Factor

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

Step 2: Domain

All reals except x = ±1

Step 3: Holes

No common factors, no holes

Step 4: Vertical asymptotes

x = 1 and x = -1

Step 5: Horizontal asymptote

Equal degrees: y = 1/1 = 1

Step 6: x-intercepts

x² - 4 = 0 → x = ±2

Step 7: y-intercept

f(0) = -4/-1 = 4

Behavior Near Vertical Asymptotes

Check sign on each side of asymptote

Determines if function goes to +∞ or -∞

Example: Analyze Behavior

f(x) = 1/(x - 2)

Near x = 2:

• As x → 2⁺ (from right): 1/(small positive) → +∞
• As x → 2⁻ (from left): 1/(small negative) → -∞

End Behavior

Behavior as x → ±∞

Determined by horizontal or slant asymptote

Example: End Behavior

f(x) = (3x² + 1)/(x² - 4)

Horizontal asymptote: y = 3

End behavior:

• As x → ∞, f(x) → 3
• As x → -∞, f(x) → 3

Special Cases

Even function: f(-x) = f(x), symmetric about y-axis

Odd function: f(-x) = -f(x), symmetric about origin

Example: Odd Rational Function

f(x) = x/(x² + 1)

Check:

f(-x) = -x/((-x)² + 1) = -x/(x² + 1) = -f(x)

Odd function, symmetric about origin

Real-World Applications

Average cost: C(x) = (fixed costs + variable costs)/x

Concentration: Amount/volume as volume changes

Physics: Combined resistance, lens equations

Biology: Michaelis-Menten enzyme kinetics

Example: Average Cost

Total cost: C(x) = 5000 + 20x Average cost per unit:

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

Vertical asymptote: x = 0 (can't make 0 units) Horizontal asymptote: y = 20 (minimum average cost)

Common Mistakes to Avoid

Don't forget to factor first - holes vs asymptotes

Don't confuse vertical and horizontal rules

Check if factor cancels - affects holes vs asymptotes

Remember domain restrictions from original function

Practice