E6

Rational Functions and Their Applications

Vertical and Horizontal Asymptotes

Rational functions are quotients of two polynomials and often exhibit asymptotic behavior. Understanding the location and nature of asymptotes is essential for analyzing and graphing these functions.

• Vertical Asymptotes: Occur at values of x where the denominator is zero (and the numerator is nonzero).

• Horizontal Asymptotes: Determined by the degrees of the numerator and denominator polynomials.

• Holes: Occur at values of x where both the numerator and denominator are zero (common factors).

Examples:

• Function: $f(x) = \frac{1}{(x-1)(x-3)}$ Vertical Asymptotes: $x=1$, $x=3$

• Function: $f(x) = \frac{1}{(x-1)(x-2)}$ Vertical Asymptotes: $x=1$ Hole: $x=2$

Determining Asymptotes in Rational Functions

To analyze rational functions, identify the types of asymptotes present:

• Vertical Asymptotes: Set denominator equal to zero and solve for x.

• Horizontal Asymptotes:

  • If degree of numerator < degree of denominator: $y=0$

  • If degrees are equal: $y=\frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$

  • If degree of numerator > degree of denominator: No horizontal asymptote (may have an oblique/slant asymptote)

Table: Types of Asymptotes in Sample Rational Functions

Function	Vertical Asymptote	Horizontal Asymptote	Slant Asymptote	Holes
$\frac{1}{x-2}$	$x=2$	$y=0$	None	None
$\frac{2x}{x-2}$	$x=2$	$y=0$	None	None
$\frac{x^2}{x-2}$	$x=2$	None	$y=x$	None
$\frac{2x^2-3x+4}{x-2}$	$x=2$	None	$y=2x+1$	None

Applications of Rational Functions

Modeling Real-World Phenomena

Rational functions are used to model various real-world scenarios, especially those involving rates, concentrations, and diminishing returns. Asymptotic behavior often reflects physical or economic limits.

Example 1: Pollution Removal Cost

The cost (in millions of dollars) for removing p percent of pollutants from a river is given by:

$C(p) = \frac{120p}{100-p}$

• Domain: $0 \leq p < 100$ (cannot remove 100% of pollutants)

• Vertical Asymptote: $p=100$ (cost becomes infinite as removal approaches 100%)

• Application: Used to estimate costs for environmental cleanup projects.

Sample Table of Values:

p (%)	10	30	50	70	90	95	99
$C(p)$	13.3	51.4	120	280	1080	2280	11880

• Interpretation: As p approaches 100, the cost increases rapidly, illustrating diminishing returns.

Example 2: Average Cost in Manufacturing

Let x represent the number of laptops produced in a week. The average cost per laptop is:

$C(x) = \frac{2000}{x} + 1200$

• Fixed Cost: $2000$ (spread over all units produced)

• Variable Cost: $1200$ per laptop

• Horizontal Asymptote: $y=1200$ (as production increases, average cost approaches $1200$)

Sample Table of Values:

x	1	10	100	1000	10,000
$C(x)$	3200	1400	1220	1202	1200.2

• Interpretation: For large x, the average cost per laptop approaches the variable cost.

Practice Problems

Sample Problems on Rational Functions

1. Drug Concentration: The concentration of a drug in the bloodstream is given by $C(t) = \frac{7}{t+2}$, where t is hours since administration.

   • Find $C(3)$: $C(3) = \frac{7}{5} = 1.4$ mg/L

   • How long until $C(t) = 1$? Solve $1 = \frac{7}{t+2} \implies t = 5$ hours

   • Graph $C(t)$ for $t = 0, 1, 2, 3, 4, 5$

2. Mixing Solutions: A 20% glucose solution is mixed with 2 L of glucose-free water. The concentration is $C(x) = \frac{20x}{x+2}$, where x is the volume of 20% solution added.

   • Find x so $C(x) = 10$: $10 = \frac{20x}{x+2} \implies x = 2$ L

   • Complete a table for $x = 0, 1, 2, 3, 4, 5$

3. Manufacturing Cost: For $C(x) = \frac{2000}{x} + 400$, find the horizontal asymptote and interpret its meaning.

   • Horizontal asymptote: $y=400$

   • Interpretation: As production increases, average cost approaches $400$ per unit.

Summary Table: Asymptotes in Rational Functions

Asymptote Type	How to Find	Interpretation
Vertical	Set denominator = 0	Function grows without bound near this value
Horizontal	Compare degrees of numerator and denominator	Long-term behavior as $x \to \infty$
Slant (Oblique)	Numerator degree is one more than denominator	Function approaches a line as $x \to \infty$

Additional info: Rational functions are widely used in science, engineering, and economics to model processes with limiting behavior, such as diminishing returns, rates of change, and concentrations over time.