Rational Functions: Polynomials Divided by Polynomials

f(x) = 1/x.

As x gets large, f(x) gets small. As x gets very large, f(x) gets very small. It approaches zero but never equals zero.

As x approaches zero, f(x) blows up. It shoots toward infinity.

That's a rational function.

A ratio of polynomials. And the behavior at the edges—where the function approaches infinity or flattens out—is captured by asymptotes.

The Unlock: Division Creates Boundaries

Polynomials grow smoothly. They have no gaps, no jumps, no forbidden values.

Rational functions are different. They have boundaries.

Dividing by zero breaks everything. So rational functions have vertical asymptotes—places where the denominator is zero and the function explodes.

As inputs get very large, the function often settles toward a constant. That's a horizontal asymptote.

Asymptotes are the lines the function approaches but never touches. They define the function's edges.

The Definition: Ratio of Polynomials

A rational function is any function of the form:

f(x) = p(x) / q(x)

where p(x) and q(x) are polynomials.

Example: f(x) = (x² + 1) / (x - 2).

Numerator: x² + 1. Denominator: x - 2.

The function is defined everywhere except where the denominator is zero.

q(x) = 0 → x = 2. So f is undefined at x = 2.

That's where the vertical asymptote is.

Vertical Asymptotes: Where the Function Blows Up

A vertical asymptote occurs at x = a if:

lim (x → a) f(x) = ±∞.

The function shoots up to positive infinity or down to negative infinity.

Rule: Set the denominator equal to zero. Solve for x. Those values are the vertical asymptotes (unless there's a removable discontinuity).

Example: f(x) = 1/(x - 3).

Denominator: x - 3. Set to zero: x = 3.

Vertical asymptote at x = 3.

As x → 3⁺ (from the right), f(x) → +∞.

As x → 3⁻ (from the left), f(x) → -∞.

The function has a vertical barrier at x = 3.

Horizontal Asymptotes: Where the Function Flattens

A horizontal asymptote is a horizontal line y = L that the function approaches as x → ±∞.

Rule: Compare the degrees of the numerator and denominator.

1. Degree of numerator < degree of denominator: Horizontal asymptote at y = 0.

Example: f(x) = 1/x².

Numerator degree: 0. Denominator degree: 2.

As x → ±∞, f(x) → 0.

Horizontal asymptote: y = 0.

2. Degree of numerator = degree of denominator: Horizontal asymptote at y = (leading coefficient of numerator) / (leading coefficient of denominator).

Example: f(x) = (3x² + 2) / (5x² - 1).

Both degrees are 2. Leading coefficients: 3 and 5.

Horizontal asymptote: y = 3/5.

As x → ±∞, f(x) → 3/5.

3. Degree of numerator > degree of denominator: No horizontal asymptote. (There may be an oblique asymptote.)

Example: f(x) = x² / x = x.

Numerator degree: 2. Denominator degree: 1.

No horizontal asymptote. The function grows without bound.

Why Horizontal Asymptotes Happen

As x gets very large, the highest-degree terms dominate.

Example: f(x) = (2x² + 3x + 1) / (x² - x + 5).

For very large x, the lower-degree terms (3x, 1, -x, 5) become negligible.

f(x) ≈ 2x² / x² = 2.

So the horizontal asymptote is y = 2.

The function settles toward 2 as x → ±∞.

Oblique (Slant) Asymptotes

If the degree of the numerator is exactly one more than the degree of the denominator, the function has an oblique asymptote—a slanted line.

To find it, perform polynomial long division.

Example: f(x) = (x² + 1) / x.

Divide: x² / x = x. Remainder: 1.

f(x) = x + 1/x.

As x → ±∞, 1/x → 0, so f(x) → x.

Oblique asymptote: y = x.

The function gets closer and closer to the line y = x.

Removable Discontinuities: Holes in the Graph

Sometimes the numerator and denominator share a common factor.

When you cancel it, the function simplifies—but there's a hole at the canceled value.

Example: f(x) = (x² - 1) / (x - 1).

Factor: (x - 1)(x + 1) / (x - 1).

Cancel: f(x) = x + 1 (for x ≠ 1).

At x = 1, the original function is undefined (0/0).

But the simplified function is f(x) = x + 1, which gives f(1) = 2.

So the graph is the line y = x + 1 with a hole at (1, 2).

This is a removable discontinuity. The function can be "fixed" by defining f(1) = 2.

Graphing Rational Functions

To graph a rational function:

1. Find vertical asymptotes: Set denominator = 0.
2. Find horizontal or oblique asymptotes: Compare degrees or divide.
3. Find intercepts:
   • x-intercepts: Set numerator = 0.
   • y-intercept: Evaluate f(0).
4. Check for holes: Factor and cancel common terms.
5. Plot a few points to determine shape.
6. Sketch the graph, respecting asymptotes and intercepts.

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

Vertical asymptote: x + 2 = 0 → x = -2.

Horizontal asymptote: Degrees equal (both 1). Leading coefficients: 1 and 1. y = 1.

x-intercept: x - 1 = 0 → x = 1.

y-intercept: f(0) = -1/2.

No holes: No common factors.

Sketch: The graph approaches x = -2 vertically and y = 1 horizontally. It passes through (1, 0) and (0, -1/2).

Behavior Near Vertical Asymptotes

Near a vertical asymptote, the function either shoots to +∞ or -∞ on each side.

To determine which, test values on each side.

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

Vertical asymptote at x = 2.

Test x = 1 (left of 2): f(1) = 1/(1 - 2) = -1. Negative.

Test x = 3 (right of 2): f(3) = 1/(3 - 2) = 1. Positive.

As x → 2⁻, f(x) → -∞.

As x → 2⁺, f(x) → +∞.

The function drops to negative infinity on the left, jumps to positive infinity on the right.

Zeros of Rational Functions

The zeros of f(x) = p(x)/q(x) are the zeros of the numerator (where p(x) = 0), provided they're not also zeros of the denominator.

Example: f(x) = (x² - 4) / (x - 1).

Numerator zeros: x² - 4 = 0 → x = ±2.

Denominator zero: x = 1.

Zeros of f: x = 2 and x = -2. (Not x = 1, because the function is undefined there.)

Sign Analysis

Rational functions change sign at zeros and vertical asymptotes.

To determine where f(x) is positive or negative:

1. Find zeros (numerator = 0) and vertical asymptotes (denominator = 0).
2. These divide the number line into intervals.
3. Test a point in each interval.

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

Zeros: x = 1. Vertical asymptotes: x = -2.

Intervals: (-∞, -2), (-2, 1), (1, ∞).

Test x = -3: f(-3) = (-4)/(-1) = 4 > 0. Positive.

Test x = 0: f(0) = (-1)/(2) = -0.5 < 0. Negative.

Test x = 2: f(2) = (1)/(4) = 0.25 > 0. Positive.

Sign chart:

• (-∞, -2): positive
• (-2, 1): negative
• (1, ∞): positive

End Behavior

End behavior describes what happens as x → ±∞.

It's determined by the horizontal or oblique asymptote.

Example: f(x) = (2x² + 1) / (x² - 1).

Horizontal asymptote: y = 2.

As x → ±∞, f(x) → 2.

Rational Functions and Rates

Rational functions model rates and proportions.

Example: Average cost.

If it costs C(x) = 1000 + 50x to produce x units, the average cost per unit is:

A(x) = C(x)/x = (1000 + 50x)/x = 1000/x + 50.

As x → ∞, A(x) → 50.

The fixed cost (1000) is spread over more units, so the average cost approaches the variable cost (50).

Example: Drug concentration.

A drug is metabolized over time. Concentration might follow:

C(t) = (100t) / (t² + 4).

As t → 0, C(t) → 0 (no time for absorption).

As t → ∞, C(t) → 0 (drug is metabolized).

There's a peak concentration at some intermediate time.

Rational Inequalities

To solve f(x) > 0 or f(x) < 0, use sign analysis.

Example: Solve (x - 1)/(x + 2) > 0.

Zeros: x = 1. Vertical asymptotes: x = -2.

Intervals: (-∞, -2), (-2, 1), (1, ∞).

Test: Positive on (-∞, -2) and (1, ∞).

Solution: x ∈ (-∞, -2) ∪ (1, ∞).

Transformations of Rational Functions

Rational functions transform like any other function.

Example: f(x) = 1/x.

g(x) = 1/(x - 2): Shift right 2.

h(x) = 1/x + 3: Shift up 3.

k(x) = -1/x: Reflect vertically.

m(x) = 1/(x - 2) + 3: Shift right 2, up 3.

Vertical asymptote shifts with horizontal shifts. Horizontal asymptote shifts with vertical shifts.

Partial Fraction Decomposition

A complex rational function can sometimes be decomposed into simpler fractions.

Example: f(x) = (3x + 5) / ((x - 1)(x + 2)).

Decompose: f(x) = A/(x - 1) + B/(x + 2).

Solve for A and B:

3x + 5 = A(x + 2) + B(x - 1).

Set x = 1: 8 = 3A → A = 8/3.

Set x = -2: -1 = -3B → B = 1/3.

f(x) = (8/3)/(x - 1) + (1/3)/(x + 2).

This is useful for integration and solving differential equations.

Rational Functions in Physics

Lens equation: 1/f = 1/d_o + 1/d_i.

Relates focal length, object distance, and image distance. Solving for one in terms of the others gives rational functions.

Ohm's law and resistors in parallel: R_total = 1 / (1/R₁ + 1/R₂).

Rational expressions model electrical resistance.

Gravitational potential: U(r) = -GMm/r.

Potential energy as a rational function of distance.

Rational functions are ubiquitous in physics.

Limits and Rational Functions

Rational functions are where you first encounter interesting limits.

Example: lim (x → ∞) (2x² + 1) / (x² - 1).

Divide numerator and denominator by x²:

lim (x → ∞) (2 + 1/x²) / (1 - 1/x²).

As x → ∞, 1/x² → 0.

Limit: 2/1 = 2.

Example: lim (x → 2) (x² - 4) / (x - 2).

Factor: (x - 2)(x + 2) / (x - 2) = x + 2 (for x ≠ 2).

Limit: 2 + 2 = 4.

Even though the function is undefined at x = 2, the limit exists.

Rational Functions Are Everywhere

Any time you model a ratio, you get a rational function.

Economics: Marginal cost, average cost, price elasticity.

Biology: Enzyme kinetics (Michaelis-Menten equation), population models.

Chemistry: Reaction rates, equilibrium constants.

Engineering: Transfer functions, frequency response.

Rational functions are the natural language of ratios, rates, and proportions.

Asymptotes as Conceptual Boundaries

Asymptotes aren't just technicalities. They encode the function's limits—literally.

A vertical asymptote says: "The function breaks here." It's a boundary you can't cross.

A horizontal asymptote says: "In the long run, the function settles here." It's the equilibrium.

An oblique asymptote says: "The function grows, but linearly." It's the long-term trend.

Asymptotes are the skeleton of the function's behavior.

Common Mistakes

Mistake 1: Confusing zeros and vertical asymptotes.

Zeros are where the numerator is zero. Vertical asymptotes are where the denominator is zero (and the numerator isn't).

Mistake 2: Forgetting to check for common factors.

If numerator and denominator share a factor, cancel it first. What looks like a vertical asymptote might be a hole.

Mistake 3: Miscalculating horizontal asymptotes.

Always compare degrees. If degrees are equal, take the ratio of leading coefficients.

Mistake 4: Ignoring oblique asymptotes.

If numerator degree is one more than denominator degree, there's an oblique asymptote. Find it with long division.

Mistake 5: Not testing sign near asymptotes.

The function can approach +∞ from one side and -∞ from the other. Test points to determine which.

The Payoff: Seeing Structure at the Edges

When you understand rational functions, you stop seeing formulas.

You see boundaries. You see where the function explodes, where it flattens, where it has gaps.

You see the long-term behavior encoded in the degrees of the polynomials.

That's the shift: from calculating values to reading structure.

Asymptotes are not obstacles. They're the map of the function's terrain.

Part 6 of the Precalculus series.

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