Pre-Calculus - Week 3

Rational Functions

Rational functions — ratios of polynomials — introduce one of the most important concepts in precalculus: asymptotes. Vertical asymptotes tell you where a function blows up; horizontal asymptotes describe long-run behavior. Calculus uses limits to define both of these precisely, so the intuition you build this week pays off immediately in Calculus I.

Learning Objectives

- Identify vertical asymptotes by finding values that make the denominator zero. - Distinguish holes (removable discontinuities) from vertical asymptotes. - Find horizontal asymptotes by comparing degrees of numerator and denominator. - Find oblique (slant) asymptotes using polynomial division. - Identify the domain of a rational function. - Sketch the graph of a rational function using asymptotes, intercepts, and sign analysis. - Solve rational equations and identify extraneous solutions.

1. What Is a Rational Function?

A rational function has the form f(x) = p(x)/q(x), where p and q are polynomials and q(x) is not the zero polynomial. The domain excludes all x where q(x) = 0. Example: f(x) = (x + 2)/(x^2 - 9). q(x) = x^2 - 9 = (x-3)(x+3). Excluded: x = 3 and x = -3. Domain: all reals except x = ±3.

2. Vertical Asymptotes

A vertical asymptote at x = a occurs when q(a) = 0 but p(a) is not 0 — that is, the factor (x - a) is NOT shared by the numerator (not a removable discontinuity). Example 1: f(x) = 1/(x - 4). Vertical asymptote at x = 4. Example 2: f(x) = (x + 2)/((x - 1)(x + 3)). Vertical asymptotes at x = 1 and x = -3. Behavior near a vertical asymptote: check the sign on each side to determine if f(x) goes to +infinity or -infinity.

3. Holes (Removable Discontinuities)

A hole occurs when the same factor cancels from both numerator and denominator. Example: f(x) = (x^2 - 4)/(x - 2) = (x-2)(x+2)/(x-2). After canceling: f(x) = x + 2 for x not equal 2. There is a HOLE at x = 2 (not an asymptote). The hole is at the point (2, 4). Always cancel common factors first. Remaining uncanceled denominator factors give vertical asymptotes. Canceled factors give holes.

4. Horizontal Asymptotes

Let n = degree of numerator, m = degree of denominator. Three cases: 1. n < m: horizontal asymptote at y = 0. 2. n = m: horizontal asymptote at y = (leading coeff of num)/(leading coeff of denom). 3. n > m: no horizontal asymptote (oblique or higher). Example 1: f(x) = 3/(x^2 + 1). n = 0 < m = 2. HA: y = 0. Example 2: f(x) = (4x^2 - 1)/(2x^2 + 3). n = m = 2. HA: y = 4/2 = 2. Example 3: f(x) = (x^3 + 1)/(x + 1). n = 3 > m = 1. No HA.

5. Oblique (Slant) Asymptotes

When the degree of the numerator is exactly 1 more than the degree of the denominator, divide to find a slant asymptote. Example: f(x) = (x^2 + 3x + 1)/(x + 1). Divide: x^2 + 3x + 1 = (x + 1)(x + 2) - 1. So f(x) = (x + 2) - 1/(x + 1). As x goes to infinity, 1/(x+1) goes to 0, so f(x) approaches x + 2. Slant asymptote: y = x + 2.

6. x- and y-Intercepts

y-intercept: evaluate f(0) (if 0 is in the domain). x-intercepts: set the numerator equal to 0 and solve (confirm denominator is not 0 there). Example: f(x) = (x - 3)/(x + 1). y-intercept: f(0) = -3/1 = -3. Point: (0, -3). x-intercept: x - 3 = 0, so x = 3. Check: denominator = 4 (not 0). Point: (3, 0).

7. Sign Analysis

To determine where f(x) is positive or negative: 1. Find all zeros (numerator = 0) and all vertical asymptotes (denominator = 0). 2. These divide the x-axis into intervals. 3. Test one point in each interval. Example: f(x) = (x - 1)/(x + 2). Critical x: x = 1 (zero), x = -2 (VA). Intervals: (-inf, -2), (-2, 1), (1, inf). Test x = -3: (-4)/(-1) = 4 > 0. Test x = 0: (-1)/(2) < 0. Test x = 2: (1)/(4) > 0.

8. Sketching Rational Graphs

Steps to sketch f(x) = rational function: 1. Cancel common factors; mark holes. 2. Find and draw vertical asymptotes (dashed lines). 3. Find horizontal or slant asymptote and draw it. 4. Find x- and y-intercepts. 5. Use sign analysis to know whether f is above or below the x-axis in each interval. 6. Sketch the curve, approaching the asymptotes. Example: f(x) = 1/(x - 2). VA at x = 2, HA at y = 0. No x-intercept. y-intercept: (0, -1/2). For x > 2: f > 0. For x < 2: f < 0.

9. Solving Rational Equations

To solve a rational equation: 1. Find the LCD of all fractions. 2. Multiply both sides by the LCD to clear fractions. 3. Solve the resulting polynomial equation. 4. CHECK each solution: reject any that make the original denominator zero (extraneous solutions). Example: Solve 1/(x - 1) + 2 = 3/(x - 1). Subtract 1/(x-1) from both sides: 2 = 2/(x-1). Multiply: 2(x - 1) = 2. So x - 1 = 1, x = 2. Check: x = 2 gives 1/1 + 2 = 3/1. True. Valid.

10. Extraneous Solutions

Example of an extraneous solution: Solve x/(x - 2) = 2/(x - 2). Multiply both sides by (x - 2): x = 2. But x = 2 makes the denominator zero — it is extraneous. No solution. This is critical: always substitute back into the ORIGINAL equation, not the simplified version. In calculus and engineering, ignoring domain restrictions leads to real errors.

11. Common Mistakes

- Declaring a vertical asymptote at a point where the factor actually cancels (that's a hole, not a VA). - Forgetting to check for extraneous solutions in rational equations. - Misidentifying horizontal asymptote when n > m (no HA in that case). - Graphing through an asymptote instead of approaching it. - Not checking both sides of a vertical asymptote for sign.