Comprehensive Guide to Rational Functions in AP Precalculus

Rational Function

A ratio of two polynomial functions, defined as r(x) = p(x)/q(x) where q(x) ≠ 0.

Domain of a Rational Function

All real numbers except those that make the denominator zero, leading to discontinuities.

Discontinuities

Points where a function is not continuous, which include removable discontinuities (holes) and infinite discontinuities (vertical asymptotes).

Removable Discontinuities

Occurs at an x-value where a factor cancels from both numerator and denominator, indicating a hole.

Vertical Asymptote (VA)

Occurs at x-values that make the denominator zero, but do not cancel with the numerator.

Limit Behavior near a Hole

The limit exists as x approaches the hole, although the function value at that point is undefined.

Finding the Coordinate of a Hole

Plug the x-value of the canceled factor into the simplified function to find the y-coordinate.

Infinite Discontinuity

A discontinuity where the graph shoots towards positive or negative infinity at a vertical asymptote.

Horizontal Asymptote (HA) - Bottom Heavy

When the degree of the numerator is less than the degree of the denominator, HA is y = 0.

Horizontal Asymptote (HA) - Balanced Degrees

When degrees of numerator and denominator are equal, HA is the ratio of leading coefficients.

Horizontal Asymptote (HA) - Top Heavy

When the degree of the numerator is greater than the degree of the denominator, there is no HA.

Slant Asymptote

Occurs when the degree of the numerator is exactly one greater than that of the denominator.

Finding a Slant Asymptote

Use Polynomial Long Division; the slant asymptote is y = quotient of division.

$y$-intercept

The value of f(0); it does not exist if 0 is not in the domain.

$x$-intercept

The solutions to p(x) = 0, considering only those values that are not holes.

Rational Equation Solution Method

Multiply both sides by the Least Common Denominator, solve the polynomial equation, and check for extraneous solutions.

Rational Inequality Solution Steps

Move terms to one side, create a single rational expression, identify critical values using zeros of numerator and denominator.

Sign Chart Method

A method used to determine the sign of a rational function across its intervals.

Extraneous Solution

A solution obtained that does not satisfy the original rational equation due to domain restrictions.

Common Mistake - Domain

Forgetting to check if a solution makes the denominator zero when solving rational equations.

Common Mistake - Holes vs VAs

Confusing removable discontinuities with infinite discontinuities.

Critical Values in Inequalities

Zeros of the numerator and denominator that help identify intervals for testing sign.

Horizontal Asymptote Behavior

A function can cross its horizontal asymptote but cannot cross a vertical asymptote.

Vertical Asymptote Condition

Occurs where the denominator equals zero, and the factor does not cancel.

Hole Condition

Occurs where the denominator equals zero and the factor cancels with the numerator.

Slant Asymptote Condition

Occurs where the degree of the numerator equals degree of the denominator plus one.

Example of a Hole

For f(x) = (x^2 - 4)/(x - 2), the hole is at coordinate (2,4).

Example of a Vertical Asymptote

Occurs at x = c where q(x) = 0, but does not cancel with p(x).