Comprehensive Guide to Rational Functions in AP Precalculus

Rational Functions: Definitions and Domain

A Rational Function is a ratio of two polynomial functions. It is generally defined as:

r(x) = \frac{p(x)}{q(x)}

Where $p(x)$ and $q(x)$ are polynomials, and $q(x) \neq 0$. The behavior of rational functions is dictated entirely by the interaction between the numerator (top) and the denominator (bottom).

Determining the Domain

The domain of a rational function consists of all real numbers except those that cause the denominator to equal zero. These excluded values result in discontinuities.

• To find the domain, solve $q(x) = 0$.
• The values obtained are strictly determining where the graph breaks (either a vertical asymptote or a hole).

Discontinuities: Vertical Asymptotes and Holes

There are two disjoint types of discontinuities in rational functions. Understanding the difference is crucial for exam questions involving limits.

1. Removable Discontinuities (Holes)

A Hole occurs at an $x$-value if a factor containing that $x$ exists in both the numerator and the denominator and can be mathematically canceled out.

Key Characteristics:

• It represents a single point where the function is undefined.
• Limit Behavior: The limit exists as $x$ approaches the hole, even though $f(c)$ is undefined.
• How to find the coordinate:
  1. Factor the numerator and denominator completely.
  2. Cancel common factors.
  3. Plug the $x$-value of the cancelled factor into the simplified version of the function to find the $y$-coordinate.

Example:
Given f(x) = \frac{x^2 - 4}{x - 2}

1. Factor: f(x) = \frac{(x-2)(x+2)}{(x-2)}
2. Cancel $(x-2)$. The simplified function is $y = x + 2$.
3. The discontinuity is at $x=2$. Plug 2 into $x+2$. The hole is at coordinate $(2, 4)$.

2. Infinite Discontinuities (Vertical Asymptotes)

A Vertical Asymptote (VA) occurs at $x$-values that make the denominator zero but do not cancel with the numerator.

Key Characteristics:

• The graph shoots towards positive or negative infinity.
• Limit Behavior: The limit does not exist (DNE) or is described as $\pm \infty$. Notationally: \lim_{x \to c} f(x) = \infty \; \text{or} \; -\infty
• To determine the direction of the curve near the asymptote, test values very close to the asymptote (e.g., $2.99$ and $3.01$).

End Behavior: Horizontal and Slant Asymptotes

End behavior describes how $f(x)$ behaves as $x \to \infty$ or $x o -\infty$. This is determined by comparing the degree of the numerator ($n$) and the degree of the denominator ($m$).

The Three Cases of Horizontal Asymptotes (HA)

1. Bottom Heavy ($n < m$):

   • The denominator grows faster than the numerator.
   • HA: y = 0

2. Balanced Degrees ($n = m$):

   • The growth rates form a constant ratio.
   • HA: Ratio of leading coefficients.
   • y = \frac{\text{leading coefficient of } p(x)}{\text{leading coefficient of } q(x)}

3. Top Heavy ($n > m$):

   • The numerator grows faster.
   • HA: None. The graph goes to $\infty$ or $-\infty$.

Slant (Oblique) Asymptotes

A special case of "Top Heavy" functions. If the degree of the numerator is exactly one greater than the degree of the denominator ($n = m + 1$), the function has a Slant Asymptote.

How to find it:
Use Polynomial Long Division or Synthetic Division (if applicable).

\frac{p(x)}{q(x)} = \text{Quotient} + \frac{\text{Remainder}}{q(x)}

• The equation of the slant asymptote is $y = \text{Quotient}$ (ignore the remainder).
• As $x \to \infty$, the remainder term approaches 0, so the graph merges with the line.

Example:
f(x) = \frac{x^2 + 3x + 1}{x - 1}
Performing division: $(x^2 + 3x + 1) \div (x - 1) = x + 4$ with a remainder.
Slant Asymptote: $y = x + 4$.

Zeros and Intercepts

• $y$-intercept: Evaluate $f(0)$. If 0 is not in the domain, there is no $y$-intercept.
• $x$-intercepts (Zeros): Set the numerator equal to zero (after canceling holes) and solve for $x$.
  • p(x) = 0
  • Note: If a value makes the numerator zero but is also a hole, it is not an intercept.

Solving Rational Equations and Inequalities

Rational Equations

To solve $\frac{A(x)}{B(x)} = \frac{C(x)}{D(x)}$:

1. Multiply both sides by the Least Common Denominator (LCD) to eliminate fractions.
2. Solve the resulting polynomial equation.
3. CRITICAL STEP: check for Extraneous Solutions. Verify that your solutions do not cause any denominator in the original equation to verify zero.

Rational Inequalities

Inequalities (e.g., $\frac{x-1}{x+3} \ge 0$) require a Sign Chart (or Test Interval) method.

Step-by-Step:

1. Move all terms to one side so the inequality compares to zero.
2. Combine terms into a single rational expression.
3. Find Critical Values:
   • Zeros of the numerator (potential solutions).
   • Zeros of the denominator (vertical asymptotes/undefined points).
4. Place these values on a number line.
5. Test an $x$-value in each interval to see if the function is positive or negative.
6. Notation Rule:
   • Numerator zeros use brackets $[ ]$ (if $\ge$ or $\le$).
   • Denominator zeros always use parentheses $( )$ because the function is undefined there.

Common Mistakes & Pitfalls

1. Ignoring Domain Restrictions: Students often solve $r(x) = 0$ and forget to check if that $x$ makes the denominator zero. Always validate your answers.
2. Confusing Holes and VAs: Just because the denominator is zero doesn't mean it's a vertical asymptote. If the factor cancels, it's a hole.
3. Cross-Multiplying Inequalities: Never cross-multiply an inequality by an expression containing $x$ unless you know the sign of $x$. Multiplying by a negative flips the inequality; since $x$ varies, you don't know when to flip. Always use the sign chart method instead.
4. Horizontal Asymptote Rigidity: Remember that a function can cross its horizontal asymptote (usually in the middle of the graph). It cannot cross a vertical asymptote.

Summary Table: Asymptote Rules