Rational Expressions and Functions: Precalculus Study Guide

Rational Expressions and Functions

Simplifying Rational Expressions

Rational expressions are quotients of polynomials. Simplifying them involves factoring and reducing common factors.

• Definition: A rational expression is an expression of the form $\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$.

• Steps to Simplify:

  1. Factor numerator and denominator completely.

  2. Cancel any common factors.

  3. State restrictions (values that make denominator zero).

• Example: $\frac{x^2 - 9}{x^2 + 3x}$ factors to $\frac{(x-3)(x+3)}{x(x+3)} = \frac{x-3}{x}$, $x \neq 0, -3$.

Multiplying and Dividing Rational Expressions

To multiply or divide rational expressions, factor all polynomials, multiply/divide numerators and denominators, and simplify.

• Multiplication: $\frac{A}{B} \cdot \frac{C}{D} = \frac{AC}{BD}$

• Division: $\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C} = \frac{AD}{BC}$

• Example: $\frac{4x^2 + 4x - 3}{6x^2 + 9x - 3} \div \frac{3x + 3}{2x^2 - 2}$

  • Factor all polynomials.

  • Multiply/divide and cancel common factors.

Adding and Subtracting Rational Expressions

To add or subtract rational expressions, find a common denominator, rewrite each expression, combine, and simplify.

• Steps:

  1. Factor denominators.

  2. Find the least common denominator (LCD).

  3. Rewrite each expression with the LCD.

  4. Add/subtract numerators, keep the LCD.

  5. Simplify the result.

• Example: $\frac{3x + 4}{x^2 + 2} + \frac{2x - 1}{x^2 + 2}$

  • Same denominator: $\frac{3x + 4 + 2x - 1}{x^2 + 2} = \frac{5x + 3}{x^2 + 2}$

Solving Rational Equations

Rational equations contain rational expressions set equal to each other or to a value. Solutions must not make any denominator zero.

• Steps:

  1. Find the LCD of all denominators.

  2. Multiply both sides by the LCD to clear denominators.

  3. Solve the resulting equation.

  4. Check for extraneous solutions (values that make any denominator zero).

• Example: $\frac{7}{x+2} = \frac{3}{2}$

  • Multiply both sides by $2(x+2)$, solve for $x$.

Domain of Rational Functions

The domain of a rational function is all real numbers except those that make the denominator zero.

• Definition: The domain of $R(x) = \frac{P(x)}{Q(x)}$ is all $x$ such that $Q(x) \neq 0$.

• Interval Notation: Exclude points where denominator is zero.

• Example: $R(x) = \frac{2x+5}{x^2+4}$, denominator never zero, so domain is $(-\infty, \infty)$.

• Example: $R(x) = \frac{3x^2 - 2x + 1}{x^2 + 3x}$, denominator zero at $x=0, x=-3$, so domain is $(-\infty, -3) \cup (-3, 0) \cup (0, \infty)$.

Vertical and Horizontal Asymptotes

Asymptotes describe the behavior of rational functions as $x$ approaches certain values or infinity.

• Vertical Asymptotes (VA): Occur at values of $x$ that make the denominator zero (and not canceled by the numerator).

• Horizontal Asymptotes (HA): Determined by comparing degrees of numerator and denominator:

  • If degree numerator < degree denominator: $y=0$ is HA.

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

  • If degree numerator > degree denominator: No HA (may be an oblique/slant asymptote).

• Example: $R(x) = \frac{x+5}{(x-4)(x+2)}$ has VAs at $x=4$ and $x=-2$, HA at $y=0$.

Transformations of Rational Functions

Transformations shift, reflect, or stretch the graph of a basic rational function.

• Basic Function: $f(x) = \frac{1}{x}$

• Transformations:

  • Horizontal shift: $f(x-h)$ shifts right by $h$ units.

  • Vertical shift: $f(x) + k$ shifts up by $k$ units.

  • Reflection: $-f(x)$ reflects over the $x$-axis.

• Example: $R(x) = \frac{1}{x+3} - 2$ shifts left by 3 units and down by 2 units.

Graphing Rational Functions

Graphing involves identifying intercepts, asymptotes, holes, and behavior near these features.

• Steps:

  1. Factor numerator and denominator.

  2. Find domain.

  3. Simplify if possible.

  4. Find $x$- and $y$-intercepts.

  5. Find vertical and horizontal asymptotes.

  6. Check for holes (common factors in numerator and denominator).

  7. Analyze behavior near asymptotes and intercepts (multiplicity affects crossing/touching).

  8. Sketch the graph.

• Example: $R(x) = \frac{5x}{x^2 - 64}$

  • Factor denominator: $(x-8)(x+8)$

  • Domain: $(-\infty, -8) \cup (-8, 8) \cup (8, \infty)$

  • $y$-intercept: $(0,0)$

  • $x$-intercept: $x=0$

  • Vertical asymptotes: $x=8$, $x=-8$

  • Horizontal asymptote: $y=0$

Solving Rational Inequalities

Rational inequalities involve finding where a rational expression is greater than, less than, or equal to zero.

• Steps:

  1. Write inequality with zero on one side.

  2. Factor numerator and denominator.

  3. Find zeros of numerator (potential $x$-intercepts) and denominator (vertical asymptotes).

  4. Divide the real line into intervals using these points.

  5. Test each interval to determine where the inequality holds.

  6. Write solution in interval notation, excluding points where denominator is zero.

• Example: $\frac{(x-3)^2}{x^2-4} \geq 0$

  • Critical points: $x=3$, $x=2$, $x=-2$

  • Test intervals and write solution: $(-\infty, -2) \cup (2, \infty)$

Summary Table: Features of Rational Functions

Feature	How to Find	Example
Domain	Set denominator $\neq 0$	$x^2+3x=0 \Rightarrow x=0, -3$
Vertical Asymptote	Zeros of denominator (not canceled)	$x=4$ for $\frac{x+5}{x-4}$
Horizontal Asymptote	Compare degrees of numerator/denominator	$y=0$ for $\frac{x+5}{x^2+3x}$
Holes	Common factors in numerator and denominator	None if no common factors
$x$-intercept	Set numerator $=0$	$x=0$ for $\frac{5x}{x^2-64}$
$y$-intercept	Set $x=0$	$(0,0)$ for $\frac{5x}{x^2-64}$

Additional info:

• Multiplicity of zeros affects whether the graph crosses (odd multiplicity) or touches (even multiplicity) the $x$-axis.

• Behavior near vertical asymptotes depends on the sign of the function on either side.

• Oblique (slant) asymptotes occur when degree of numerator is exactly one more than denominator.