E1

Rational Functions and Operations

Definition of Rational Functions

A rational function is any function of the form $f(x) = \frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomials and $q(x) \neq 0$ for all $x$ in the domain. The domain of $f$ is all real numbers except those that make the denominator zero.

• Example: $f(x) = \frac{x^2 + 1}{x - 3}$ is a rational function. The domain is all real numbers except $x = 3$.

Review of Fraction Operations

Reducing Fractions

• Factor numerator and denominator.

• Cancel common factors.

• Example: $\frac{57}{69} = \frac{3 \times 19}{3 \times 23} = \frac{19}{23}$

Multiplying Fractions

• Multiply numerators together.

• Multiply denominators together.

• Reduce the result if possible.

• Example: $\frac{3}{5} \times \frac{4}{7} = \frac{3 \times 4}{5 \times 7} = \frac{12}{35}$

Dividing Fractions

• Multiply by the reciprocal of the divisor.

• Follow multiplication rules.

• Example: $\frac{4}{5} \div \frac{2}{3} = \frac{4}{5} \times \frac{3}{2} = \frac{12}{10} = \frac{6}{5}$

Adding/Subtracting Fractions

• Find the least common denominator (LCD).

• Rewrite each fraction with the LCD.

• Add or subtract numerators, keep the LCD as denominator.

• Simplify if possible.

• Example: $\frac{2}{5} + \frac{3}{7} = \frac{2 \times 7}{35} + \frac{3 \times 5}{35} = \frac{14 + 15}{35} = \frac{29}{35}$

Operations on Rational Expressions

Simplifying Rational Expressions

To simplify a rational expression, factor both numerator and denominator and cancel any common factors. Always state any restrictions on the variable (values that make the denominator zero).

• Example: $\frac{4x}{2x} = \frac{2 \times 2x}{2x} = 2$, $x \neq 0$

• Example: $\frac{x^2 - 9}{x^2 + 6x + 9} = \frac{(x - 3)(x + 3)}{(x + 3)^2} = \frac{x - 3}{x + 3}$, $x \neq -3$

Caution: You cannot cancel terms that are added or subtracted, only factors that are multiplied.

Multiplying and Dividing Rational Expressions

• Multiply numerators and denominators as with fractions.

• Factor and reduce if possible.

• State restrictions on the variable.

• Example: $\frac{x}{x+2} \times \frac{x-2}{x} = \frac{x(x-2)}{x(x+2)} = \frac{x-2}{x+2}$, $x \neq 0, -2$

Adding and Subtracting Rational Expressions

• Find the LCD of all denominators.

• Rewrite each expression with the LCD.

• Add or subtract numerators, keep the LCD as denominator.

• Simplify and state restrictions.

• Example: $\frac{2}{x} + \frac{3}{x+1} = \frac{2(x+1) + 3x}{x(x+1)} = \frac{2x+2+3x}{x(x+1)} = \frac{5x+2}{x(x+1)}$, $x \neq 0, -1$

Solving Rational Equations

Steps for Solving Rational Equations

1. Clear the fractions by multiplying both sides by the LCD.

2. Solve the resulting equation.

3. Check your solutions in the original equation. Exclude any values that make the denominator zero (extraneous solutions).

• Example: Solve $\frac{2}{x} = \frac{3}{x+4}$

  • LCD is $x(x+4)$

  • Multiply both sides: $2(x+4) = 3x$

  • $2x + 8 = 3x \implies x = 8$

  • Check: $x \neq 0, -4$

Example: Solving More Complex Rational Equations

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

  • LCD is $(x-2)(x+3)$

  • Multiply both sides: $3(x+3) + 1(x-2) = 2(x-2)(x+3)$

  • Simplify and solve for $x$

  • Check for extraneous solutions

Practice Problems

• Simplify rational expressions and state restrictions on the domain (values that make the denominator zero).

• Solve rational equations, checking for extraneous solutions.

Sample Practice Problems

• Write $\frac{5x-2}{x^2} - \frac{3}{x}$ as a single rational expression.

• Simplify $\frac{x^2-2x-3}{x^2-9}$ and state restrictions.

• Solve $\frac{2}{x} = \frac{3}{x+4}$ for $x$.

Summary Table: Steps for Operations with Rational Expressions

Operation	Steps	Restrictions
Simplify	Factor numerator and denominator, cancel common factors	Denominator $\neq 0$
Multiply/Divide	Multiply numerators and denominators, reduce, for division multiply by reciprocal	Denominator $\neq 0$ for all expressions
Add/Subtract	Find LCD, rewrite each with LCD, add/subtract numerators, simplify	Denominator $\neq 0$ for all expressions
Solve Equation	Multiply both sides by LCD, solve, check for extraneous solutions	Exclude values making denominator zero

Additional info: The notes emphasize the importance of stating domain restrictions and checking for extraneous solutions when solving rational equations. These skills are foundational for further study in algebra and calculus.