E1

Rational Functions and Rational Expressions

Definition and Domain

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

• Example: $f(x) = \frac{x^2 + 1}{x - 3}$ has domain $x \neq 3$.

• Example: $g(x) = \frac{x^2 - 4}{x^2 + 1}$ has domain $x \in \mathbb{R}$ (since $x^2 + 1 \neq 0$ for real $x$).

Operations on Rational Expressions

Reducing Fractions

Reducing (simplifying) fractions involves dividing both numerator and denominator by their greatest common factor.

• Step 1: Factor numerator and denominator.

• Step 2: Cancel common factors.

• Example: $\frac{57}{51} = \frac{3 \times 19}{3 \times 17} = \frac{19}{17}$

Multiplying Fractions

To multiply fractions, multiply numerators together and denominators together, then simplify.

• Step 1: Cancel common factors if possible.

• Step 2: Multiply remaining factors in numerator and denominator.

• Example: $\frac{3}{5} \times \frac{4}{9} = \frac{3 \times 4}{5 \times 9} = \frac{12}{45} = \frac{4}{15}$

Dividing Fractions

To divide fractions, multiply by the reciprocal of the divisor.

• Step 1: Flip the second fraction (reciprocal).

• Step 2: Multiply as usual.

• Example: $\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8}$

Adding and Subtracting Fractions

To add or subtract fractions, first find a common denominator (usually the least common denominator, LCD).

• Step 1: Find the LCD of all denominators.

• Step 2: Rewrite each fraction with the LCD as the denominator.

• Step 3: Add or subtract numerators.

• Step 4: 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

When simplifying rational expressions, factor both numerator and denominator and cancel any common factors. Always note any restrictions on the domain (values that make the denominator zero).

• Example: $\frac{x^2 - 4}{x^2 - 2x} = \frac{(x-2)(x+2)}{x(x-2)} = \frac{x+2}{x}$, with $x \neq 0, 2$

• Important: You cannot cancel terms unless they are factors. For example, $\frac{x^2 + x}{x}$ can be written as $\frac{x(x+1)}{x} = x+1$ for $x \neq 0$.

Multiplying and Dividing Rational Expressions

Follow the same rules as for numerical fractions: factor, cancel common factors, multiply numerators and denominators, and note domain restrictions.

• Example: $\frac{x^2 - 4}{x^2 + 2x} \times \frac{x + 2}{x - 2} = \frac{(x-2)(x+2)}{x(x+2)} \times \frac{x+2}{x-2} = \frac{(x+2)^2}{x(x+2)} = \frac{x+2}{x}$, $x \neq 0, -2, 2$

Adding and Subtracting Rational Expressions

Find the LCD of all denominators, rewrite each expression with the LCD, combine numerators, and simplify.

• Example: $\frac{2x}{x^2 - 4} + \frac{3}{x + 2}$ Factor denominators: $x^2 - 4 = (x-2)(x+2)$ LCD: $(x-2)(x+2)$ Rewrite: $\frac{2x}{(x-2)(x+2)} + \frac{3(x-2)}{(x-2)(x+2)} = \frac{2x + 3(x-2)}{(x-2)(x+2)} = \frac{2x + 3x - 6}{(x-2)(x+2)} = \frac{5x - 6}{(x-2)(x+2)}$

Solving Rational Equations

Steps for Solving Rational Equations

To solve equations involving rational expressions:

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

2. Solve the resulting equation (often linear or quadratic).

3. Check for extraneous solutions by substituting back and ensuring no denominator is zero.

• Example: Solve $\frac{2}{x-2} = \frac{3}{x+4}$ Multiply both sides by $(x-2)(x+4)$: $2(x+4) = 3(x-2)$ $2x + 8 = 3x - 6$ $x = 14$ Check: $x \neq 2, -4$ (domain restrictions)

Solving More Complex Rational Equations

• Example: Solve $\frac{3}{x^2 - 2x} = \frac{1}{x}$ Multiply both sides by $x(x-2)$: $3x = x-2$ $2x = -2$ $x = -1$ Check: $x \neq 0, 2$

• Example: Solve $\frac{2}{x-3} - \frac{1}{x+2} = 1$ Multiply both sides by $(x-3)(x+2)$: $2(x+2) - 1(x-3) = (x-3)(x+2)$ $2x + 4 - x + 3 = x^2 - x - 6$ $x + 7 = x^2 - x - 6$ $0 = x^2 - 2x - 13$ Solve quadratic equation for $x$.

Practice Problems and Domain Restrictions

Practice Problems

Practice simplifying rational expressions and solving rational equations. Always list any restrictions on the domain (values that make the denominator zero).

• Example: Simplify $\frac{x^2 - 2x + 1}{x^2 - 2x}$ Factor: $\frac{(x-1)^2}{x(x-2)}$ Domain: $x \neq 0, 2$

• Example: Solve $\frac{2x}{x^2 - 4} = \frac{3}{x + 2}$ Multiply both sides by $(x-2)(x+2)$, solve for $x$, and check restrictions.

Summary Table: Operations on Rational Expressions

Operation	Steps	Example
Multiplication	Factor, cancel common factors, multiply numerators and denominators	$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$
Division	Multiply by reciprocal of divisor	$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$
Addition/Subtraction	Find LCD, rewrite each fraction, combine numerators, simplify	$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$
Simplification	Factor numerator and denominator, cancel common factors, note domain restrictions	$\frac{x^2 - 4}{x^2 - 2x} = \frac{(x-2)(x+2)}{x(x-2)} = \frac{x+2}{x}$

Additional info: These operations and techniques are foundational for understanding rational functions, their domains, and solving equations involving rational expressions in College Algebra.