Rational Expressions and Functions - Study Notes

6.1 Rational Expressions and Functions

• Rational Expression: A fraction with both numerator and denominator as polynomial functions.

  - Examples:
    - $\frac{3}{2x}$
    - $\frac{x + 4}{x^{2} - 4x + 4}$
    - $\frac{-5x}{x^{2} + 2x - 3}$

• Recall DOMAIN: All x-values yielding a valid y-value.
    - In rational functions, the domain excludes values that make the denominator zero.

• Definition of a Rational Function:
    - If u(x) and v(x) are polynomial functions,
    $f(x) = \frac{u(x)}{v(x)}$ is a rational function. The domain of f consists of all real numbers for which v(x) ≠ 0.

Example 1: Finding Domains of Rational Functions

• Objective: Identify the domain of the following functions in interval notation.
    - Technology: Use graphing calculators for visualization.

Part a

• Function: $f(x) = \frac{x - 2}{x - 2}$
    - Denominator = 0 condition:
      - $x - 2 = 0$
      - This implies $x = 2$
    - Domain: $(-\infty, 2) \cup (2, \infty)$
    - Graph Analysis: Examine graph behavior near x = 2.

Part b

• An additional function is introduced for evaluation.

• Steps to Evaluate Domain:
    1. Factor any numerator or denominator.
    2. Set the denominator to zero and solve.

Checkpoint

• Determine domains for the following:
    1. $f(x) = \frac{2x + 5}{8}$
    2. $f(x) = \frac{3x}{x^{2} - 2x - 3}$
      - Factor denominator:
        - $x^{2} - 2x - 3 = (x - 3)(x + 1)$
      - Determine domain:
        - Solve for values that make the denominator zero.
        - Combine remaining intervals.

Example 2: Application Involving a Restricted Domain

• Scenario: A small business manufacturing lamps.

• Initial Investment: $120,000

• Cost per lamp: $15

• Total Cost Function:
    - $C = 15x + 120,000$
    - Here, x is the quantity of lamps produced.

• Average Cost Function:
    - $\text{Average cost} = \frac{C}{x} = \frac{15x + 120,000}{x}$

• Domain Considerations:
    - The function has a division by x, so x cannot be zero.
    - Also, x must represent realistic production amounts (must be a positive integer).

Example 3: Simplifying Rational Functions

• Steps for Simplification:
    1. Completely factor both numerator and denominator.
    2. Cancel any common factors (not terms).
    - Key Idea: The success of simplifying rational expressions relies on complete polynomial factorization.

Sample Problem to Simplify:

• Factor and simplify: $\frac{2x^{3} - 6x}{6x^{2}}$
    - Factorization:
      - Numerator: $2x(x^{2} - 3)$
      - Denominator: $6x^{2}$
    - Simplified result: $\frac{2(x^{2} - 3)}{3x}$

• Important Note: Keep domain restrictions from the original function in the simplified version.