Rational Equations and Functions

Rational Equations

• Definition: An equation that involves rational expressions (fractions where the numerator and/or denominator are polynomials).
• Form: Typically expressed as ( \frac{P(x)}{Q(x)} = 0 ) or ( \frac{P(x)}{Q(x)} = R ), where ( P(x) ) and ( Q(x) ) are polynomials.
• Finding Solutions:
  • Set the numerator equal to zero (for ( \frac{P(x)}{Q(x)} = 0 )).
  • Solve ( P(x) = 0 ) to find potential solutions.
  • Ensure solutions do not make the denominator zero.
• Restrictions: Identify values for ( x ) that would make the denominator ( Q(x) = 0 ), as these are excluded from the solution set.
• Example:
  • For ( \frac{x+3}{x-2} = 0 ):
    • Set ( x+3 = 0 ) → ( x = -3 ).
    • Check ( x-2 \neq 0 ) → valid solution since (-3 \neq 2).

Domain and Range

• Domain:

  • Definition: The set of all possible input values (x-values) for a function.
  • For rational functions, exclude values that make the denominator zero.
  • Notation: Usually represented in interval notation or set-builder notation.
  • Example:
    • For ( f(x) = \frac{1}{x-1} ), domain is ( x \in (-\infty, 1) \cup (1, \infty) ).

• Range:

  • Definition: The set of all possible output values (y-values) for a function.
  • Determined by the possible values of the function after considering the domain.
  • Can often be found by analyzing the function's behavior (e.g., asymptotes, intercepts).
  • Example:
    • For ( f(x) = \frac{1}{x} ), range is ( y \in (-\infty, 0) \cup (0, \infty) ).

• Finding Domain and Range:

  • Step 1: Identify any restrictions on x (for domain) and y (for range).
  • Step 2: Use graphical methods or algebraic analysis to understand behavior near restrictions.
  • Step 3: Express the domain and range in appropriate notation.

Rational Equations

• Rational Expressions: Equations involving fractions where the numerator and/or denominator are polynomials.
• General Form: Expressed as ( \frac{P(x)}{Q(x)} = 0 ) or ( \frac{P(x)}{Q(x)} = R ).
• Finding Solutions:
  • Solve ( P(x) = 0 ) to find potential solutions for equations set to zero.
  • Ensure potential solutions do not simultaneously make ( Q(x) = 0 ).
• Restrictions: Identify values of ( x ) that make the denominator ( Q(x) = 0 ) to exclude them from the solution set.
• Example Calculation:
  • For ( \frac{x+3}{x-2} = 0 ), set ( x+3 = 0 ) leading to ( x = -3 ).
  • Verify that the solution ( -3 ) does not make the denominator zero (( -3 \neq 2 )), confirming it as a valid solution.

Domain and Range

• Domain:

  • Definition: The set of all possible input values (x-values) for a function.
  • Exclude values causing the denominator to equal zero.
  • Notation: Often represented in interval notation or set-builder notation.
  • Example Domain:
    • For ( f(x) = \frac{1}{x-1} ), the domain is ( x \in (-\infty, 1) \cup (1, \infty) ) due to exclusion of ( x = 1 ).

• Range:

  • Definition: The set of all possible output values (y-values) for a function.
  • Determined from the behavior of the function, considering the domain.
  • Example Range:
    • For ( f(x) = \frac{1}{x} ), the range is ( y \in (-\infty, 0) \cup (0, \infty) ) since it never equals zero.

• Finding Domain and Range Steps:

  • Step 1: Identify restrictions on ( x ) for the domain and on ( y ) for the range.
  • Step 2: Use graphs or algebraic analysis to analyze behavior near any restrictions.
  • Step 3: Express the domain and range using proper notation.