Rational Equations and Functions Quiz

Rational Equations and Inequalities

• Rational Expression: A fraction of polynomial expressions, formatted as ( \frac{A}{B} ) where ( A ) and ( B ) are polynomials and ( B \neq 0 ). Example: ( \frac{ab}{7 - x} ).
• Rational Equation: An equation that includes at least one rational expression. Example: ( \frac{ab}{7 - x} = x ).
• Rational Inequality: Involves rational expressions combined with inequality symbols (≤, ≥). Example: ( \frac{ab}{7 - x} > 2x ).

Rational Functions

• Polynomial Function: Defined by ( f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0 ), where ( a_n \neq 0 ) and ( n ) is a non-negative integer.
• Linear Function: A special polynomial function of degree 1, expressed as ( f(x) = mx + b ), where ( m ) can be zero.
• Quadratic Function: A polynomial of degree 2, formatted as ( f(x) = ax^2 + bx + c ) with ( a \neq 0 ).
• Rational Function: Expressed as ( f(x) = \frac{N(x)}{D(x)} ) where both ( N(x) ) and ( D(x) ) are polynomials and ( D(x) ) is not the zero polynomial.
• Domain of a Rational Function: Comprises all values of ( x ) that do not result in a zero denominator.

Finding the Domain of Functions

• To determine the domain, identify values of ( x ) that cause the denominator to equal zero and exclude them.
• Examples:
  • For ( f(x) = \frac{2x + 5}{x - 6} ): The restricted value is ( x = 6 ). Domain is all real numbers except ( 6 ).
  • For ( f(x) = \frac{x + 3}{x + 3} ): The restricted value is ( x = -3 ). Domain is all real numbers except ( -3 ).
  • For ( f(x) = \frac{x^2 - 9}{1} ): The restricted values are ( x = 3 ) and ( x = -3 ). Domain excludes ( ±3 ).

Activity

• Students are instructed to find the domain of specified functions on a whole sheet of paper and show their solutions.