Asymptotes, Domain, and Range

Asymptotes

• Horizontal Asymptote (HA): Line that the graph approaches as x approaches infinity.

  • Example: For a rational function, if the degrees of the polynomial in the numerator and denominator are equal, the horizontal asymptote can be found by taking the ratio of the leading coefficients.
  • Notation: ( y = K ) where K is a constant.

• Vertical Asymptote (VA): Line that the graph approaches as x approaches a certain value where the function is undefined.

  • Occurs at values of x that make the denominator zero (and the numerator non-zero).
  • Notation: ( x = a ) where a is the value causing the function to be undefined.

Domain and Range

• Domain (D): Set of all possible input values (x-values) for the function.

  • Example for the function ( f(x) = \frac{1}{x-2} ): Domain is all real numbers except ( x = 2 ).

• Range (R): Set of all possible output values (y-values) for the function.

  • Example for a horizontal asymptote ( y = K ): The range might be all real numbers except K.

Specific examples from the transcript:

• Given the notations:
  • VA: ( x = -5 )
  • HA: ( y = 4 )
  • Domain: Values where the function is defined, such as all real numbers except ( x = -5 ).
  • Range can be specified based on the horizontal asymptote and behavior of the function.

Intersection and Relationships

• Understanding how vertical and horizontal asymptotes affect the graph's behavior at extremities of the domain (as x approaches infinite values and undefined points).
• Analyze specific points like intercepts, and consider local behavior around vertical asymptotes.