Rational Functions and Asymptotes

This video lecture on pre-calculus covers the properties of rational functions, including their definition, domain, and asymptotes. It provides examples of identifying domains using set builder and interval notation. The lecture explains vertical and horizontal asymptotes, their significance, and how to find them. It also covers graphing rational functions using transformations and discusses the importance of simplifying functions to identify asymptotes correctly.

What is a rational function?

• A function that has no variables

• A function that is always linear

• A function that is always quadratic

• A function that is a ratio of two polynomials

Which of the following is NOT a restriction for the domain of a rational function?

• Values that are not real numbers

• Values that make the denominator zero

• Values that are complex numbers

• Values that make the numerator zero

In set builder notation, how would you express the domain of a function where x cannot be 6 or 2?

• {x | x ≠ 6 and x ≠ 2}

• {x | x = 6 and x = 2}

• {x | x = 6 or x = 2}

• {x | x ≠ 6 or x ≠ 2}

What is the domain of the function f(x) = x/(x^2 + 3)?

• All real numbers except x = -3

• All real numbers except x = 3

• All real numbers

• All real numbers except x = 0

What is a vertical asymptote?

• A line that is part of the graph

• A line that the graph crosses

• A line that the graph approaches but never touches

• A line that the graph never approaches

How do you find the vertical asymptotes of a rational function?

• By finding the derivative of the function

• By setting both numerator and denominator equal to zero

• By setting the denominator equal to zero

• By setting the numerator equal to zero

What happens if the degree of the numerator is less than the degree of the denominator?

• The function has a slant asymptote

• The horizontal asymptote is x = 0

• The horizontal asymptote is y = 0

• There is no horizontal asymptote

If the degrees of the numerator and denominator are equal, how do you find the horizontal asymptote?

• There is no horizontal asymptote

• The horizontal asymptote is y = 0

• Divide the leading coefficients

• The function has a slant asymptote

What is an oblique asymptote?

• A vertical line that the graph approaches

• A slanted line that the graph approaches

• A line that the graph crosses

• A horizontal line that the graph approaches

How do you find the equation of an oblique asymptote?

• By setting the numerator equal to zero

• By using polynomial long division

• By setting the denominator equal to zero

• By using synthetic division