Rational Functions Review

Front

What is a rational function?

Back

A rational function is a function that can be expressed as the quotient of two polynomials, i.e., \( f(x) = \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomials.

Front

What are the key characteristics of rational functions?

Back

Key characteristics include: points of discontinuity, vertical asymptotes, horizontal asymptotes, zeros, and the domain.

Front

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

Back

Vertical asymptotes occur where the denominator is zero and the numerator is not zero. Solve \( Q(x) = 0 \) to find these points.

Front

What is a horizontal asymptote?

Back

A horizontal asymptote describes the behavior of a function as \( x \) approaches infinity. It can be found by comparing the degrees of the numerator and denominator.

Front

How do you determine the domain of a rational function?

Back

The domain of a rational function is all real numbers except where the denominator is zero.

Front

What is a point of discontinuity?

Back

A point of discontinuity occurs where a function is not continuous, often where the denominator is zero.

Front

How do you find the zeros of a rational function?

Back

Zeros of a rational function are found by setting the numerator equal to zero and solving for \( x \).

Front

What is the process for dividing polynomials?

Back

To divide polynomials, use long division or synthetic division to simplify the expression.

Front

What is a hole in a rational function?

Back

A hole occurs at a point where both the numerator and denominator are zero, indicating a removable discontinuity.

Front

How do you identify horizontal asymptotes for rational functions?

Back

1. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \). 2. If they are equal, the asymptote is \( y = \frac{a}{b} \) where \( a \) and \( b \) are the leading coefficients.