Operations with Rational Functions

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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

How do you simplify a rational function?

Back

To simplify a rational function, factor both the numerator and the denominator and then cancel any common factors.

Front

What is 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 does it mean to perform operations on rational functions?

Back

Performing operations on rational functions includes addition, subtraction, multiplication, and division of two or more rational functions.

Front

How do you add two rational functions?

Back

To add two rational functions, find a common denominator, rewrite each function with that denominator, and then combine the numerators.

Front

What is the process for subtracting rational functions?

Back

Subtracting rational functions involves finding a common denominator, rewriting each function, and then subtracting the numerators.

Front

How do you multiply two rational functions?

Back

To multiply two rational functions, multiply the numerators together and the denominators together: \( \frac{P(x)}{Q(x)} \cdot \frac{R(x)}{S(x)} = \frac{P(x)R(x)}{Q(x)S(x)} \).

Front

What is the method for dividing rational functions?

Back

To divide rational functions, multiply by the reciprocal of the divisor: \( \frac{P(x)}{Q(x)} \div \frac{R(x)}{S(x)} = \frac{P(x)}{Q(x)} \cdot \frac{S(x)}{R(x)} \).

Front

What is a vertical asymptote in the context of rational functions?

Back

A vertical asymptote occurs at values of \( x \) that make the denominator zero, indicating that the function approaches infinity.

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What is a horizontal asymptote?

Back

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