Rational Function

Definition

A rational function is a function that can be expressed as the ratio of two polynomial functions. It is a function that can be written in the form $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomial functions and $Q(x)$ is not equal to zero.

5 Must Know Facts For Your Next Test

1. The domain of a rational function is the set of all real numbers $x$ such that the denominator $Q(x)$ is not equal to zero.
2. Rational functions can have vertical asymptotes, which occur when the denominator $Q(x)$ has a factor that makes the function undefined at that value of $x$.
3. Rational functions can have horizontal asymptotes, which occur when the degree of the numerator $P(x)$ is less than the degree of the denominator $Q(x)$.
4. Rational functions can be used to model a variety of real-world situations, such as inverse variation and other types of algebraic relationships.
5. The limit of a rational function as $x$ approaches a value where the denominator is zero is often undefined, leading to a vertical asymptote.

Review Questions

• Explain how the domain and range of a rational function are related to the structure of the function.
  • The domain of a rational function is the set of all real numbers $x$ such that the denominator $Q(x)$ is not equal to zero. This means that the domain of a rational function excludes the values of $x$ that make the denominator zero, as the function would be undefined at those points. The range of a rational function is often more complex, as it can depend on the specific form of the numerator and denominator polynomials, as well as the presence of any asymptotes or holes in the graph of the function.
• Describe the role of polynomial division in the analysis of rational functions.
  • Polynomial division is an important tool in the study of rational functions, as it can be used to determine the behavior of the function, including the presence of asymptotes and the factorization of the numerator and denominator. By dividing the numerator polynomial by the denominator polynomial, one can identify any common factors, which can lead to the presence of holes in the graph of the rational function. Additionally, the results of the polynomial division can provide insights into the behavior of the function near its asymptotes, both vertical and horizontal.
• Explain how rational functions can be used to model real-world situations involving variation, and discuss the implications of the model in terms of the function's properties.
  • Rational functions are often used to model real-world situations involving variation, such as inverse variation, where one quantity is inversely proportional to another. In these cases, the rational function can provide a mathematical representation of the relationship between the two variables, allowing for predictions and analysis. The properties of the rational function, such as its domain, range, asymptotes, and behavior near critical points, can have important implications for the interpretation and application of the model in the real-world context. Understanding these properties is crucial for making accurate inferences and drawing meaningful conclusions from the rational function model.

Related terms

Polynomial Function: A polynomial function is a function that can be written in the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$, where $a_n, a_{n-1}, ..., a_1, a_0$ are real numbers and $n$ is a non-negative integer.

Asymptote: An asymptote is a line that a graph of a function approaches but never touches. Rational functions can have horizontal, vertical, or oblique asymptotes.

Hole: A hole in the graph of a rational function occurs when the numerator and denominator have a common factor, causing the function to be undefined at that point.