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. Rational functions can be multiplied and divided by following the rules for multiplying and dividing fractions, which involve multiplying or dividing the numerators and denominators separately.
2. When adding or subtracting rational expressions, the least common denominator (LCD) must be found, and the expressions must be rewritten with the LCD before performing the addition or subtraction.
3. Rational inequalities can be solved by using the sign chart method, which involves finding the critical points of the inequality, determining the sign of the function on each interval, and then using the sign information to determine the solution set.
4. The graph of a rational function will have vertical asymptotes at the values of $x$ that make the denominator equal to zero, and horizontal asymptotes at $y = 0$ or $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator, respectively.
5. Rational functions can be used to model a wide variety of real-world situations, such as inverse variation, population growth, and the behavior of electrical circuits.

Review Questions

• Explain the process of multiplying and dividing rational expressions.
  • To multiply rational expressions, you multiply the numerators together and multiply the denominators together. To divide rational expressions, you invert the divisor and then multiply the dividend by the inverted divisor. This follows the rules for multiplying and dividing fractions, where you multiply or divide the numerators and denominators separately. The key is to ensure that the denominators are not equal to zero, as that would make the expression undefined.
• Describe the steps involved in adding and subtracting rational expressions.
  • When adding or subtracting rational expressions, the first step is to find the least common denominator (LCD) of the expressions. Once the LCD is found, you rewrite each expression so that it has the LCD in the denominator. Then, you can perform the addition or subtraction operation on the numerators, keeping the common denominator. This process ensures that the resulting expression is in simplest form and can be easily evaluated.
• Analyze the process of solving rational inequalities and explain how the sign chart method is used.
  • To solve a rational inequality, you first identify the critical points of the inequality, which are the values of $x$ that make the denominator equal to zero. Then, you use the sign chart method to determine the sign of the function on each interval between the critical points. Based on the sign information, you can determine the solution set of the inequality, which may include one or more intervals where the inequality is true. The sign chart method is a systematic approach that allows you to visualize the behavior of the rational function and make informed decisions about the solution set.

Related terms

Polynomial Function: A polynomial function is a function that can be written as the sum of a finite number of non-negative integer powers of the independent variable, with coefficients.

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

Domain and Range: The domain of a rational function is the set of all real numbers except the values of $x$ that make the denominator equal to zero. The range of a rational function can be determined by analyzing the behavior of the function and its asymptotes.