Excluded Values

Definition

Excluded values refer to the specific input values that are not allowed or considered valid within the domain or range of a function. These values are typically excluded because they would result in an undefined or meaningless output, or they may represent points where the function is not defined.

5 Must Know Facts For Your Next Test

1. Excluded values are important to consider when determining the domain and range of a function, as they define the valid input and output values.
2. Functions with rational expressions or square roots often have excluded values, as division by zero or taking the square root of a negative number would result in an undefined output.
3. Graphically, excluded values are represented by vertical asymptotes, which the function approaches but never touches.
4. Identifying and understanding excluded values is crucial for analyzing the behavior and properties of a function, such as its continuity and differentiability.
5. Excluded values can also arise from constraints or limitations imposed on the function, such as the domain of a logarithmic function being restricted to positive real numbers.

Review Questions

• Explain how excluded values relate to the domain and range of a function.
  • Excluded values are input values that are not allowed within the domain of a function, as they would result in an undefined or meaningless output. These excluded values are not part of the function's domain and, consequently, are also not included in the function's range. Identifying and understanding excluded values is crucial for determining the valid input and output values for a function, which is essential for analyzing its behavior and properties.
• Describe the graphical representation of excluded values and their significance.
  • Graphically, excluded values are represented by vertical asymptotes, which are lines that the function approaches but never touches. These asymptotes indicate the points where the function is not defined, and the function's behavior changes dramatically as it approaches these values. Understanding the location and significance of these excluded values is important for sketching the graph of a function and interpreting its behavior, such as its continuity and differentiability.
• Analyze the role of excluded values in the context of rational functions and square root functions.
  • Rational functions and square root functions often have excluded values due to the mathematical operations involved. For rational functions, excluded values correspond to the input values that would result in division by zero, which is undefined. For square root functions, excluded values correspond to the input values that would result in taking the square root of a negative number, which is also undefined. Identifying and understanding these excluded values is crucial for determining the valid domain and range of these types of functions, as well as for analyzing their properties and behavior.

Related terms

Domain:

The set of all possible input values for a function.

Range:

The set of all possible output values for a function.

Asymptote:

A line that a function approaches but never touches, indicating where the function is not defined.