Relations and Functions Basics

Relations and Functions

Relations

• A relation is a set of ordered pairs.
• It can be represented as:
  • A set of ordered pairs: {(a, b), (c, d), ...}
  • A table or matrix
  • A graph on a coordinate plane
• The domain of a relation is the set of all first elements (x-coordinates) in the ordered pairs.
• The range of a relation is the set of all second elements (y-coordinates) in the ordered pairs.

Functions

• A function is a relation in which every element in the domain corresponds to exactly one element in the range.
• It can be represented as:
  • A set of ordered pairs: {(a, b), (c, d), ...}
  • A table or matrix
  • A graph on a coordinate plane
• The domain of a function is the set of all input values (x-coordinates) in the ordered pairs.
• The range of a function is the set of all output values (y-coordinates) in the ordered pairs.
• A function can be represented algebraically using a formula, such as f(x) = x^2 + 3x - 4.

Characteristics of Functions

• Domain: The set of input values for which the function is defined.
• Range: The set of output values of the function.
• One-to-one: A function is one-to-one if each element in the range corresponds to exactly one element in the domain.
• Onto: A function is onto if every element in the range is the output of at least one element in the domain.

Operations with Functions

• Function Composition: The result of applying one function to the output of another function.
• Inverse Functions: A function that "reverses" the operation of another function.
• Even and Odd Functions: A function is even if f(-x) = f(x) and odd if f(-x) = -f(x).

Relations and Functions

Relations

• A relation is a set of ordered pairs, which can be represented in different ways, including a set of ordered pairs, a table or matrix, or a graph on a coordinate plane.
• The domain of a relation is the set of all first elements (x-coordinates) in the ordered pairs.
• The range of a relation is the set of all second elements (y-coordinates) in the ordered pairs.

Functions

• A function is a special type of relation in which every element in the domain corresponds to exactly one element in the range.
• A function can be represented in various ways, including a set of ordered pairs, a table or matrix, or a graph on a coordinate plane.
• The domain of a function is the set of all input values (x-coordinates) in the ordered pairs.
• The range of a function is the set of all output values (y-coordinates) in the ordered pairs.
• A function can be represented algebraically using a formula, such as f(x) = x^2 + 3x - 4.

Characteristics of Functions

Domain and Range

• The domain of a function is the set of input values for which the function is defined.
• The range of a function is the set of output values of the function.

One-to-One and Onto Functions

• A function is one-to-one if each element in the range corresponds to exactly one element in the domain.
• A function is onto if every element in the range is the output of at least one element in the domain.

Operations with Functions

Function Composition

• Function composition is the result of applying one function to the output of another function.

Inverse Functions

• An inverse function is a function that "reverses" the operation of another function.

Even and Odd Functions

• A function is even if f(-x) = f(x).
• A function is odd if f(-x) = -f(x).