Functions are the building blocks of algebra, connecting inputs to outputs in a unique way. They're like mathematical machines that take numbers in and spit out results, following specific rules that define their behavior.

Function notation is a shorthand way to write these rules. It helps us understand how functions work and lets us easily calculate outputs for different inputs. This knowledge is crucial for solving real-world problems and understanding more complex math concepts.

Function Notation and Terminology

Functions and Function Notation

• A function is a relation that assigns to each element x in a set A exactly one element, called f(x), in a set B
• Function notation is written as f(x), where f is the name of the function and x is the input value
  • For example, if f(x) = 2x + 1, then f(3) = 2(3) + 1 = 7

Domain, Codomain, and Range

• The set A is called the domain of the function, and the set B is called the codomain
• The range is the set of all output values of the function
• The vertical line test can be used to determine if a relation is a function
  • If any vertical line intersects the graph more than once, the relation is not a function
  • For instance, a circle is not a function because a vertical line would intersect it twice

Domain and Range of Functions

Determining Domain and Range

• The domain of a function is the set of all possible input values (x-values) for which the function is defined
• The range of a function is the set of all possible output values (y-values) that the function can produce
• The domain and range can be determined by examining the graph, equation, or table of values for a function
  • For a graph, the domain is all x-values that have a corresponding y-value, and the range is all y-values that have a corresponding x-value
  • For an equation, consider any restrictions on the input values (such as avoiding division by zero or taking the square root of a negative number)

Domain Restrictions

• Restrictions on the domain may occur due to the nature of the function
  • The denominator of a rational function cannot be zero (e.g., f(x) = 1/(x-2) is undefined when x = 2)
  • The radicand of a square root function cannot be negative (e.g., f(x) = √(x+1) is only defined for x ≥ -1)
• These restrictions can be determined by setting the denominator equal to zero or the radicand greater than or equal to zero and solving for x

Function Classifications

One-to-One Functions (Injective)

• A function is one-to-one (injective) if each element in the codomain is paired with at most one element in the domain
  • In other words, no two distinct elements in the domain map to the same element in the codomain
• The horizontal line test can be used to determine if a function is one-to-one
  • If any horizontal line intersects the graph more than once, the function is not one-to-one
  • For example, f(x) = x^2 is not one-to-one because a horizontal line y = 4 would intersect the graph at x = -2 and x = 2

Onto Functions (Surjective)

• A function is onto (surjective) if each element in the codomain is paired with at least one element in the domain
  • In other words, every element in the codomain is mapped to by at least one element in the domain
• To determine if a function is onto, check if every element in the codomain has a corresponding element in the domain that maps to it

Bijective Functions

• A function is bijective if it is both one-to-one and onto
• Each element in the codomain is paired with exactly one element in the domain
• A bijective function has an inverse function that is also bijective

Functions vs Relations

Defining Relations

• A relation is a set of ordered pairs (x, y) that defines a relationship between two sets, where x is an element of the domain and y is an element of the codomain
• A relation can be represented using a graph, equation, or table of values

Determining if a Relation is a Function

• For a relation to be a function, each element in the domain must be paired with exactly one element in the codomain
• The vertical line test can be applied to determine if the relation is a function
  • If a relation is given as a set of ordered pairs, check that no two ordered pairs have the same first coordinate (x-value) with different second coordinates (y-values)
  • For example, the relation {(1,2), (2,3), (1,4)} is not a function because the x-value 1 is paired with both 2 and 4 in the codomain