Understanding Functions: Definitions, Properties, and Types

Function

A function defines the relationship between an initial set and a final set, so that each element of the initial set (independent variable) corresponds to a single element of the final set (dependent variable).

Domain of the Function

The domain of a function is the set of possible values that the independent variable (e.g., coins) can take.

Range of the Function

The range of a function is the set of possible values that the dependent variable (e.g., drinks) can represent.

A function can be represented by tables, graphs, and algebraic formulas.

Increasing and Decreasing Functions

• A function is increasing on an interval if for any pair of values a and b in this interval, where a < b, the rate of change is positive.
• A function is decreasing on an interval if for any pair of values a and b in this interval, where a < b, the rate of change is negative.

Maximum and Minimum

• A function f has a relative maximum at x = a if there is an environment around the point where the function takes values that are less than or equal to f(a).
• A function f has a relative minimum at x = a if there is an environment around the point where the function takes values that are greater than or equal to f(a).
• A function f has an absolute maximum at x = a if f(a) is greater than or equal to the value of f(x) anywhere in the domain of the function.
• A function f has an absolute minimum at x = a if f(a) is less than or equal to the value of f(x) anywhere in the domain of the function.

Periodic Functions

A function is periodic if there exists a real constant value T such that f(x + T) = f(x) for all values of x in the domain.

Bounded Functions

• A function is bounded below if there is a real number k such that, for all x, f(x) > k. The number k is called the lower bound.
• A function is bounded above if there is a real number k' such that, for all x, f(x) < k'. The number k' is called the upper bound.

Symmetric Functions

• A function f is symmetric about the ordinate axis when, for all x in the domain, f(-x) = f(x). These functions are also called even functions.
• A function f is symmetric about the origin if, for all x in the domain, f(-x) = -f(x). These functions are also called odd functions.