Domain Definition

The domain of a function is the set of all possible input values for which the function is defined. It helps to determine the behavior of a function and its applicability to specific problems or scenarios.

For example, consider the function , which represents the square root of . The domain of this function is all non-negative real numbers, as the square root is not defined for negative numbers.

When analyzing a function, it is essential to determine its domain, as this can provide valuable information about the function’s behavior and limitations. In some cases, the domain can be easily determined by examining the function’s equation. For example, the domain of a polynomial function is all real numbers, as polynomials are defined for any input value.

In other cases, the domain might be restricted by the function’s definition or by the context of the problem. For example, a function that models the height of a projectile as a function of time would have a domain restricted to non-negative time values.

The domain of a function is often represented using interval notation or set notation. For instance, the domain of the function can be represented as or .

Range Definition

The range of a function is the set of all possible output values generated by the function for a given domain. It is a measure of how the output values of a function are distributed and can be used to analyze the behavior of the function and its applications in various contexts.

Interval Definition

An interval is a set of real numbers that lies between two given numbers. The two numbers that define the interval are often called the endpoints. Intervals can be open (not including the endpoints), closed (including the endpoints), or a mix of both.

Function Definition

A function is a mathematical relationship between two sets, called the domain and the codomain, in which each element in the domain corresponds to exactly one element in the codomain.