Explain what is meant by a function.

A function is a mathematical concept that describes a particular relationship between two sets of objects, typically input values and corresponding output values. A function takes an input from one set (called the domain) and assigns to it an output from another set (called the codomain), such that each input value has exactly one output value. This relationship can be expressed in the form of an equation, a graph, a table, or even simple rules.

Functions usually have the following components: 1. Domain: The set of all possible input values. 2. Codomain: The set from which output values are derived. 3. Rule: The procedure or formula used to generate an output value for each input value.

Functions are commonly represented using the notation "f(x)" or "g(x)," where "f" and "g" denote the function's name and "x" represents the input variable. The notation "f(x) = y" indicates that for a given input value x, the function f assigns the output value y. Functions can have multiple input variables (e.g., "f(x,y)") or be defined using other variables (e.g., "f(t)" for time).

Example 1: A linear function. A linear function can be written as f(x) = mx + b, where m and b are constants representing the slope and y-intercept, respectively. The domain of a linear function is usually all real numbers. Example 2: A quadratic function. A quadratic function is a polynomial function of degree 2, represented as f(x) = ax^2 + bx + c, where a, b, and c are constants. The domain of a quadratic function is also typically all real numbers. Example 3: A trigonometric function. A trigonometric function, such as the sine function, can be written as f(x) = sin(x), where x represents an angle. The domain of the sine function consists of all real numbers, while its codomain ranges from -1 to 1. In summary, a function is a mathematical concept that describes a relationship between two sets of values, with each input value corresponding to exactly one output value. A function can be defined using various notation and can represent a wide range of mathematical relationships.