Functions and graphs: Relations and functions
An implicit relationship
In the previous example about blood alcohol concentration (BAC) at time after alcohol consumption, BAC is a function of , denoted as
The thin lens formula for a thin lens with focal length is
This is called an implicit relation between quantities. Such a relation is sometimes a function, but not always (e.g., think of the equation of the unit circle , for which cannot be defined by a single function of ).
A relation between two variables and , in which is taken as independent variable, is a function if every acceptable value of leads to exactly one value of . Every acceptable value of is called an input value; every corresponding value of is then called the output value or the function value of the given input. A function relates a given input value to exactly one output values. All acceptable values of the independent variable together form the domain of the function and all possible function values together form the range of the function.
In case of the absolute value of a real number we simply neglect the sign of that number. The absolute value function, denoted by vertical bars , can be defined as follows: