Functions and graphs: Relations and functions
An implicit relationship
In the previous example about blood alcohol concentration (BAC) at time \(t\) after alcohol consumption, BAC is a function of \(t\), denoted as \[\mathrm{BAC}=\frac{D}{r\cdot G}-\beta\cdot t\] The dependent variable is explicitly defined in the above example, i.e., the dependent variable is isolated on the left-hand side of the formula. You are probably used to this in mathematics lessons, in which for example the square function is often denoted by \(y=x^2\), or in physics lessons, in which for example the distance \(s\) travelled during time \(t\) is given by the formula \(s=v\times t.\) However, many relations are not explicitly given in the form of a function. An example:
The thin lens formula for a thin lens with focal length \(f\) is \[\frac{1}{b}+\frac{1}{v}=\frac{1}{f},\] where \(v\) is the distance from the object to the lens and \(b\) is the distance from the image to the lens.
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 \(x^2+y^2=1\), for which \(y\) cannot be defined by a single function of \(x\)).
A relation between two variables \(x\) and \(y\), in which \(x\) is taken as independent variable, is a function \(y=y(x)\) if every acceptable value of \(x\) leads to exactly one value of \(y\). Every acceptable value of \(x\) is called an input value; every corresponding value of \(y\) 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 \(|\dots|\), can be defined as follows: \[|x|=\left\{\begin{array}{rl} x & \text{if }x\ge 0\\ -x & \text{if }x\lt 0\end{array}\right.\] The domain of this function is the set of all real numbers (\(\mathbb{R}\)) and its range consists of all nonnegative real numbers.
