### 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.