A function is the mathematical way of describing a relationship between two sets of variables. In usual notation, the set of the independent variables x, is called the domain and the set of the dependent variables y is termed the range.
Most commonly encountered functions are
The linear function y = x, the quadratic function y = x2, the cubic function
y = x3, the exponential function y = ex, the logarithmic function y = log(x), and periodic functions such as y = cos x, y = sin x, the set of the variable quantity x is the domain, and the set of the corresponding values of the variable y constitutes the range.
Below I reproduce (courtesy of the mathsisfun website,
www.mathsisfun.com/sets/functions-common.html), in descending order,
the linear, quadratic,
the sine, and the exponential functions graphs.
Carbon dating is the method whereby the age of very old objects is estimated.
The method consists of measuring the current radioactivity of the specimen due to its unstable (and rare) carbon isotope 14C content and comparing it with the average background radioactivity of the atmosphere. (Willard Libby 1947). The specimen must contain remains of a plant. So long as the plant was alive its radioactivity due to the intake of CO2 from the atmosphere was equal to the radioactivity of the atmosphere however when it died, the amount of 14C in it began to fall due to the radioactive decay of this element into nitrogen (14N). The half-life of 14C, defined by T_(1/2)=(ln(2))/λ, can be measured experimentally and is found to have the value 1.808 ×〖10〗^11 seconds. The above expression results from the fact that nuclear decay, and therefore the radioactivity of a radioactive element, is an exponential function of time because the activity at any time is proportional to the number of the nuclei present at that time. And half-life is the time it takes for a radioactive element to fall into half its initial amount. So in the usual notation we have N(t)/N(0) = ½ when 〖t=T〗_(1/2). And because, from the exponential decay law, we have N(t)/N(0) = e – λt, where λ is the decay constant, the above expression for T_(1/2) follows.
In this way, when we have the function N(t) from real life measurements from biological or geological specimens, we can calculate the t value from the known function and the known constant.
Carbon dating has been used to argue against the “young earth” hypothesis and for estimating the age of some scrolls and ancient texts by observing the number of decays in one hour in the specimen and comparing it with the average atmospheric activity before the year 1900, and using the above formulas. (Halliday, Resnick, and Walker, 2006; Bowman, 1999.)
The inverse of an exponential function must be such that it maps x back to itself according to the relation f – 1 f (x) = x, with f (x) = e x. Therefore
f – 1 (x) = ln (x), because ...