%path = "maths/functions" %kind = kinda["texts"] %level = 9
A function (= mapping) can be seen as set of value pairs \((x,y)\) with \(x\in X\) (domain, preimage) and \(y\in Y\) (codomain, image, range). Important is the uniqueness of the image: for any \(x\) there is exactly one \(y\). \(x\) can be freely chosen. \(x\) is independent value, preimage, argument or position. \(y\) is determined by \(x\). No new information is needed to select \(y\). This is why functions are important. \(y\) is called (dependent or function) value or image.
If the uniqueness is not satisfied, then it is called a relation.
A function \(f\) as a direction from the set of all \(x\) (\(X\)) to the set of all \(y\) (\(Y\)). This is notated as \(f:X\rightarrow Y\).
The value pairs normally cannot be written down because too many or infinite. Therefore one describes the function via an analytic expression, i.e. \(y=x^2+1\). This basically is an algorithm, a program.
Basic Concepts:
domain
codomain
mapping
If we do not want to have a separate letter \(f\) for the mapping, we can write: \(y(x)\), i.e. the parentheses say that \(y\) follow from \(x\), i.e. is function of \(x\). Sometimes \(f\) can mean both the mapping or the function value.
If we concentrate only on the mapping, instead of \(g(f(x))\) we can write \(g\circ f\) meaning: first we map via \(f\), then via \(g\), i.e. the ordering is the same in both writings.
There can be more \(x\) with the same \(y\) and it is still a function. If there is only one preimage \(x\) for a \(y\), then the function is injective, i.e. it keeps the distinction or does not loose information. If in addition every element of \(Y\) is reached (surjective), then the function is bijektive. In this case by choosing \(y\) we also choose \(x\) (\(x(y)\), inverse function).
If the images of elements that are close together are also close together, then the function is continuous. Close is intuitive, but still needs a formal definition. This is done via a metric \(d\) (\(d(x,y)\ge 0\), \(d(x,y)=d(y,x)\) and \(d(x,z)\le d(x,y)+d(y,z)\), e.g. \(d(x,y)=|y-x|\)) (or in a more abstract way in topology via sets of nested open sets).
Continuity at \(x\)
For every \(\varepsilon > 0\) there is a \(\delta\), such that for all \(y\) with \(d(x,y)<\delta\) we have \(d(f(x),f(y))<\varepsilon\).
For every \(\varepsilon\)-neighbourhood there is a \(\delta\)-neighbourhood.
A function does not presuppose order of domain and codomain. But if it is given, then a function is said (strictly) monotonically increasing, if \(x\le y\) (\(x<y\)) makes \(f(x)\le f(y)\) (\(f(x)<f(y)\)). Analogously one defines (strictly) monotonically decreasing.
Morphisms are related to functions (see {{!util.a("r.cs")}}).
Regarding graphical representation of a function in a coordinate system:
First values of variables \(X\) and \(Y\) are mapped via units to numbers
A unit for the graph is chosen (e.g. cm). The ratio of unit in graph to unit in reality (kg, km, m/s,…) is the scale.
For a value of the independent variable \(X\) one goes the number of graph units to the left (\(x\)-coordinate, abscissa).
For a value of the dependent variable \(Y\) one goes the number of graph units upward (\(y\)-coordinate, ordinate).
This one repeats for a few pairs of values. These \((x,y)\)-pairs can also be written into a table as a intermediate step (value table).
Since usually it will be a continuous function, one can connect the points with a continuous line. If the line is linear then it is called a linearen function.
Examples of graphs for fundamental types of functions of one variable can be found here: {{!util.a("r.cf")}}.